From Business Intelligence To Competitive Intelligence Inferring Competitive Measures Usin

Description
In the early part of the 20th century, Arthur C. Nielsen revolutionized the field of marketing research by inventing and commercializing the concept of market share.

Information Systems Research
Vol. 23, No. 3, Part 1 of 2, September 2012, pp. 698–720
ISSN 1047-7047 (print) ISSN 1526-5536 (online)http://dx.doi.org/10.1287/isre.1110.0385
©2012 INFORMS
From Business Intelligence to Competitive Intelligence:
Inferring Competitive Measures Using Augmented
Site-Centric Data
Zhiqiang (Eric) Zheng
School of Management, University of Texas at Dallas, Dallas, Texas 75080, [email protected]
Peter Fader
The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104, [email protected]
Balaji Padmanabhan
College of Business, University of South Florida, Tampa, Florida 33620, [email protected]
M
anagers routinely seek to understand ?rm performance relative to the competitors. Recently, competitive
intelligence (CI) has emerged as an important area within business intelligence (BI) where the emphasis
is on understanding and measuring a ?rm’s external competitive environment. A requirement of such systems
is the availability of the rich data about a ?rm’s competitors, which is typically hard to acquire. This paper
proposes a method to incorporate competitive intelligence in BI systems by using less granular and aggregate
data, which is usually easier to acquire. We motivate, develop, and validate an approach to infer key competitive
measures about customer activities without requiring detailed cross-?rm data. Instead, our method derives these
competitive measures for online ?rms from simple “site-centric” data that are commonly available, augmented
with aggregate data summaries that may be obtained from syndicated data providers. Based on data provided
by comScore Networks, we show empirically that our method performs well in inferring several key diagnostic
competitive measures—the penetration, market share, and the share of wallet—for various online retailers.
Key words: business intelligence; competitive intelligence; competitive measures; probability models;
NBD/Dirichlet
History: Vallabh Sambamurthy, Senior Editor; Siva Viswanathan, Associate Editor. This paper was received
August 8, 2009, and was with the author 11 months for 2 revisions. Published online in Articles in Advance
November 3, 2011.
1. Introduction
In the early part of the 20th century, Arthur C.
Nielsen revolutionized the ?eld of marketing research
by inventing and commercializing the concept of
“market share.” Before ACNielsen Inc. began its
store census process it was virtually impossible for
?rms to obtain timely, complete, and accurate mar-
ket intelligence about competing brands. Today it is
well recognized that knowledge of the overall com-
petitive landscape is important for any business, and
in response to this, there are many ?rms that special-
ize in the task of collecting and disseminating such
information in various industries. Likewise, there are
dozens of different kinds of measures that research
?rms (and their clients) use to characterize the com-
petitive landscape (Davis 2007, Farris et al. 2006).
Recognizing the signi?cance of competitive mea-
sures, a trend in the business intelligence (BI) ?eld
is the increasing importance given to competitive intel-
ligence (CI), i.e., the information that a ?rm knows
about its external competitive environment (Kahaner
1998). Although current BI dashboards are versatile
and can pull data from different sources, most of the
information in these dashboards is typically about
the internal environment of the ?rm. Boulding et al.
(2005, p. 161) consider this myopic view to be one of
the pitfalls of current CRM practice; they suggest that
successful implementation of CRM requires ?rms to
incorporate knowledge about competition and com-
petitive reaction into their CRM processes. Hence,
methods that can provide useful competitive intelli-
gence will be vital for the design of next-generation BI
dashboards. As one example, Google Trends recently
started providing some competitive intelligence docu-
menting the volume of search queries across different
competitors.
Current BI dashboards often fall short of providing
CI capabilities, largely due to the fact that detailed
information on competitors is hard to obtain. For
most ?rms, these competitive measures are obtained
solely through third-party data providers such as
ACNielsen and other industry-speci?c syndicated
data sources. Although such data can be integrated
into BI systems to provide CI, it comes at the cost of
698
Zheng et al.: From Business Intelligence to Competitive Intelligence
Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS 699
being expensive to acquire, and is also often based on
historical data (rather than real time). In contrast to
this approach, research in marketing (Park and Fader
2004) and information systems (Padmanabhan et al.
2001, 2006) has shown that ?rms may bene?t from
data augmentation, where they might augment their
own internal data (which has been referred to as site-
centric data
1
in the ecommerce context) with limited
amounts of external data to achieve similar goals. This
paper follows the same approach in spirit, and devel-
ops a method to infer competitive measures using
augmented site-centric data.
Our focus is mainly on competitive measures
that capture customer visits and purchasing behav-
ior across competitors in e-commerce. Our model
builds on the rich repeat-buying literature in market-
ing that developed and studied the classic Dirichlet
model (Ehrenberg 1959, Goodhart et al. 1984) and
its various extensions. The Dirichlet model assumes
that each customer makes two independent purchas-
ing decisions: she ?rst decides on the total number
of purchases within a product category, and then
makes a choice on which competing ?rm’s prod-
uct to purchase. These two behavioral processes are
captured through two probabilistic mixture models:
the negative binomial (for category incidence) and
the Dirichlet-multinomial (for brand choice). Taken
together, the overall modeling framework is known
as the “NBD/Dirichlet” (hereafter referred to as
Dirichlet) in this literature (Winkelmann 2008). This
research dates back to Ehrenberg (1959) with key
updates (Goodhardt et al. 1984, Schmittlein et al. 1985,
Ehrenberg 1988, Fader and Schmittlein 1993, Uncles
et al. 1995) over the years. Numerous studies have
documented the success of the Dirichlet model (Sharp
2010). It has been said that the Dirichlet model may
be the best-known example of an empirical general-
ization in marketing, with the possible exception of
the Bass model (Uncles et al. 1995).
A de?ning feature of the Dirichlet model is that for
each customer it is necessary to know the number of
transactions conducted with each of the competing
?rms in the market. We refer to this as the complete
information requirement, which in reality is hard to
ful?ll because a ?rm needs to know the purchases
their customers make across all competing ?rms. This
requires the ?rm either to ?nd a way to convince
its competitors to share customer data or to convince
their customers in the market to disclose their pri-
vate purchasing data to the ?rm. Neither is an easy
1
The term site-centric data was ?rst introduced by Padmanabhan
et al. (2001) in the e-commerce context to refer to the data captured
by an individual ?rm (site) on its customers’ transactions with the
?rm. The counterpart, user-centric data, refers to the user-level data
that capture these customers’ transactions across ?rms (sites).
task. More often than not, a ?rm only has data on its
own customers (i.e., site-centric data), and therefore is
unable to implement the NBD/Dirichlet model.
Is there a middle ground where ?rms can obtain
some data about their competitors, but not at the
individual customer level? There is some recent work
(e.g., Fader et al. 2007, Yang et al. 2005) that examines
the possibility of sharing summaries of data instead of
detailed customer transactions. Sharing such aggre-
gate data rather than individual transactions can be
a practical approach for retailers, and in some cases
may be the only way to obtain competitive intelli-
gence. At the same time, however, for many ?rms it
is not enough to rely on aggregate summaries alone—
they would like to make the best use of their inter-
nal data in conjunction with these external aggregate
summaries in order to obtain the most complete pic-
ture of the competitive environment. These desires,
constraints, and concerns bring us to the main point
of this paper: can ?rms combine their own customer-level
data with commonly available aggregate summary statis-
tics to infer important competitive measures?
In this paper we show that this can be done by an
implementation of the Dirichlet model in the limited-
information scenario. Towards this end, we develop
a realistic model termed as the |imited í nformation
NBD/Dirichlet (LIND) that improvises
2
the standard
Dirchlet model. Our model aims to capture some of
the power of the Dirichlet model, but with far less
information, speci?cally with individual ?rms having
access only to their own data plus the aggregate num-
bers (e.g., market share) from the other ?rms in the
industry.
A key strength of the Dirichlet family of models is
that they capture individual-level customer behavior
and then derive the distributions of various aggregate
statistics of interest, such as penetration, frequency,
market share (MS), and share of wallet (SoW).
3
The
2
Note that this objective differs fundamentally from a stream of
papers that aim to improve the Dirichlet model by relaxing the model
assumptions and incorporating additional marketing-mix variables
such as price and promotion (e.g., Danaher et al. 2003, Bhattacharya
1997.) We thank the AE for making this important observation.
3
Penetration is de?ned as the percentage of customers who trans-
acted with the focal store. Frequency is de?ned here as the average
number of purchases among the buyers of a product. Share of wal-
let is the percentage of purchases made to a speci?c store among
those customers who actually transacted with the store (Uncles
et al. 1995.) As an example, suppose that the total number of pur-
chases to all online apparel stores is 1 million, and that a focal store,
e.g., L.L.Bean, observes 100,000 of them. The market share for this
retailer is 10%. Next, suppose that L.L. Bean’s customers accounted
for 300,000 of the total purchases to any apparel site. Their SoW is
therefore 33%. SoW is also referred to as share of category require-
ments (SCR) in different settings. Please refer to the electronic com-
panion for a brief de?nition of these marketing terminologies. The
electronic companion is available as part of the online version athttp://dx.doi.org/10.1287/isre.1110.0385.
Zheng et al.: From Business Intelligence to Competitive Intelligence
700 Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS
performance of the Dirichlet model is often evaluated
in terms of how well it infers these aggregate statis-
tics, i.e., competitive measures (Ehrenberg et al. 2004,
Bhattacharya 1997). We focus particularly on estimat-
ing penetration, the market share, and share of wallet
for the full set of competing ?rms in a given market.
Our results from data on various online retailers show
that the limited-information model performs almost
as well as the full-information NBD/Dirichlet model,
with far less data, in inferring these key competitive
measures.
Our research addresses a fundamental, and largely
new, business problem and has implications to both
the marketing and information systems ?elds. Our
motivations, as well as implications of our results,
relate to effective design of CI systems, data suf?-
ciency, information sharing, and privacy—all areas in
which the information systems community has strong
interests. Our primary contribution to the market-
ing ?eld is essentially a new model that works in a
limited-information scenario that is commonly faced
by marketers. Speci?cally, the method presented here
and the associated empirical results are particularly
important to information systems theory and practice
for the following reasons:
• Our method provides a solution for CI when
detailed transactional data on competitors are not
available. This approach can be readily integrated into
the design of BI dashboards due to the relatively sim-
ple inputs required from a data perspective.
• The competitive measures derived here can form
the basis for sense-and-respond BI capabilities. Specif-
ically, on a strategic level knowledge of the com-
petitive landscape may clearly inform initiatives
such as mergers and acquisitions. At the tactical
level, the Internet presents unique opportunities for
?rms to act in real time on competitive informa-
tion. For instance, carefully designed personalized
promotions and online advertising strategies can
often be implemented immediately, on a customer-
by-customer basis, when faced with eroding market
share or other competitive threats.
• From an information privacy context, we present
an approach to sharing data that does not require
individual transactions to be revealed. This is partic-
ularly relevant given numerous cases of identity rev-
elation from supposedly anonymized data, which has
made many ?rms far more cautious when it comes
to sharing transactional data outside the boundaries
of the ?rm. This is also related to recent research
in information systems and management science on
privacy-preserving information sharing (Fader et al.
2007, Menon and Sarkar 2007, Yang et al. 2005).
• It raises a more general question of how much
data are really needed when there are speci?c objec-
tives for which these data are sought. This is related
to recent research in management science on infor-
mation acquisition and active learning (Zheng and
Padmanabhan 2006). Our results should also be
viewed in the context of recent results in the IS litera-
ture that examine the innovative use of aggregate data
to improve prediction models. For instance, Umyarov
and Tuzhilin (2011) show that combining aggregated
movie review data (i.e., IMDB data) with individual
customer data from Net?ix can enhance the movie
recommendation accuracy.
The rest of this paper is structured as follows. In §2
we review related work from the research areas men-
tioned brie?y above. Section 3 provides the theoret-
ical background and discusses the full-information
NBD/Dirichlet model that can be used to infer com-
petitive measures when complete user-centric data
are available. The limited-information NBD/Dirichlet
model is then presented in §4. Results from applying
the limited and full-information models are then pre-
sented in §5, followed by a discussion of limitations
and future work in §6. We conclude in §7.
2. Related Work and Context
Recent work in a variety of disciplines, including
?nance (Kallberg and Udell 2003), marketing (Park
and Fader 2004), economics (Liu and Serfes 2006),
and information systems (Padmanabhan et al. 2001,
2006), have demonstrated the value of using customer
data across ?rms for a variety of important problems.
Kallberg and Udell (2003) demonstrate that lenders’
sharing credit information on borrowers adds value
to all participating lenders. In the case of online retail,
Park and Fader (2004) show that customer browsing
behavior can be modeled more accurately by using
cross-site data, suggesting the value of sharing infor-
mation between online ?rms. Padmanabhan et al.
(2006) quantify the bene?ts that may be expected from
using the more complete user-centric data compared
to traditional site-centric data that individual online
retailers typically observe, for predicting purchases
and repeat visits. However, unlike our work here,
which uses aggregate information, these approaches
require integrating individual customer data across
?rms. Often a fairly large amount of such data has
to be collected by a ?rm before any bene?ts can be
reaped, and thus, Padmanabhan et al. (2006) caution
against acquiring too little data. Unlike the above
approaches, this paper focuses on learning competi-
tive measures at the ?rm level, the inference of which
may not need complete individual customers’ across
?rm data, as we demonstrate later.
As we pointed out earlier, the research on com-
petitive measures dates back to the early part of
the 20th century, when Arthur C. Nielsen revolution-
ized the ?eld of marketing research with the mea-
sure of “market share.” Since then, marketers have
Zheng et al.: From Business Intelligence to Competitive Intelligence
Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS 701
developed a variety of competitive measures. Farris
et al. (2006) enumerates more than 50 competitive
measures, including market share, market penetra-
tion, purchase frequency, etc., and show how to use a
“dashboard” of these metrics to gauge a ?rm’s busi-
ness from various perspectives, such as promotional
strategy, advertising, distribution, customer percep-
tions, and competitors’ power. Davis (2007) de?ned
over a hundred such metrics that marketers need
to know to understand the competitive landscape.
Both lists encompass all the competitive metrics pro-
posed in the Dirichlet literature (Goodhardt et al.
1984, Ehrenberg 1988). Besides those we introduced
before (i.e., penetration, market share, frequency, and
SoW), other common measures include the percent-
age of a ?rm’s customers who transacted with that
?rm only once (“once only”), the percentage of a
?rm’s customers who only transact with that ?rm in
the category (“100% loyal”), and the percentage of a
?rm’s customers who also transacted with a speci?ed
other ?rm (“duplication”). Our proposed model can
be used to derive all these measures analytically.
Many of the CI measures are also being used
in ?rms as KPIs (key performance indicators),
4
as
opposed to the conventionally inward-looking BI
measures in a typical dashboard. However, the seem-
ingly appealing notion of incorporating CI is often
hindered by the lack of necessary data. Con?dential-
ity and privacy issues are often major obstacles to
obtaining the relevant data. This has spawned the
rise of privacy-preserving data mining (e.g., Menon and
Sarkar 2007), a set of methods that enable managers to
share data while concealing any speci?c individual-
level patterns or other potentially sensitive charac-
teristics. Still, most of these approaches assume that
?rms are willing to share individual-customer level
data, an assumption that is increasingly unrealistic
in the context of some recent high-pro?le disclosure
cases. As one example, when a prominent online
portal released some of its supposedly anonymized
search query data, reporters were able to link a spe-
ci?c individual to many of the search queries, spark-
ing a major outcry.
5
As mentioned in the introduction, one key mea-
sure we attempt to infer is SoW. The SoW mea-
sure has been noted to be one of the best ways to
gauge customer loyalty as well as the overall effec-
tiveness of a customer relationship management strat-
egy (Uncles et al. 1995, Fox and Thomas 2006, Du
et al. 2007). Beyond the academic literature, practi-
tioners also acknowledge its importance (see, e.g., a
4
Seehttp://en.wikipedia.org/wiki/Competitive_intelligence.
5
Source: AOL search data scandal athttp://en.wikipedia.org/
wiki/AOL_search_data_scandal.
recent white paper
6
that focuses on the online apparel
industry). When comparing MS and SoW, it becomes
quite clear that the latter is much harder for indi-
vidual ?rms to determine, because it requires them
to have information speci?cally on their customers’
transactions at competing stores. However, we will
show how it can be inferred from readily available
summary statistics combined with the ?rm’s site-
centric customer records.
There is recent work (Du et al. 2007, Fox and
Thomas 2006) that directly addresses the estimation
of share of wallet from augmented data or from infor-
mation sharing. Du et al. (2007) consider the case of
a focal ?rm that desires to estimate the share of wal-
let for each of its customers. In this setting, initially
the ?rm observes only their customers’ purchases.
Their method involves obtaining detailed survey data
for a sample of customers regarding their activities
at competing ?rms, and using these acquired val-
ues to impute those unknown values for other cus-
tomers. The imputed values are then used to compute
each customer’s SoW. However, as recent research
(Zheng and Padmanabhan 2006) in management sci-
ence has shown, such list augmentation approaches
may require extensive complete data before the impu-
tation models work well. Whereas the approach in Du
et al. (2007, p. 102) estimates SoW well for a validation
set of 10,000+ customers, these results are based on
acquiring survey data for approximately 24,000 cus-
tomers. In practice, acquiring detailed data for such a
large percentage of customers can be prohibitive. One
possibility is to augment the above approach with
active learning methods for list augmentation (Zheng
and Padmanabhan 2006), but this has not been com-
prehensively studied in this literature as yet. Another
recent approach (Fox and Thomas 2006) considered
using customers’ shopper loyalty card data to predict
customer-level SoW. This approach models spending
at competing retailers uses multioutlet panel data,
and uses this model on loyalty card data to predict
expected spending at other retailers, thereby generat-
ing SoW for each customer. In their experiments they
use detailed panel data on 210 households to make
SoW predictions for 148 households. Hence, as in the
previous work, the acquired data in these experiments
is considerably large.
Compared to these approaches, our method uses
substantially less data (speci?cally the penetration or
market share for each ?rm in an industry) to make
SoW inferences. Second, whereas the above literature
only estimates SoW for a focal ?rm, our approach
6
This report (http://www.techexchange.com/thelibrary/ShareOf
-Wallet.html) begins with the following sentence: “Maximizing
share of wallet is among the most important issues facing the part-
ners in any consumer oriented value chain.”
Zheng et al.: From Business Intelligence to Competitive Intelligence
702 Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS
permits a focal ?rm to infer the SoW for all ?rms
in its industry. Finally, acquiring individual customer-
level data across ?rms may raise speci?c privacy
concerns and will therefore require effective privacy-
preserving approaches such as the ideas described
in Menon and Sarkar (2007). We avoid this issue by
using only aggregate data about competitors, an effec-
tive way of preserving individual privacy and min-
imizing data-handling complexity. Aggregated data,
when used appropriately, can be a feasible and valu-
able alternative to more granular data. For exam-
ple, the model proposed in Umyarov and Tuzhilin
(2011), which combines aggregated movie review data
from IMDB with the individual user’s movie rat-
ing, enhances Net?ix’s movie recommendation sys-
tem signi?cantly.
Finally, there is broader interest in making infer-
ences using data sets that combine limited informa-
tion about individuals with comprehensive aggregate
data. Recent papers such as Chen and Yang (2007) and
Musalem et al. (2008, 2009) offer sophisticated new
approaches that enable analysts to infer individual-
level choice patterns from aggregate data. Although
these papers seem similar in spirit to the method
we develop here, there are several key differences
worth emphasizing. First, although those papers
utilize aggregate data, they assume that period-by
period (e.g., weekly) sales information is available
for all competitors. In many industries, however, this
is not the case in practice. Second, although these
papers provide valuable insights about individual-
level choice processes, they are not built upon a rich,
well-validated tradition such as the NBD/Dirichlet.
By no means does this imply that their models are
inferior, but our ability here to link the model (and
the resulting output measures) back to an exten-
sive and well-validated literature is a signi?cant pos-
itive. Finally, these alternative approaches all require
computationally intensive simulation-based estima-
tion methods for model implementation. Ours, in
contrast, is quite simple to implement—even in a
familiar spreadsheet environment such as Excel. This
can be an advantage for practitioners grappling with
the real-world problem that we focus on in this paper.
3. Theoretical Background and the
NBD/Dirichlet Model
The theory that we build is based on the extensively
researched area of repeat buying in the marketing
literature. The basic problem is to model customers,
each of whom has a variable number of transactions
within a product category, and the associated deci-
sions about which competing ?rm to choose each
time. A key strength of these models is that they
capture individual-level customer behavior and then
derive the distributions of various aggregate statistics
of interest. In this case, we are going to work “back-
wards” in order to make individual-level inferences
from some aggregate statistics. First, we brie?y review
the traditional NBD/Dirichlet model.
3.1. The NBD Model for Category Purchasing
The negative binomial distribution (NBD) is one of
the best-known “count models” (Winkelmann 2008).
A complete analysis of this model is provided by
Schmittlein et al. (1985). A notable aspect of the
NBD model is that just simple histogram data are
needed for parameter estimation. The NBD model
assumes that the overall number of (category) pur-
chases (denoted as N, with the individual subscript
suppressed for ease of exposition) for any customer
during a unit time period can be modeled as a Pois-
son process with purchase-rate parameter \. How-
ever, each individual may have a different purchase
rate. To accommodate this difference (heterogeneity)
among customers, these rate parameters are assumed
to be gamma distributed ] (\) = (o
r
\
r?1
c
?\o
),!(r)
with shape parameter r and scale parameter o, where
|(\) = r,o. It is common knowledge that these two
assumptions combine to yield the NBD model for the
aggregate number of customer purchases. In other
words, the probability of observing n purchases in
any ?xed unit time period for the random count vari-
able N is:
P(N =n r, o) =
_

0
P(N =n \)] (\ r, o) d\
=
!(r +n)
!(r)n!
_
o
o+1
_
r
_
1
o+1
_
n
. (1)
The key feature de?ning the NBD model is the lin-
earity of the conditional mean—the purchase propen-
sity as speci?ed in Equation (1) does not change
over time, known as the stationary market assump-
tion (Schmittlein et al. 1985). Through comprehen-
sive empirical and simulation studies, Dunn et al.
(1983, p. 256) show that the NBD assumptions are rea-
sonable for the majority of buyers, both for brand-
and store-level purchases. They conclude that “for
most purposes in brand purchasing studies, the NBD
tends to be accepted as robust to most observed
departures from its (stationary) Poisson assumption.”
Morrison and Schmittlein (1988, p. 151) also hold that
the NBD model is very robust and the combination
of the Poisson and Gamma processes tend to work
very well. Further details about the NBD as a stand-
alone model of product purchasing can be found in
Ehrenberg (1959), Morrison and Schmittlein (1988),
and Schmittlein et al. (1985).
Zheng et al.: From Business Intelligence to Competitive Intelligence
Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS 703
3.2. The Dirichlet-Multinomial Model for
Brand Choice
Every time a customer makes a purchase in the
category, she also makes a brand-choice decision.
As in the case of the NBD model for category
purchasing, one ?rst makes an assumption about
the individual-level choice process, and then brings
in a mixing distribution to capture heterogeneity
across customers. In this case, a standard multi-
nomial distribution is used for the former, and a
Dirichlet distribution for the latter. This leads to
the well-known Dirichlet-multinomial (or “Dirichlet,”
for short) model, which was discussed in detail in
Ehrenberg (1988), Goodhardt et al. (1984), and also in
Fader and Schmittlein (1993).
Suppose there are | brands, let u
¡
be the Dirichlet
parameter indicating the attraction of brand ¡, and
let the sum of u
¡
be s =

|
¡=1
u
¡
. That is, s represents
the overall attractiveness of the entire category (to a
random customer) and u
¡
,s can be interpreted as the
share of attractiveness (i.e., market share) of brand ¡ in
the market. Then the probability of a customer mak-
ing x
1
purchases of brand 1 and x
2
purchases of brand
2, . . . , and x
|
purchases of brand |, given n category
purchases has been shown to be:
P(X
1
=x
1
, . . . , X
|
=x
|
n, u
1
, . . . u
|
)
=
_
1?

P
¡
0
_
1
0
· · ·
_
1
0
|

¡=1
P(X
¡
=x
¡
n, p
¡
)
· g(p
¡
u
1
, . . . u
|
)dp
1
. . . dp
|
=
_
n
x
1...
x
|
_
!(s)

|
¡=1
!(u
¡
)

|
¡=1
!(u
¡
+x
¡
)
!(s +n)
. (2)
Ehrenberg et al. (2004) point out that the Dirichlet
model works because it nicely captures regularities
prevalent in a variety of markets such as (1) smaller
brands not only have fewer customers, but they tend
to be purchased less frequently by their customers
(referred to as “double jeopardy” as if smaller brands
are punished twice); (2) large heterogeneity of cus-
tomers, with some purchasing very few and some
purchasing very frequently; and (3) much the same
proportion of any particular brand’s customers also
bought another brand of interest, i.e., the so-called
constant duplication phenomenon (p. 1310). These
“lawlike” patterns have been con?rmed from soup to
gasoline, prescription drugs to aviation fuel, where
there are large and small brands, and light and heavy
buyers, in geographies as diverse as the United States,
United Kingdom, Japan, Germany, and Australasia,
and for over three decades (Sharp 2010).
The Dirichlet model is not without limitations. It
is best applied in what Ehrenberg et al. (2004) refer
to as a stable market where (1) customers do not
change their pace of purchases of a product over time
(a result of the stationary NBD process of category
purchases) and (2) the brands are not functionally dif-
ferentiated and they show no special partitioning of
certain brands (the so-called nonsegmented market as
a result of the multinomial choice process). Over the
years, there have been attempts to extend the model
to nonstable markets by considering the existence of
nonstationarity (Fader and Lattin 1993), change of
pace and niche markets where a brand specializes
in attracting a particular group of customers (Kahn
et al. 1988), a large amount of customers who make
zero purchases of a brand (spike at zero), and viola-
tions of the distributional assumptions (Morrison and
Schmittlein 1988) and in segmented markets (Danaher
et al. 2003). In the third online appendix (of the elec-
tronic companion), we provide more-detailed expla-
nations for all these marketing terminologies.
3.3. The NBD/Dirichlet for Full Information
In a typical application of the NBD/Dirichlet model,
researchers assume that the two aforementioned
behavioral processes are independent of each other
(i.e., there is no linkage between category inci-
dence and brand choice). Assuming that complete
data are available to observe both processes, stan-
dard estimation approaches (e.g., maximum likeli-
hood) can be used to obtain the two parameters for
the NBD component and the | parameters for the
Dirichlet component. In the full-information world,
it is unnecessary for a researcher to estimate all
| + 2 parameters simultaneously—thus, the “full-
information” NBD/Dirichlet is not truly an integrated
model; it is just the concatenation of two separate,
independent submodels. In contrast, our “limited-
information” approach, to be discussed shortly, is
fully integrated and allows for the simultaneous esti-
mation of the entire model.
Once the | + 2 parameter estimates are available,
it is possible to derive a broad array of summary
brand performance measures, including a variety of
competitive indicators, such as market share, share of
wallet, and the number of 100% loyal customers. In
contrast to more traditional econometric approaches,
model evaluation is often judged by how well a given
model can capture the various diagnostic measures
mentioned above (Goodhardt et al. 1984, Ehrenberg
1995, Fader and Schmittlein 1993, Uncles et al. 1995),
in addition to overall model ?tness. We will follow
this approach in this paper, and will similarly evalu-
ate our limited-information model based on how well
the model can be used to derive these important com-
petitive measures.
We conclude this section by reminding the reader
about the overall data requirements for the typical
full-information NBD/Dirichlet model. Speci?cally,
Zheng et al.: From Business Intelligence to Competitive Intelligence
704 Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS
for each customer it is necessary to know the num-
ber of transactions conducted with each of the com-
peting ?rms in the market. As noted at the outset
of the paper, this is often a very dif?cult data struc-
ture to obtain. More frequently, ?rms have complete
information only for their brand, and therefore are
unable to implement the NBD/Dirichlet model. We
begin to address this “limited-information” scenario
in the next section.
3.4. The BB/NBD Model for Limited Information
Although full information on each customer’s cross-
brand purchasing may not be available in practice
most of the time, many ?rms still wish to tease apart
the two underlying behavioral processes (i.e., cate-
gory incidence and brand choice) from each other.
Although it may be hard to sort out separate NBD
and Dirichlet models when the observed data con-
found the two processes, there is an elegant model
that, in theory, allows this “teasing apart” to be
accomplished even without complete data for each cus-
tomer. This model, known as the beta-binomial/NBD
(or BB/NBD) was ?rst introduced by Schmittlein et al.
(1985) and has been used in a variety of settings to
try to pull apart two integrated processes that cannot
be separately observed (often due to data limitations
as described earlier). The BB/NBD is a special two-
brand case of the NBD/Dirichlet model that lumps all
competing brands into a single “other brand” com-
posite. Here the setting is that of a random customer
who ?rst decides whether to visit the category accord-
ing to a NBD process and then decides whether to
purchase the focal brand according to a beta-binomial
process (Schmittlein et al. 1985).
Like the NBD/Dirichlet, the BB/NBD continues to
assume that category incidence is governed by an
NBD submodel, but instead of using the complete
|-brand Dirichlet-multinomial, the BB/NBD uses a
dichotomous beta-binomial in its place. In other
words, the decision of whether to purchase the focal
brand is now a binomial process (as opposed to
the multinomial process in the Dirichlet case) with
choice propensity p. However, customers are allowed
to differ from each other in their choice propensi-
ties, and this distribution of p across the population
is assumed to follow a beta distribution where g(p) =
(1,8(u, |))p
u?1
(1 ?p)
|?1
with mixing parameters as u
and | (the beta distribution is the two-alternative spe-
cial case of the more-general Dirichlet distribution).
This leads to the BB/NBD distribution, the detailed
derivation of which is presented in the ?rst online
appendix (of the electronic companion).
P(X =x)
=
_
1
0
_

0
p(X =x \, p)g(p)] (\) d\dp
=
!(r +x)
!(r)x!
_
o
o+1
_
r
_
1
o+1
_
x
·
!(u+x)
!(u)
!(u+|)
!(u+|+x)
2
í
1
_
r,|+x,u+|+x,
1
o+1
_
, (3)
where X represents the observed number of pur-
chases of the focal brand and
2
í
1
( ) is the Gaussian
hypergeometric function.
7
Note that this model needs
four parameters to be estimated (r and o parameters
for the NBD model of category purchase, and u and |
for the beta distribution of brand choice).
The elegance of the BB/NBD model lies in its ability
to make inferences about each of the model compo-
nents without being able to observe themseparately. A
bene?t of this model is that most of the summary mea-
sures obtained from the full NBD/Dirichlet can still
be obtained despite the data limitations. For instance,
the expected reach or penetration (i.e., the percentage
of customers who made at least one purchase of the
focal brand) can be derived as Penetration =1 ?P(X =
0) =1 ?(o,(1 +o))
r
2
í
1
(r, |, u +|, 1,(1 +o)). This and
other key summary measures are derived in detail by
Fader and Hardie (2000).
However, despite the appeal of the BB/NBD, its
ability to really sort out the underlying processes of
interest is questionable. Given that a typical BB/NBD
data set is just a simple histogram (capturing the dis-
tribution of purchase frequencies for the focal brand
alone), it is hard to uniquely identify each of the four
parameters in a reliable manner. The empirical anal-
yses performed by researchers such as Bickart and
Schmittlein (1999) and Fader and Hardie (2000) show
several problems, including: (1) a high degree of sen-
sitivity to initial settings in the parameter estimation
process; (2) a very ?at likelihood surface indicating
the presence of many local optima; (3) limited ability
to outperform simpler speci?cations, such as the ordi-
nary NBD by itself; and (4) managerial inferences that
do not have a high degree of face validity. The extent
of these problems would likely become even more
acute when trying to make inferences on competitive
summary statistics (e.g., SoW) because the data lacks
any information at all about the various competing
brands.
With these problems in mind, we need to aug-
ment the basic BB/NBD model by introducing some
information about competitive brands. This will give
the data set more “texture,” and such data augmen-
tation beyond the simple histogram can allow the
reliable estimation of multiple parameters with the
7
The Gaussian hypergeometric function
2
í
1
( ) is discussed in
detail in Johnson et al. (1992) and the Wolfram functions site
athttp://functions.wolfram.com/HypergeometricFunctions/Hyper
-geometric2F1/02/. It solves the integral
2
í
1
(u, |, c, x) =
(!(c),(!(|)!(c ?|)))
_
1
0
|
|?1
(1 ?|)
c?|?1
(1 ?|x)
?u
d|.
Zheng et al.: From Business Intelligence to Competitive Intelligence
Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS 705
Table 1 Overview of Notation Used
Notation Explanation
X
ij,
x
ij
X
ij
is a random variable and x
ij
is the actual number of purchases of customer i to site j.
r , u The parameters that capture customers’ category-level purchase behavior according to an NBD process,
where r is the shape parameter and u is the scale parameter.
a
j
The Dirichlet parameter a
j
captures customers’ multinomial choice propensity to site j.
s and t
j
Summary statistics (used for convenience) where s =

k
j=1
a
j
and t
j
=s ?a
j
.
2
F
1
( ) The Gaussian hypergeometric function detailed in Footnote 7.
M
j
The total number of customers for the focal site j.
A
i
, r
i
A
i
is a random variable and r
i
is the total number of category purchases for customer i.
ability to make direct linkages to (and inferences
about) the focal brand’s competitors. The key idea
is to move from the pure BB/NBD back towards a
NBD/Dirichlet, but using only commonly available
summary statistics instead of the complete individual-
level transaction histories that it usually requires. We
demonstrate how to do so in the next section.
4. The Limited-Information
NBD/Dirichlet (LIND) Model
There are three characteristics for the data we utilize
in our limited-information setting:
1. Each ?rm has access to its own customer data,
i.e., site-centric data for an online store, but knows
that there may exist customers that only purchase
with its competitors in the same category (whom they
may therefore never observe).
2. Each ?rm knows exactly one piece of aggre-
gated information (e.g., penetration or market share)
for each of the other relevant ?rms in the category.
For example, if Amazon is the focal store to examine,
then it uses as inputs the count information on all
its customers (e.g., the number of purchases that each
customer made at Amazon) as well as just the pene-
tration (or market share) for each of the other leading
online stores. This kind of competitive information is
widely available at low cost.
8
3. At the category level, we make a realistic
assumption that we can only observe those customers
who made at least one purchase at the category. That
is, we assume we do not know how many customers
may be interested in a category but have not pur-
chased anything yet.
We believe that these are reasonable assumptions,
supported by actual data collection practices. We
8
Such data are routinely reported by many research ?rms, both
online and of?ine. For instance, Information Resources, Inc. has
regularly reported penetration statistics for grocery categories for
many years (e.g., Fader and Lodish 1990), and comScore Networks
publishes online ?rms’ reach as part of its Media Metrix service
(http://www.comscore.com/metrix/).
make these assumptions strictly because of their real-
ism, as opposed to mathematical convenience. Fur-
thermore, they apply equally well to an online or
of?ine context, and are not tied to any particular
industry/sector.
4.1. The Model
We propose a model that integrates the NBD
model for category purchasing and the Dirichlet-
multinomial model for brand choice, yet with a
considerably easier data requirement than the full-
information NBD/Dirichlet relies upon. This limited-
information NBD/Dirichlet (LIND) model takes one
speci?c ?rm (e.g., Amazon) as the focal one and mod-
els the entire market from its perspective. We summa-
rize the notation to be used in Table 1.
First we assume that the usual NBD process applies
to category-level purchase patterns, with one differ-
ence: because we only observe customers with at least
one purchase in the category (data assumption 3),
we model the number of category purchases, N, as a
shifted Poisson
9
with rate \:
P(N =n \) =
\
n?1
c
?\
(n?1)!
, n =1, 2. . . , \ >0. (4)
Then, as with the regular NBD, we use a gamma
distribution with parameters r and o to characterize
the differences in these rates. This yields a commonly
used Shifted NBD (sNBD) model for category purchas-
ing. Next, given the number of category purchases
for a random customer, the choice of the focal brand
is modeled (just as before) using a binomial distri-
bution with rate p. Following standard assumptions
(Schmittlein et al. 1985, Fader and Hardie 2000), \ and
p are assumed to follow a gamma and a beta distri-
bution, respectively.
Altogether this yields a new model that we term as
BB/sNBD (beta-binomial/shifted negative binomial
9
Another common way to deal with nonzero data is the truncated
Poisson. We derive the truncated LIND model and present the
results in Online Appendix 2 of the electronic companion as a pos-
sible alternative. Guidelines for choosing between the shifted and
truncated models are also provided there.
Zheng et al.: From Business Intelligence to Competitive Intelligence
706 Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS
distribution) model. The detailed derivation of this
model is provided in Appendix 1. Below we directly
present the marginal distribution of x.
P(X =x) =
_
o
o+1
_
r
|
u +|
·
2
í
1
_
r,|+1,u+|+1,
1
o+1
_
, x=0 (5.1)
P(X =x)
=
o
r
(o+1)
(x+r?1)
!(r +x ?1)
(x ?1)!!(r)
!(u +x)!(u +|)
!(u)!(u +| +x)
·
2
í
1
_
x +r ?1, |, u +| +x,
1
o+1
_
+
o
r
(o+1)
(x+r)
!(r +x)
x!!(r)
!(u +x)!(| +1)!(u +|)
!(u +| +1 +x)!(u)!(|)
·
2
í
1
_
x+r,|+1,u+|+x+1,
1
o+1
_
, x?1 (5.2)
where x represents the number of purchases of a cus-
tomer to a site. Note that Equation (5.1) yields the
expected penetration:
Penetration
=1 ?P(X =0)
=1?
_
o
o+1
_
r
|
u+|
2
í
1
_
r,|+1,u+|+1,
1
o+1
_
. (6)
If we only had data from a single site, the BB/sNBD
would be a logical model to use, and we will examine
it as a benchmark to help evaluate the performance
of the LIND model. In order to go from the BB/sNBD
towards LIND, we replace the beta-binomial com-
ponent with the multibrand Dirichlet-multinomial
choice process, so that each site ¡ is characterized by
its own attraction parameter, u
¡
. The focal site thus
has K + 2 parameters to estimate: two parameters
(r and o) for category purchasing and K parame-
ters u
1
, u
2
, . . . , u
|
representing the cross-?rm Dirichlet
parameters. Now, as shown in Equation (6), if these
parameters are known, then each ?rm’s expected pen-
etration can be derived analytically. Because in our
setting we know the actual value of these penetra-
tions (see data assumption 2), we introduce K con-
straints on the LIND model that restrict the expected
penetration computed for each site to equal the actual
observed values in Equation (7). We would like to
note that LIND is not restricted to using only pene-
tration as the input (aggregate measure); it is a ?ex-
ible model that can accommodate other inputs (e.g.,
market share, frequency, etc.) in a similar fashion. We
will also demonstrate how to use market share as the
inputs instead later in §5.2, and provide guidelines in
§6 on how to choose appropriate inputs to best esti-
mate the competitive measures of interest.
Adding constraints turns the usual unconstrained
maximum-likelihood optimization problem into a
constrained optimization task, but a relatively
straightforward one. This constrained optimization
formulation (Equation (7)) solves the problem from
the focal site’s perspective. Denote x
i¡
as the num-
ber of purchases of customer i to site ¡. The objective
function is simply the sum of the log-likelihood based
on the density function shown in Equation (A5). Let
A
¡
be the number of customers of site ¡. We have:
Max
r, o, u
1
...u
|
A
¡
_
i=1
ln|P(X
i¡
=x
i¡
)]
s.|
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
Penetration
1
=1 ?
_
o
o+1
_
r
|
1
u
1
+|
1
·
2
í
1
_
r, |
1
+1, u
1
+|
1
+1,
1
o+1
_
. . .
Penetration
|
=1 ?
_
o
o+1
_
r
|
|
u
|
+|
|
·
2
í
1
_
r, |
|
+1, u
|
+|
|
+1,
1
o+1
_
r, o, u
1
. . . u
|
>0.
(7)
The right-hand side (RHS) of each constraint is the
expected penetration from Equation (6) and the left-
hand side (LHS) is the observed actual penetration for
each site. Note that each ?rm in the product category
would solve formulation (7) using its own observed
customer data, with a separate constraint for itself and
each of its competitors.
Estimating the K + 2 parameters may appear to
be somewhat daunting because the likelihood func-
tion and constraints include the Gaussian hypergeo-
metric function, noted to be cumbersome (Fader and
Hardie 2000). However, Appendix 2 presents the VBA
(Visual Basic for Applications) code we wrote for the
2
F
1
component, which can be integrated into Excel as
a customized function. With this customized function,
the constrained optimization problem can be easily
implemented in Excel using its built-in Solver tool.
4.2. Deriving Competitive Measures
After estimating the parameters for (7), a focal
site can then use them to derive any traditional
NBD/Dirichlet measures of interest in order to char-
acterize the competitive landscape of the overall cate-
gory. As noted in the introduction, two popular ones
are the market share and the share of wallet for each
?rm. An important result here is that a focal site
can derive these measures analytically for all other
?rms without requiring any individual customer data
from them.
Zheng et al.: From Business Intelligence to Competitive Intelligence
Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS 707
The expected market share for each ?rm is straight-
forward and given in (8). This is essentially the ratio
of the propensity of a random customer to purchase
at a speci?c site to the propensity to purchase at any
site in the category. Let s =

|
¡=1
u
¡
and |
¡
=s ?u
¡
, the
market share of site ¡ is:
AS
¡
=
u
¡
s
. (8)
Note that MS
¡
is often available from the same
sources that provide the penetration data, so this
might not be a big deal by itself. However, we include
it here both for generality as well as a way to check
the validity of our model. The share of wallet for any
?rm is shown in (9). Let x
¡
be the number of pur-
chases of a random customer to site ¡ and n be the
total number of purchases to all ?rms in this cate-
gory. The share of wallet can be estimated for any
?rm as |(x
¡
,n x
¡
> 0). Unlike the market share, the
conditioning is important because share of wallet for
a ?rm is only computed based on its own customers’
across-?rm purchases (x
¡
> 0). The numerator in (9)
represents a random customer’s number of purchases
to the site of interest. Suppose the customer made
n purchases to the category, with the probability of
P, i.e., the density of the shifted NBD. From the
market share de?nition, given n, we know the num-
ber of purchases the random customers make at a
speci?c site is n × u,s. Because n is a random vari-
able, the numerator thus represents the expectation
of this number of purchases. The denominator repre-
sents the expectation of the total number of category
purchases a random customer who has transacted at
the site (the conditioning) makes. The probability of
the random customer buying at least once at the site
is 1 ?P(X
¡
= 0 n), and the probability of making n
category purchases is P. Integrating the expected
category purchases over n, we derive the denomina-
tor as

n=1
Pn(1 ?P(x
¡
=0 n). Thus, SoW of site ¡
becomes:
SoW
¡
=|
_
x
¡
n
¸
¸
¸
¸
x
¡
>0
_
=

n=1
Pnu
¡
,s

n=1
nP(1 ?P(x
¡
=0 n)
=((u
¡
,s)(r,o+1)) ·
_
(u
¡
,s)(r,o+1) ?

_
n=1
n(!(r +n?1)
,!(r)(n?1)!)(o,(o+1))
r
(1,(o+1))
n?1
(!(s)!(|
¡
+n)
,!(|)!(s +n)
_
?1
. (9)
Appendix 2 presents detailed derivations and the
VBA code for customizing SoW as a function with
parameters r, o, and u
1
, . . . , u
|
as inputs.
Whereas we focus on two principal measures (MS
and SoW), there are other competitive measures
(Ehrenberg 1988, Ehrenberg et al. 2004) that may also
be derived. Appendix 2 also provides the deriva-
tions for “frequency,” “once only,” “100% loyal,” and
“duplication.” We would like to note that several of
the key formulas we derived (e.g., SoW, 100% loyal
and duplication) are new to the Dirichlet literature.
5. Results
The data we use are provided by comScore Networks,
made available through The Wharton School’s WRDS
service (wrds.wharton.upenn.edu). The raw data con-
sist of 50,000 panelists’ online visiting and purchasing
activities. This “academic” data set provides detailed
session-level data at the customer level for every site
within a category (in contrast to the purely site-centric
data to which most ?rms are limited). This enables
us to treat each site as the “focal” site and run the
model separately for each of them. This also provides
us with a much more comprehensive test of the model
than if we had only chosen a single site. Furthermore,
after ?rst presenting detailed results for one category
(online air travel agents), we will then present a sum-
mary of results across multiple categories.
The online travel agent category is one of the early
adopters and one of most successful industries in
e-commerce. Determining various competitive mea-
sures is of particular value given the ?erce competi-
tion in this industry. We selected the top ?ve (based
on the number of customer purchases) online travel
agents in this category in the entire year of 2007—
Expedia (EP), Orbitz (OB), Cheaptickets (CT), Trave-
locity (TL), and Priceline (PL),
10
and hereafter we use
the abbreviations in parentheses to refer to these sites.
Because we are interested in estimating competitive
measures regarding customer purchases to the vari-
ous sites, the preprocessed data representing the full-
information case (used by the NBD/Dirichlet model)
is a count data set where each record represents a
speci?c customer and has six variables—the user ID
and ?ve variables representing the total number of
purchases that this customer made to each of the ?ve
online agents.
5.1. Detailed Results for the
Online Air Travel Industry
Table 2 provides a summary of this data including
reach (the number of customers who made at least
one purchase), penetration, frequency (the average
number of purchases per customer who made at least
one purchase at the focal site), market share, and SoW.
In the spirit of the “double jeopardy” phenomenon
in marketing (Goodhardt et al. 1984), we see that
?rms with higher reach tend to also have higher val-
ues for most of the other measures. As seen (and
10
These sites account for 94% of total visits, 85% of unique users,
and 92% of total purchases in the category.
Zheng et al.: From Business Intelligence to Competitive Intelligence
708 Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS
Table 2 Some Observed Competitive Measures for Online Travel
EP OB CT TL PL Category
Reach 1,930 1,578 1,410 1,250 646 6,132
Penetration (%) 31.5 25.7 22.9 20.4 10.5 100
Frequency 1.40 1.33 1.26 1.41 1.37 1.51
Market share (%) 29.3 22.8 19.2 19.2 9.6 100
SoW (%) 84.0 79.9 78.0 81.4 77.6 100
emphasized) quite often in the NBD/Dirichlet liter-
ature, the cross-site penetrations have much higher
variance than the purchase frequencies.
We ?rst estimate the parameters of the full-
information NBD/Dirichlet model using maximum
likelihood estimation (MLE). There are K +2 param-
eters to be estimated for each focal site: the r and
o parameters for the shifted NBD model of cat-
egory purchase, and K parameters (u
1
, . . . , u
|
) for
the brand-choice probabilities across these K sites.
The estimated parameters of this full-information
model are reported in Table 3. As the expected mar-
ket shares (u,s) show, the full-information Dirichlet
model’s market share inferences correspond reason-
ably well to the actual observed market share mea-
sures reported in Table 2.
5.1.1. The LIND Results. The LIND model takes
a given ?rm as the focal site for parameter estima-
tion, but we repeat this estimation procedure for each
?rm. Thus, there are ?ve separate sets of parameters
reported in Table 4, depending on which site is con-
sidered as the focal one (in other words, the model
is run separately for each one). Each set (i.e., column)
consists of the parameter estimates for r, o, and u
1
through u
5
for all ?ve sites. The order of the esti-
mated parameters is consistent across all ?ve sites,
regardless of which one is being used as the focal
site. At ?rst glance, the magnitudes of the estimated
Dirichlet parameters (u
1
to u
5
) seem to vary across
different focal sites, but this is largely due to incon-
sequential scaling effects. As we will show shortly
(in Table 5), the estimates of market shares, which
automatically adjust for these scaling differences (as
shown in Equation (8)), are virtually identical across
the ?ve focal sites. The magnitude of r and o are
adjusted accordingly to re?ect the scaling. Also note
Table 3 Parameter Estimates from the Shifted NBD Model and the
Dirichlet Model
NBD
r 0.398
u 0.788
EP OB CT TL PL
Dirichlet
a 0.174 0.140 0.124 0.111 0.057
Market share (a,s) (%) 28.7 23.1 20.5 18.3 9.3
Table 4 LIND Parameter Estimation with Each Site (Each Column) as
the Focal Site
Focal site EP OB CT TL PL
r 0.306 0.281 0.495 0.331 0.513
u 0.570 0.590 1.248 0.577 0.933
a (EP) 0.171 0.202 0.225 0.156 0.148
a (OB) 0.139 0.164 0.183 0.126 0.120
a (CT) 0.123 0.146 0.163 0.113 0.107
a (TL) 0.109 0.129 0.144 0.099 0.094
a (PL) 0.056 0.066 0.073 0.051 0.048
that these estimated parameters are comparable to
those of the full-information NBD/Dirichlet, which
suggests that the proposed LIND model does a good
job of recovering the underlying customer behaviors
even with a drastically smaller set of input data.
However, as noted above, merely examining the
closeness of the parameter estimates does not reveal
the performance of individual LIND models. Thus,
we further examine the overall ?tness by plotting the
histogram charts of the observed customer purchases
versus the expected purchases from LIND in Figure 1.
It is clear that each of the ?ve models works quite
well for each focal site.
Based on these parameters shown in Table 4, we
then use the formulae described in §4.2 to estimate the
market share and share of wallet measures for these
?ve retailers. These results are presented in Tables
5 and 6.
These tables should be interpreted as follows.
Each column under “LIND” represents the view of
the entire category from each site’s perspective. For
instance, the results corresponding to the column EP
correspond to building a LIND model with Expedia
as the focal site (i.e., estimating the model parame-
ters from EP’s data plus the constraints) and using
this model to compute MS and SoW for the other
sites. Hence, the results under column EP would rep-
resent Expedia’s view of the world as it related to MS
and SoW, and so on for the columns OB, CT, TL, and
PL. The last two rows of Tables 5 and 6 compute the
mean absolute deviation (MAD)
11
of the LIND results
11
We use MAD as the measure here, following the convention of the
literature in NBD/Dirichlet (e.g., Goodhardt et al. 1984, Fader and
Schmittlein 1993, Uncles et al. 1995). We also tried some other mea-
sures such as average percentage deviation, but the results do not
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Table 5 Market Share Results
LIND (%)
Brand Observed Dirichlet (%) EP OB CT TL PL
EP 29.3% 28.7 28.7 28.6 28.6 28.5 28.5
OB 22.8% 23.1 23.2 23.2 23.3 23.2 23.2
CT 19.2% 20.5 20.5 20.7 20.7 20.8 20.7
TL 19.2% 18.3 18.2 18.3 18.2 18.2 18.3
PL 9.6% 9.3 9.3 9.2 9.2 9.3 9.3
MAD Vs. observed 0.68 0.75 0.75 0.75 0.75 0.75
MAD Vs. Dirichlet 0.07 0.07 0.07 0.07 0.07
Figure 1 The Fitness of LIND: Histograms of the Observed vs. Expected
Fitness of the LIND model Fitness of the LIND model Fitness of the LIND model
5,000
4,000
3,000
2,000
1,000
0
6,000
5,000
4,000
2,000
1,000
0
6,000
5,000
4,000
2,000
1,000
0
0 1 2 3 4 5 6 7 8 9 10+ 0 1 2 3 4 5 6 7 8 9 10+ 0 1 2 3 4 5 6 7 8 9 10+
0
0 1 2 3 4 5 6
Observed
Expected
7 8 9
5,000
4,000
3,000
2,000
1,000
0
0 10+ 1 2 3 4 5 6 7 8 9 10+
500
1,000
1,500
2,000
2,500
3,000
3,500
4,000
4,500
Fitness of the LIND model on Expedia Fitness of the LIND model on Orbitz
on Priceline on Travelocity on Cheaptickets
versus the observed values and the LIND results ver-
sus the Dirichlet results, respectively.
As these two tables show, the full-information
Dirichlet model captures the observed MS and SoW
measures well for all ?rms. This is consistent with
prior research that applies the Dirichlet model for
other retail markets (Goodhardt et al. 1984, Ehrenberg
1988). For our purpose, we use these results as
an upper bound that the use of full information
can achieve. For the MS estimation, LIND’s results
are very close to those of NBD/Dirichlet (within
0.07%), and not far from the actual (within 0.8%).
For the SoW results, LIND compares well with the
NBD/Dirichlet model (within 1.5%) and the observed
values (within 2.0%).
In summary, for the online travel industry, the
LIND model performs well compared to the full-
information NBD/Dirichlet model, while at the same
change qualitatively (e.g., the LIND model estimates are still within
1% of the Dirichlet estimates). Furthermore, we only use MAD
to illustrate the performance of LIND on one category because of
the insuf?cient degrees of freedom to perform a formal statistical
analysis. Later, in Table 12, we formally test the differences using
multicategory data, where we do not rely on the MAD measure.
time using signi?cantly less data than what the
NBD/Dirichlet model uses.
5.1.2. Comparing LIND with BB/NBD and
BB/sNBD. We further compare the LIND model’s
overall ?tness statistically with those of two bench-
mark models: BB/NBD and BB/sNBD. We use the
Bayesian Information Criteria (BIC) to compare the
three models. Notice that all three models were run on
the same customer data (in total 6,132 customers) for
each of the ?ve sites. Both the BB/NBD and BB/sNBD
models use four parameters (r, o, u, |), whereas LIND
uses seven parameters (r, o, and ?ve u).
Table 7 presents the BIC values for the three mod-
els under each site. Based on the BIC numbers,
it is clear that LIND consistently outperforms both
BB/sNBD and BB/NBD; the traditional BB/NBD per-
forms worst among the three.
5.2. Robustness Checks
In this section, we conduct two robustness checks.
5.2.1. Results Using Market Share as the Input.
So far we considered using penetration to be the
inputs for LIND. However, LIND is a ?exible model,
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Table 6 Share of Wallet Results
LIND (%)
Brand Observed (%) Dirichlet (%) EP OB CT TL PL
EP 84.0% 82.0 81.0 80.7 82.8 81.3 82.9
OB 79.9% 80.7 79.7 79.4 81.6 80.0 81.7
CT 78.0% 80.1 79.0 78.7 81.0 79.4 81.1
TL 81.4% 79.6 78.4 78.1 80.5 78.8 80.6
PL 77.6% 77.5 76.2 75.8 78.5 76.6 78.7
MAD Vs. observed 1.35 1.72 1.92 1.52 1.54 1.57
MAD Vs. Dirichlet 1.15 1.48 0.83 0.81 0.99
Table 7 The Overall Fitness (BIC) of the Three Models
Site LIND BB/sNBD BB/NBD
EP 10,671.2 10,745.9 10,860.9
OB 9,201.2 9,272.5 9,354.5
CT 8,395.4 8,408.1 8,430.1
TL 8,229.4 8,261.3 8,291.7
PL 5,133.2 5,167.7 5,178.1
and other aggregate measures can also be utilized
instead; a natural candidate is market share due to its
common availability. Here we further investigate the
performance of LIND when market share is used as
the aggregated inputs. The only modi?cation we need
to make in this case is to change the constraints in
Equation (7) to ensure the expected market share (for-
mula of which is provided in the appendix) is equal
to the actual observed market share. Then we use the
estimated LIND parameters to infer penetration and
SoW, the results of which are reported in Tables 8 and
9, respectively.
As these two tables show, LIND still performs well,
con?rming the ?exibility and generality of LIND.
5.2.2. Summary Results for Multiple Categories.
We also report results from applying the model to
a broader set of ?ve online retail categories: online
apparel, wireless services, books, of?ce supplies, and
travel services. Among the different categories of pur-
chases that comScore Networks identify in their data,
these categories are among the most-visited retail cat-
egories in the panel and are featured by comScore
among key industries
12
for online retailing. Speci?-
cally, these ?ve categories reached 73% of unique cus-
tomers and accounted for 36% of total transactions in
the panel.
One caveat concerning the comScore panel data is
that there are not many customers who make pur-
chases across multiple sites in these additional cat-
egories. In Table 10, we report the percent of 100%
loyal customers (those customers who only purchase
at a single website) in each category. For the ?ve
12
Seehttp://www.comscore.com/solutions/is.asp.
categories of interest, on average only a little more
than 10% of customers purchase at multiple sites.
Although this is the reality of the online business,
it does not provide a particularly tough test for the
LIND model. Accordingly, we decided to switch from
purchasing data to site-visit data for these further
tests. Competitive measures of visits are important in
their own right, e.g., share of visits has a direct impact
on advertising revenue. Google’s new tool “Google
insights for search” has added “share of search” as a
measure to analyze searching patterns of users across
different competitors. Further, visit patterns capture
prepurchase information search and consideration set
formation of the consumer. It is well recognized that
consideration set formation is important in under-
standing and explaining consumer-choice behavior
(Roberts and Lattin 1997). This is especially critical
for the online world because it is possible to capture
what sites an individual considered before arriving at
a purchase decision. Google Analytics offers a tool for
sites to funnel down from site visits to purchasing.
Finally, this alternative measure gives us a chance to
see the robustness of the LIND model, which helps
demonstrate its general applicability to different kinds
of behavioral settings.
Table 10 shows that there are plenty of cus-
tomers who visit multiple websites in each category
(along with some interesting and believable differ-
ences across the ?ve categories). We apply the LIND
model to the visit data from comScore in exactly the
same way as we did with the purchase data, using
penetration as the inputs. For each category we con-
tinue to use the top ?ve sites to maintain consistency.
The comScore data show that visits to various online
retail categories is highly concentrated, with the top
?ve sites in each category accounting for more than
90% of total visits in the category. In one extreme case
(wireless services) the top ?ve sites account for 99.5%
of all visits.
To compare the LIND and the Dirichlet estimates,
reconsider the results shown for online travel in
Tables 5 and 6. Here note that every LIND cell in these
tables—and there are 25 of them corresponding to ?ve
focal sites—can be compared to the corresponding
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Table 8 Penetration Results (with MS as the Input)
LIND (%)
Brand Observed Dirichlet (%) EP OB CT TL PL
EP 31.5% 31.6 31.7 32.8 34.3 31.2 31.6
OB 25.7% 25.7 24.9 25.9 27.1 24.4 24.7
CT 23.0% 22.9 21.0 21.9 23.0 20.6 20.9
TL 20.4% 20.5 21.0 21.9 23.0 20.6 20.9
PL 10.5% 10.6 10.6 11.1 11.8 10.4 10.6
MAD Vs. observed 0.07 0.76 0.94 1.63 0.88 0.74
MAD Vs. Dirichlet 0.68 0.88 1.61 0.85 0.67
Table 9 Share of Wallet Results (with MS as the Input)
LIND (%)
Brand Observed Dirichlet (%) EP OB CT TL PL
EP 84.0% 82.0 81.0 80.7 82.8 81.3 82.9
OB 79.9% 80.7 79.7 79.4 81.6 80.0 81.7
CT 78.0% 80.1 79.0 78.7 81.0 79.4 81.1
TL 81.4% 79.6 78.4 78.1 80.5 78.8 80.6
PL 77.6% 77.5 76.2 75.8 78.5 76.6 78.7
MAD Vs. observed 1.35 1.67 1.53 1.50 1.21 2.03
MAD Vs. Dirichlet 1.76 1.55 1.02 0.18 2.18
Table 10 Percentage of 100% Loyal Customers
Apparel Book Of?ce Travel Wireless
(%) (%) (%) (%) (%) Average
Purchases 83.8 89.2 90.2 90.0 93.6 89.4
Visits 63.9 70.1 67.1 34.2 67.3 60.5
Dirichlet estimates. Extending this to ?ve categories,
we obtain 125 speci?c comparisons for each measure.
Figures 2 and 3 summarize the LIND/Dirichlet com-
parison, with the LIND values on the Y axis and a
trend line plotted for comparison.
The market share estimates of the LIND model
are very closely aligned with the estimates from the
full-information model, and the SoW estimates also
line up well, albeit with slightly higher variance.
Figure 2 Market Share Plot for All Five Categories
Market share scatterplot
0
10
20
30
40
50
60
70
(
%
)
80
90
100
0 10 20 30 40 50
(%)
60 70 80 90 100
LIND
Linear (LIND)
Trendline slope = 1.069
The estimated slopes are signi?cant at 1.06 and
0.96, respectively, again close to the ideal values
obtained when the LIND estimates exactly corre-
spond to the Dirichlet estimates.
13
These results are
very signi?cant, considering the fact that the full-
information Dirichlet model uses detailed customer-
level cross-site data for the entire category, whereas
LIND only uses one simple, often publicly available,
aggregate measure for each competing ?rm. In the
stream of work on estimating competitive measures,
to our knowledge this is the ?rst result that shows
such good performance with so little data.
Next we present results that compare the Dirichlet,
observed, and LIND estimates against each other.
Again, we obtain 125 points where LIND can be com-
pared against Dirichlet and observed values. For each
comparison, we compute an absolute deviation repre-
senting the absolute value of the difference between
13
Compared to the market share results, the SoW estimates exhibit
higher variance. This is expected, as is evident from Equations (8)
and (9) (for estimating market share and SoW, respectively). The
market share variance is only a function of the variance of param-
eter u, whereas the SoW variance is ampli?ed by the variance of r
and o. Thus, the variability associated with these additional param-
eter estimates will in?ate the variability for the SoW statistics.
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712 Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS
Figure 3 SoW Plot for All Five Categories
SoW scatterplot
0
10
20
30
40
50
(
%
)
60
70
80
90
100
0 10 20 30 40 50 60
(%)
70 80 90 100
LIND
Linear (LIND)
Trendline slope = 0.959
the LIND and Dirichlet/observed values. Across the
125 points we compute a mean absolute deviation
(MAD). Instead of reporting this entire table, we
group these into ?ve based on the categories and
report the MAD for each category.
Grouping the results into categories also has the
advantage that the Dirichlet/observed comparisons
will not have “repeats.” Speci?cally, for each category
the Dirichlet estimates (and observed values) will
always be the same for any focal site. For instance,
referring back to Tables 5 and 6, note that the ?rst
two columns (Dirichlet and observed values) are inde-
pendent of which focal site is chosen for parameter
estimation.
Table 11 presents the summary of results. First, note
that the market share results are particularly good.
The MADs for all comparisons are low. The SoW esti-
mates are also good, but not as close as the market
share estimates. Speci?cally, the MAD of LIND ver-
sus Dirichlet is only 2.7%. The MAD comparison is an
effective method for determining how close the esti-
mates are in absolute terms (Goodhardt et al. 1984). To
compute closeness to optimality in relative percent-
age terms, we compared the percentage differences
between the LIND and Dirichlet estimates across all
cases. For the market share estimates LIND is within
Table 11 Mean Difference Results and Signi?cance Tests by Category
Market share SoW
Observed vs. Observed vs. Dirichlet vs. Observed vs. Observed vs. Dirichlet vs.
Categories Dirichlet LIND LIND Dirichlet LIND LIND
Apparel (%) 1.1 2.4 2.7 4.5 5.0 2.1
Book (%) 0.2 0.4 0.5 3.4 5.3 2.7
Of?ce (%) 1.3 1.3 0.1 6.7 7.5 3.9
Travel (%) 1.2 2.0 0.8 1.9 3.3 2.2
Wireless (%) 1.8 1.9 0.3 7.1 7.6 2.7
Average (r =125) (%) 1.1 1.6 0.9 4.7 5.7 2.7
Deviation (r =125) 0.009 0.014 0.015 0.045 0.046 0.022
0.9% of the Dirichlet estimates on average, and for the
SoW estimates it is within 2.7%.
We further use a repeated measure ANOVA to test
if the results from the LIND “treatment” are signif-
icantly different from that of the Dirichlet’s. Hence,
the LIND and Dirichlet treatments are the two within-
subjects factors each site receives. Moreover, all the
sites are grouped into ?ve categories, and thus the
categories form the between-subjects factors. Table 12
presents the two within-subjects tests for market share
and SoW results separately. Both tests show that there
is no signi?cant difference under the Dirichlet ver-
sus the LIND treatments for inferring market share
(p-value = 0.984) and SoW (p = 0.583). In addition,
LIND’s market share estimates approach the observed
values (p = 0.987), but the SoW estimates are still
signi?cantly different than the actual observed val-
ues (p = 0.007). This is also true even for the full-
information Dirichlet model (p = 0.006), suggesting
the dif?culty of estimating SoW precisely even under
the classic Dirichlet model.
6. Discussion
The model presented in this paper, LIND, is a spe-
cial case of the NBD/Dirichlet, applicable to a limited
data scenario that is prevalent in business. We show
that by formulating this as a constrained optimization
problem, the derived parameter estimates can be used
to intelligently infer several useful competitive mea-
sures. Our analysis of online retail data shows that
LIND is an effective model that improvises on the
well-known Dirichlet model when only limited data
are available. We are not aware of any studies that
show how transactional data can be combined with
aggregate summaries to make important inferences in
such a case. This paper presents one novel approach
towards this end.
This study is not without limitations. Understand-
ing these limitations is important to know the scope
of LIND and how to apply it effectively. Given its lin-
eage, LIND inherits the strengths as well as the lim-
itations of the Dirichlet model. We ?rst discuss the
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Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS 713
Table 12 Results from Repeated-Measure ANOVA
Market share SoW
Observed vs. Observed vs. Dirichlet vs. Observed vs. Observed vs. Dirichlet vs.
Categories Dirichlet LIND LIND Dirichlet LIND LIND
Mean square error 7.04E?10 6.40E?08 6.40E?08 1.57E?02 1.81E?02 1.55E?04
l -value 0.000 0.000 0.000 7.924 7.657 0.304
F -value 1.000 0.987 0.984 0.006 0.007 0.583
limitations common to the Dirichlet family of models
and the conditions under which these models work
well (or not). We then discuss the limitations speci?c
to LIND.
6.1. Limitations Common to the
Dirichlet Family of Models
As we discussed in §3.2, the Dirichlet model requires
the market to be stable, two key conditions of which
are (1) the market is stationary and (2) the market is
nonsegmented (Ehrenberg et al. 2004, Goodhardt et al.
1984, Ehrenberg 1988). The resultant market exhibits
the prevalent “double jeopardy” phenomenon.
First, the stationarity condition is mainly a result
of the Poisson assumption for the category purchase.
Poisson distribution is memoryless, i.e., a customer’s
purchase propensity does not change across differ-
ent time periods and the last-period purchase has no
in?uence on a customer’s next purchase. In a non-
stable market, a customer may change her pace of
purchase across different time periods. Thus the con-
ditional mean of the future-period purchase may no
longer be a linear projection of the previous periods,
the de?ning feature of the NBD model (Schmittlein
et al. 1985). This is less of a problem when the mar-
keter’s interest is to perform diagnostic analysis on
the current period of data as we do in this paper.
When the endeavor moves to predicting future-period
customer behavior, violation of this condition looms
larger and will have an adverse effect on the Dirich-
let model’s (and LIND’s) performance. Unfortunately,
there is no standard treatment of nonstationarity in
the literature. Some early attempts include Fader and
Lattin (1993) and Jeuland et al. (1980). For exam-
ple, Jeuland et al. (1980) replaced the stationary Pois-
son assumption with the Erlang-2 distribution, which
processes records at every second arrival to allow
for change of pace among customer purchases over
time. However, the trade-off is that we pay a big sta-
tistical price for the added generality with limited
improvement of these complex extensions (Morrison
and Schmittlein 1988).
Second, the performance of the Dirichlet family
of models is subject to the nonsegmented market
condition. However, a fully disparate, segmented
market may not be best treated as a whole market in
the ?rst place. We may need to rede?ne the relevant
market to be each individual segment (i.e., applying a
separate Dirichlet model to each segment). If we force
these segmented markets to be one big market, the
double jeopardy property may not hold. For example,
in a niche market, the segment may cater to exces-
sive loyal customers who purchase more frequently
than what is expected by the Dirichlet model in a
normal market. There are some attempts to extend
the Dirichlet model to account for segmented mar-
kets. Danaher et al. (2003) used a latent class model
approach where latent segments are modeled as a
?nite mixture model as a function of additional infor-
mation such as price, promotion, etc. These are excel-
lent approaches that shed light on future extensions
of LIND to such markets.
Third, it is not easy to incorporate market-mix vari-
ables (e.g., price and promotion) or the individual
customer-level variable (e.g., demographics) into the
Dirichlet-type models (e.g., Bhattacharya 1997). This
work avoids the complex task of directly integrating
covariates into the Dirichlet model by using a two-
step modeling procedure. First, a regular Dirichlet
model is built and then competitive measures such
as SCR (share of category requirement) are estimated
using the Dirichlet parameters. Deviations of these
estimates from the actual SCRs are then calculated.
Subsequently, in the second step, a linear regression
analysis is performed by regressing these deviations
on the marketing-mix variables. Bhattacharya (1997,
p. 431) maintains that “it is better to study and explain
these deviations outside the Dirichlet model, rather
than try to make this ?exible, elegant and parsimo-
nious model more complex.” To our knowledge, a
generalized Dirichlet model that directly integrates
market mix or customer level of data has not been
developed in the literature, making it an interesting
topic for future research.
Fourth, the Dirichlet model is elegant, but rather
complex, because it mixes Poisson, Gamma, multi-
nomial, and Dirichlet distributions. It is hard to
tweak the model with other alternative distributional
assumptions. Even a small extension often ends up
with an intractable model. Although there are some
early attempts, none supersedes the NBD/Dirichlet
model and it still stands as one of the most popu-
lar and successful models in the repeat-buying liter-
ature. Some notable attempts include the condensed
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714 Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS
NBD model (Jeuland et al. 1980) with an Erlang-2
distribution (instead of Poisson); the log-normal mix-
ing distribution (Lawrence 1980); and the generalized
inverse Gaussian (Sichel 1982) to replace the Gamma
mixing distribution. An analysis of the Dirichlet
model’s sensitivity to these distributional assumption
is provided in Fader and Schmittlein (1993).
We would like to conclude this section with the
argument of Ehrenberg et al. (2004, p. 1312) that
“such known discrepancies (with the stable-market
assumptions) seldom curtail the model’s practical
use. Attempts to improve or elaborate the Dirich-
let have so far not resulted in major gains in either
predictive power or parsimony.” Just as Morrison
and Schmittlein (1988, p. 152) elegantly argued that
although there is only one way for a process to remain
stable, there are of course an in?nite variety of pos-
sible nonstable behaviors. Thus, not very much can
be said in general about the effects of instability. It
is beyond the scope of a single paper to thoroughly
examine these generalizations. In future research we
plan to extend LIND to such a market.
Moreover, even in nonstable markets, the Dirich-
let model still serves as an important benchmark
model for managers to detect whether there is devi-
ation from the norm predicted by the model and
where the deviations come from. For example, Fader
and Schimittlein (1993) use the Dirichlet benchmark
to identify niche-market brands—those exhibit higher
loyalty than predicted by the Dirichlet. Kahn et al.
(1988, p. 385) assert that although deviations from
norms are plausible, “they are dif?cult to apply with-
out a relevant benchmark for comparison.”
6.2. LIND-Speci?c Issues
LIND can be conceived as a realistic version of the
Dirichlet model for the limited data scenario. In terms
of implementing LIND, the user needs to ?rst choose
appropriate known inputs to infer the unknown out-
puts (i.e., competitive measures) of interest. Not all
inputs are equally useful. We identify three conditions
to ensure the proper selection of an input.
First, the chosen input should be indicative of the
market. Measures re?ecting the attractiveness of the
brand, such as penetration and market share, are
informative of the structure of the market. These mea-
sures tend to have higher variation across brands
(e.g., the penetration of the biggest brand Expedia
is almost three times that of Priceline, the smallest
one in our data). Measures on the intensity of cus-
tomer purchase (e.g., frequency) and the loyalty of
customers (e.g., SoW, 100% loyalty and duplication)
tend to vary less and hence are less indicative of the
market structure (Ehrenberg et al. 2004). For example,
the frequency of Expedia is 1.4 versus 1.37 for Price-
line; SoW of expedia is 84% versus 77% for Priceline.
Thus, inputs such as frequency and SoW, which do
not vary much across brands, may not serve as good
candidate inputs. This is echoed in Ehrenberg (2000,
p. 188) that “it is penetration, not the average pur-
chase level (i.e., frequency) that determines the sales
level of a brand. . . .”
Second, inputs that are expected to have high
correlation with the outputs are preferred. Not
surprisingly, our results indicate that the higher the
correlation, the better the model performs. For exam-
ple, in the travel industry, the (Spearman) correla-
tion between penetration and market share is 0.97,
that between market share and SoW is 0.82, and that
between penetration and SoW is 0.7. The results show
that LIND works best when using penetration to esti-
mate market share, followed by using market share
to estimate SoW and then using penetration to esti-
mate SoW.
Third, in a stable market, the rank order of the
outputs is generally expected to follow that of the
inputs. In a stable market exhibiting “double jeop-
ardy,” the rank order of several key measures such as
penetration and frequency are expected to be exactly
the same. This is described in the seminal paper of
Goodhardt et al. (1984, p. 623) that “this is known as
a Double Jeopardy pattern (for two brands X and Z):
Z not only has fewer buyers than X, but they also
buy it (slightly) less often.” The double-jeopardy rule
is further summarized in Ehrenberg (2000, p. 186) as
.(1 ? |) = c, where . is the frequency of a brand
(among all customers in the market), | is the pene-
tration of the brand, and c is a constant. This implies
that the larger the brand (in terms of penetration), the
larger the frequency and, in turn, the larger the mar-
ket share (because it is a function of . ×|). In other
words, market share is expected to exhibit the same
rank order as penetration in the Dirichlet world.
The same expected rank order of penetration, mar-
ket share, and SoW (the three measures demonstrated
in this paper) can be shown analytically. Everything
else being equal, a bigger brand in LIND (also Dirich-
let) will yield a higher value in parameter u
¡
(an indi-
cation of customer purchase propensity to brand ¡). It
is easy to verify that the following measures would
be a monotonic-increasing function of u
¡.
(i)
MarketShare =
u
¡

|
¡=1
u
¡
=
u
¡
s
,
clearly the ?rst derivative with respect to u
¡
is
positive.
(ii)
Penetration
¡
= 1 ?P(X
¡
=0)
= 1?
_
o
o+1
_
r
|
¡
s
2
í
1
_
r,|
¡
+1,s +1,
1
o+1
_
,
Zheng et al.: From Business Intelligence to Competitive Intelligence
Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS 715
because
2
í
1
(u
¡
) is a monotonic-decreasing function of
u
¡
, and |
¡
,s is also decreasing as a function of u
¡,
, pen-
etration increases as u
¡,
increases.
(iii)
SoW = ((u
¡
,s)(r,o+1)) ·
_
(u
¡
,s)(r,o+1)
?

_
n=1
(!(r +n?1)),!(r)(n?1)!)
· (o,(o+1))
r
(1,(o+1))
n?1
· n(!(s)!(|
¡
+n),!(|)!(s +n))
_
?1
is also an increasing function of u
¡
. Let
c1 =
_
r
o
+1
_
,
c2 =

_
n=1
!(r +n?1)
!(r)(n?1)!
_
o
o+1
_
r
_
1
o+1
_
n?1
· n
!(s)!(|
¡
+n)
!(|)!(s +n)
.
Then SoW

(u
¡
) =|C1,(C1 ?C2(1 +|
¡
,u
¡
))]

>0.
In sum, all these three measures exhibit the same
rank order. The ideal market for the Dirichlet world
is a triple-jeopardy world: a large brand not only
attracts more purchases per customer, but more-loyal
customers as well.
7. Conclusion
In this paper we addressed the problem of estimating
important competitive measures from a single focal
site’s point of view. Given a realistic assumption that
each ?rm has its own data and is able to obtain the
penetration or market share numbers of its leading
competitors, we develop a constrained optimization
model based on a well-established theory of con-
sumer behavior. Solving this problem from each focal
site’s perspective provides parameter estimates for
the proposed LIND model. We then show how these
parameter estimates can be used to derive analytical
expressions for the market share and share of wallet.
Testing our approach on various online retail cate-
gories shows that the LIND model performs surpris-
ingly well considering how little information it uses.
In summary, in this paper we developed a method
for making competitive inferences that is (i) practi-
cal, in that it relies on sharing simple aggregate data;
(ii) effective, in that it can enable the determination
of useful competitive measures such as market share
and share of wallet; and (iii) explainable in that it is
based on a well-studied theory.
The problem of making inferences about these com-
petitive measures is one that is important in any
industry. Although our application domain here was
online retail, our methods are certainly not restricted
to it, and can be applied in more traditional markets
as well. Dif?culties in obtaining cross-?rm panel data
are widespread, as are sources of aggregate compet-
itive measures such as SoW. The emergent competi-
tive intelligence ?eld has emphasized the importance
of competitive measures. We expect to see the next-
generation dashboard to incorporate some of these
measures. Google’s new tool “Google insights for
search,” released in 2008, has added competitive intel-
ligence analysis, focusing on the searching patterns
of users across different competitors (http://www
.google.com/insights/search/#). Their concept of
“share of search” is what we demonstrate in the
robustness check section. Microsoft’s “adCenter Labs”
also provides some Web analytics functions that
track a ?rm’s performance relative to its com-
petitors. The social media analytics tool release in
2010 has added the functionalities to analyze the
share of buzz in the social media across differ-
ent brands (http://www.sas.com/software/customer
-intelligence/social-media-analytics/).
As one more example, the Lundberg Survey is an
authoritative source for information about gasoline
prices and competitive measures in the petroleum
industry, used heavily by investors and industry
experts to understand and forecast purchasing pat-
terns in the industry. Data collection is extremely
complex, based on a laborious biweekly sampling
of 7000 of the 133,000 gas stations in the United
States. However, even this survey cannot obtain
SoW estimates because that would require linking
customer-level information across gas stations. Thus,
the method presented in this paper is of particular
managerial relevance given the prevalence of such
scenarios in this and many other industries.
Electronic Companion
An electronic companion to this paper is available as
part of the online version athttp://dx.doi.org/10.1287/
isre.1110.0385.
Appendix A. Deriving the BB/sNBD Model
A.1. Deriving the Conditional Probability P(X =x \, p)
Based on the data assumption (see §4) that ?rms only
observe those users making at least one purchase to the
category, we model the category purchase N as a shifted
Poisson with parameter \ as the purchase rate of a random
customer in the unit time interval.
P(N =n \) =
\
n?1
c
?\
(n?1)!
, n =1, 2. . . , \ >0 (A1)
Zheng et al.: From Business Intelligence to Competitive Intelligence
716 Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS
Given the category purchase n of a random customer, the
choice of a focal brand is modeled as a binomial distribution
with rate p.
P(X =x n, p) =
_
n
x
_
p
x
(1 ?p)
n?x
,
x =0, 1, . . . n, 0 ?p ?1. (A2)
It is assumed that an individual’s choice rate p is inde-
pendent of the purchase rate \. Thus, P(X = x \, p),
the conditional probability of a customer making a pur-
chase to the focal site given \ and p, is a BB/sNBD (beta-
binomial/shifted negative binomial distribution) model.
First consider the special case where x =0. Because ?rms
share the aggregated data—the number of customers they
reached—each ?rm knows the number of customers who
purchased the category, but not the focal ?rm. We have
P(x =0 \, p) =

_
n=1
\
n?1
c
?\
(n?1)!
(1 ?p)
n
= c
?\p
(1 ?p). (A3)
Notice that the sum in (3) starts with n =1 (because the
data are truncated at n ? 1) instead of n = x. In the more
general case when x ?1, we have
P(x \, p) =

_
n=x
\
n?1
c
?\
(n?1)!
_
n
x
_
p
x
(1 ?p)
n?x
=
c
?\
p
x
x!

_
n=x
\
n?1
(n?1)!
n!(1 ?p)
n?x
(n?x)!
=
c
?\
p
x
\
x?1
x!

_
n=x
n\
n?x
(1 ?p)
n?x
(n?x)!
. (A4)
Let j =n?x,. Equation (A4) is transformed into
P(x \, p) =
c
?\
p
x
\
x?1
x!

_
j=0
(x +j)\
j
(1 ?p)
j
j!
=
c
?\p
p
x
\
x?1
x!

_
j=0
(x +j)
|\(1 ?p)]
j
c
?\(1?p)
j!
.
Notice that

_
j=0
|\(1 ?p)]
j
c
?\(1?p)
j!
=1 and

_
j=0
j
|\(1 ?p)]
j
c
?\(1?p)
j!
=|(j) =\(1 ?p),
thus, when x ?1,
P(x \, p) =
c
?\p
p
x
\
x?1
x!
|x +\(1 ?p)]
= o
x>0
c
?\p
p
x
\
x?1
(x ?1)!
+
c
?\p
p
x
\
x
(1 ?p)
x!
(A5)
where o
x>0
=1 if x >0 and 0 otherwise.
A.2. Deriving the Marginal Probability P(X =x)
To capture consumer heterogeneity, the distribution of the
purchase frequency \ across the population is assumed to
be distributed as gamma with pdf
] (\) =
o
r
\
r?1
c
?\o
!(r)
, \ >0. (A6)
Similarly, the distribution of consumer choice p across the
population is distributed as beta with pdf
g(p) =
1
B(u, |)
p
u?1
(1 ?p)
|?1
, 0 ?p ?1 (A7)
where B(u, |) is the beta function.
Further, the following derivation uses the Euler’s inte-
gral representation of the Gaussian hypergeometric function
2
í
1
( · ) as explained in Footnote 4.
_
1
0
|
u
(1?|)
|
(u+.|)
?c
d|
=
_
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
_
B(u+1,|+1)u
?c
2
í
1
_
c,u+1,u+|+2,?
.
u
_
,
. -u
B(u+1,|+1)(u+.)
?c
2
í
1
_
c,|+1,u+|+2,
.
u+.
_
,
. ?u
(A8)
Further, the Gaussian hypergeometric function converges
when the last argument is no greater than 0 and diverges
otherwise.
A.2.1. Case 1: X =0. In this case, from (3), the marginal
probability becomes
p(X =0) =
_
1
0
_

0
c
?\p
(1 ?p)
\
r?1
o
r
c
?\o
!(r)
p
u?1
(1 ?p)
|?1
B(u, |)
d\dp
=
o
r
!(r)B(u, |)
_
1
0
p
u?1
(1 ?p)
|
_
_

0
c
?\(p+o)
\
r?1
d\
_
dp
=
o
r
!(r)B(u, |)
_
1
0
p
u?1
(1 ?p)
|
(p +o)
?r
dp.
Further necessary transformation is needed to ensure the
convergence of the Gaussian hypergeometric function as
noted above. Let q =1?p, which implies dp =?dq, we have
_
1
0
p
u?1
(1 ?p)
|
(p +o)
?r
dp
=?
_
0
1
(1 ?q)
u?1
(q)
|
(1 ?q +o)
?r
dq
=
_
1
0
(1 ?q)
u?1
(q)
|
(1 +o?q)
?r
dq.
It is clear that (1 +o) >1, and based on the Euler’s inte-
gral given in (8), we have
p(X =0)
=
o
r
!(r)
!(r)B(u, |)
B(u +x, | +1)
(o+1)
r
2
í
1
_
r, | +1, u +| +1,
1
o+1
_
=
_
o
o+1
_
r
|
u +|
×
2
í
1
_
r, | +1, u +| +1,
1
o+1
_
. (A9)
Zheng et al.: From Business Intelligence to Competitive Intelligence
Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS 717
A.2.2. Case 2: X > 0. In the more general case when
X >0,
P(X =x) =
_
1
0
_

0
p(X =x \, p)] (\)g(p) d\dp
=
_
1
0
_

0
_
o
x>0
c
?\p
p
x
\
x?1
(x ?1)!
+
c
?\p
p
x
\
x
(1 ?p)
x!
_
·
\
r?1
o
r
c
?\o
!(r)
p
u?1
(1 ?p)
|?1
B(u, |)
d\dp. (A10)
For convenience, we separate the following derivation for
Equation (10) into two terms. Term 1 consists of
_
1
0
_

0
o
x>0
c
?\p
p
x
\
x?1
(x ?1)!
\
r?1
o
r
c
?\o
!(r)
p
u?1
(1 ?p)
|?1
B(u, |)
d\dp,
and term 2 consists of
_
1
0
_

0
c
?\p
p
x
\
x
(1 ?p)
x!
\
r?1
o
r
c
?\o
!(r)
p
u?1
(1 ?p)
|?1
B(u, |)
d\dp.
Term 1 derivation.
Term 1
=
o
r
(x ?1)!!(r)B(u, |)
_
1
0
p
(x+u?1)
(1 ?p)
|?1
·
_
_

0
c
?\(p+o)
\
(x+r?2)
d\
_
dp
=
o
r
(x ?1)!!(r)B(u, |)
_
1
0
p
(x+u?1)
(1 ?p)
|?1
· |?(p +o)
?(x+r?1)
!(r +x ?1, (p +o)\]

0
] dp
=
o
r
!(r +x ?1)
(x ?1)!!(r)B(u, |)
_
1
0
p
(x+u?1)
(1 ?p)
|?1
(p +o)
?(r+x?1)
dp
where !(u, x) =
_

x
|
u?1
c
?|
d| is the incomplete Gamma func-
tion. Again let q = 1 ? p, which implies dp = ?dq; we
then have
_
1
0
p
(x+u?1)
(1 ?p)
|?1
(p +o)
?(r+x?1)
dp
=?
_
0
1
(1 ?q)
(x+u?1)
q
|?1
(1 +o?q)
?(r+x?1)
dq
=
_
1
0
(1 ?q)
(x+u?1)
q
|?1
(1 +o?q)
?(r+x?1)
dq.
Clearly, we have (1 +o) >1, and based on (8) the above
integral yields
_
0
1
(1 ?q)
(x+u?1)
q
|?1
(1 +o?q)
?(r+x?1)
dq
=B(|, x +u)(1 +o)
?(r+x?1)
2
í
1
_
x +r ?1, |, u +| +x,
1
o+1
_
,
and thus
Term 1 =
o
r
!(r +x ?1)
(x ?1)!!(r)B(u, |)
B(u +x, |)
(o+1)
(x+r?1)
·
2
í
1
_
x +r ?1, |, u +| +x,
1
o+1
_
=
o
r
(o+1)
(x+r?1)
!(r +x ?1)
(x ?1)!!(r)
!(u +x)!(u +|)
!(u)!(u +| +x)
·
2
í
1
_
x +r ?1, |, u +| +x,
1
o+1
_
. (A11)
Term 2 derivation.
Term 2
=
_
1
0
_

0
c
?\p
p
x
\
x
(1 ?p)
(x)!
\
r?1
o
r
c
?\o
!(r)
p
u?1
(1 ?p)
|?1
B(u, |)
d\dp
=
o
r
x!!(r)B(u, |)
_
1
0
p
u+x?1
(1 ?p)
|
_
_

0
c
?\(p+o)
\
x+r?1
d\
_
dp
=
o
r
!(r +x)
x!!(r)B(u, |)
_
1
0
p
u+x?1
(1 ?p)
|
(p +o)
?(x+r)
dp
=
o
r
!(r +x)
x!!(r)B(u, |)
8(u +x, | +1)
(o+1)
(x+r)
·
2
í
1
_
x +r, | +1, u +| +x +1,
1
o+1
_
=
o
r
(o+1)
(x+r)
!(r +x)
x!!(r)
!(u +x)!(| +1)!(u +|)
!(u +| +1 +x)!(u)!(|)
·
2
í
1
_
x +r, | +1, u +| +x +1,
1
o+1
_
. (A12)
The ?nal result of the marginal probability P(X =x) is a
combination of the two terms, which is
P(X =x) =
o
r
(o+1)
(x+r?1)
!(r +x ?1)
(x ?1)!!(r)
!(u +x)!(u +|)
!(u)!(u +| +x)
·
2
í
1
_
x +r ?1, |, u +| +x,
1
o+1
_
+
o
r
(o+1)
(x+r)
!(r +x)
x!!(r)
!(u +x)!(| +1)!(u +|)
!(u +| +1 +x)!(u)!(|)
·
2
í
1
_
x +r, | +1, u +| +x +1,
1
o+1
_
. (A13)
This is the shifted BB/sNBD model, which is the baseline
model of LIND.
Appendix B. Derivation of the Various
Competitive Measures
Here we derive the various competitive measures for the
LIND model. The notations used here follow the de?nitions
in Table 1.
B.1. Market Share
Market share is equal to the ratio of the focal site’s pur-
chases to the category purchases, and thus:
MarketShare =
u
¡

|
¡=1
u
¡
=
u
¡
s
.
B.2. Share of Wallet (SoW)
SoW is equal to the ratio of the total purchases at a focal
brand over the total category purchases of those customers
who made purchase at the focal site. Suppose the cate-
gory purchase n is known, then the total purchases at a
focal is simply n×u
i
,s, where s =

¡
u
¡
. Moreover, P fol-
lows a shifted NBD process. Hence, the numerator of the
ratio is simply

n=1
P ×n ×u
¡
,s. The probability of cus-
tomers who made at least one purchase at the focal site
is 1 ? P(x
¡
= 0 n), where P(x,n) follows a binomial pro-
cess. The denominator of the ratio thus is

n=1
P ×n ×
(1 ?P(x
¡
=0 n)).
Zheng et al.: From Business Intelligence to Competitive Intelligence
718 Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS
SoW =|
_
x
¡
n
,x
¡
>0
_
=

_
n=1
Pnu
¡
,s

_
n=1
Pn(1 ?P(x
¡
=0 n))
=

_
n=1
!(r +n?1)
!(r)(n?1)!
_
o
o+1
_
r
_
1
o+1
_
n?1
nu
¡
,s

_
n=1
!(r +n?1)
!(r)(n?1)!
_
o
o+1
_
r
_
1
o+1
_
n?1
n
_
1 ?
B(u
¡
+0, |
¡
+n+0)
B(u
¡
, |
¡
)
_
=
u
¡
s
_
r
o
+1
_
u
¡
s
_
r
o
+1
_
?

_
n=1
!(r+n?1)
!(r)(n?1)!
_
o
o+1
_
r
_
1
o+1
_
n?1
n
!(s)!(|
¡
+n)
!(|)!(s +n)
.
Solving the above expression analytically is nontrivial.
We use a numerical approach instead that integrates over
n from 1 to a suf?ciently large number (determining what
large number is suf?cient depends on the desirable preci-
sion one needs. In our implementation, we use 100, which
reaches a precision 10
?8
). This approach can be easily
implemented in Excel using a VBA Macro. The following
VBA Macro de?nes a customized function SoW(u, |, r, o)
in excel.
Public Function SoW(a As Double, b As Double, r As
Double, alpha As Double) As Double
Dim n, x As Integer
Dim numerator, denominator, pn, px0 As Double
Temp, pn_sum, numerator, denominator = 0
For n = 1 To 100
With Application
pn = Exp(.GammaLn(r + n-1) - .GammaLn(r)) /
.Fact(n-1) * (alpha / (alpha + 1)) r * (1 / (alpha +
1)) (n-1)
numerator = numerator + pn * n * a / (a + b)
px0 = Exp(.GammaLn(b + n) + .GammaLn(a + b) -
.GammaLn(a + b + n) - .GammaLn(b))
denominator = denominator + pn * n * (1-px0)
End With
Next n
SoW = numerator / denominator
End Function
B.3. Penetration
Site j’s expected penetration is equal to 1 minus the expected
percentage of customers who did not purchase the site
and thus
Penetration
¡
= 1 ?P(X
¡
=0)
= 1 ?
_
o
o+1
_
r
|
¡
s
2
í
1
_
r, |
¡
+1, s +1,
1
o+1
_
.
B.4. Frequency
Expected frequency of all customers with respect to site ¡
is given below. Notice that the numerator is the same as
the numerator in the SoW expression above, which repre-
sents a random customer’s number of purchases to the site
of interest. The denominator represents the probability that
a random customer (in the category) would purchase the
focal site.
Frequency = |(X
¡
)
=

_
n=1
Pnu
¡
,s
=

_
n=1
!(r +n?1)
!(r)(n?1)!
_
o
o+1
_
r
_
1
o+1
_
n?1
nu
¡
,s.
Letting n =m+1, we can transform the above summation
into
|(x
¡
) =

_
m=0
!(r +m)
!(r)m!
_
o
o+1
_
r
_
1
o+1
_
m
(m+1)u
¡
,s
=
u
¡
s
×
_

_
m=0
!(r +m)
!(r)m!
_
o
o+1
_
r
_
1
o+1
_
m
m
+

_
m=0
!(r +m)
!(r)m!
_
o
o+1
_
r
_
1
o+1
_
m
_
=
u
¡
s
_
r
o
+1
_
.
Note that the frequency of buyers of site ¡ (those cus-
tomers who made at least one purchase) is simply the above
frequency divided by penetration. That is,
|(x
¡
x
¡
>0) =
|(x
¡
)
Penetration
=
(u
¡
,s)(r,o+1)
1?(o,(o+1))
r
(|
¡
,s)
2
í
1
(r,|
¡
+1,s +1,1,(o+1))
B.5. Once Only
This represents the marginal probability that a random cus-
tomer purchased the focal site exactly once.
P(X
¡
=1)
=
o
r
(o+1)
r
!(u +1)!(u +|)
!(u)!(u +| +1)
2
í
1
_
r, |, u +| +1,
1
o+1
_
+
o
r
(o+1)
(1+r)
!(r +1)
!(r)
!(u +1)!(| +1)!(u +|)
!(u +| +2)!(u)!(|)
·
2
í
1
_
1 +r, | +1, u +| +2,
1
o+1
_
=
_
o
o+1
_
r
u
u +|
2
í
1
_
r, |, u +| +1,
1
o+1
_
+
_
o
o+1
_
r
r
o+1
u|
(u +|)(u +| +1)
·
2
í
1
_
1 +r, | +1, u +| +2,
1
o+1
_
.
B.6. 100% Loyal
This measure represents the expected percentage of cus-
tomers that only purchase the focal site in the category. This
Zheng et al.: From Business Intelligence to Competitive Intelligence
Information Systems Research 23(3, Part 1 of 2), pp. 698–720, ©2012 INFORMS 719
is equivalent to saying that the purchases to all other brands
are zero.
P((X
¡
=n (N =n, X
¡
>0))
=
P(X
¡
=n N =n, X
¡
>0)
P(X
¡
>0)
=
P(X
¡
=n N =n) ?P(X
¡
=n N =n, X
¡
=0)
P(X
¡
>0)
=
P(X
¡
=n N =n) ?0
P(X
¡
>0)
=

_
n=1
!(r +n)
n!
_
1
o+1
_
n
!(s)!(u
¡
+n)
!(s +n)!(u
¡
)

_
n=1
!(r +n)
n!
_
1
o+1
_
n
(1 ?
!(s)!(|
¡
+n)
!(s +n)!(|
¡
)
)
.
Duplication Between Two Brands X and Y: Customers
Who Bought X Also Bought Y
Duplication(X,Y) = P(Y >0 X>0)
=
P(Y >0,X>0)
P(X>0)
=
1?P(XY =0)
P(X>0)
=
1?P(X=0)?P(Y =0)+P(X+Y =0)
P(X>0)
.
The only dif?cult part in the above expression is P(X +
Y =0). This can be done by creating a composite (and ?c-
titious) X +Y brand. Because the Dirichlet world assumes
independence between X and Y, the parameters u of this
(X +Y) brand are u
X+Y
=u
X
+u
Y
and |
X+Y
=

5
i=1
u
i
?u
X+Y
.
Hence, from the equation of marginal distribution P(X =x),
we have
P(X +Y =0) =
_
o
o+1
_
r
|
X+Y
S
2
í
1
_
r, |
X+Y
+1, S +1,
1
o+1
_
.
Let s =

5
i=1
u
i
for the market of ?ve sites. From expressions
6.1 and 6.2, we derive duplication as follows:
Duplication(X, Y)
=
1 ?P(X =0) ?P(Y =0) +P(X +Y =0)
1 ?P(X =0)
=1 ?
P(Y =0) ?P(X +Y =0)
1 ?P(X =0)
=1 ?
_
o
o+1
_
r
((|
Y
,S)
2
í
1
(r, |
Y
+1, S +1, 1,(o+1))
?(|
X+Y
,S)
2
í
1
(r, |
X+Y
+1, S +1, 1,(o+1)))
· (1 ?(o,(o+1))
r
(|
X
,S)
2
í
1
(r, |
X
+1, S +1, 1,(o+1)))
?1
.
B.7. The VBA Code for Customizing the
2
í
1
Function in
Excel
Public Function GHF(a As Double, b As Double, c As
Double, z As Double) As Double
Dim i As Integer
Dim j As Integer
Dim temp As Double
GHF = 1
temp = 1
For j = 1 To 500
temp = temp * (a + j - 1) * (b + j - 1) * z /
((c + j - 1) * j)
GHF = GHF + temp
Next j
End Function
B.8. Algorithm LIND
We present algorithm LIND below to compute the market
share and SoW for a focal ?rm ¡.
Input: Firm j’s customer purchase data, other K ? 1
firms’ penetration.
Output: Market share and SoW estimation for all K
firms.
1 LL=0 /*initial log-likelihood is set to 0 */
2. Forall customers (i ? (1…N) {
3 Compute BB/sNBD probability p(X
i
=x
i
) according to
Equation (A5)
4 LL= LL + Ln(p)
5 }
6 Maximize likelihood according to the formulation
(7)
7 Obtain the K+2 parameters as the solution to (7)
8 Compute Market Share according to Equation (8)
9 Compute SoW according to Equation (9)
10 Output: MS and SoW
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