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A Consumer Purchasing Model with Learning and Departure Behaviour

Author(s): C. Wu and H.-L. ChenSource: The Journal of the Operational Research Society, Vol. 51, No. 5 (May, 2000), pp. 583-591Published by: Palgrave Macmillan Journalson behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/254189.

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584 Journal

f he

Operational

esearch

ociety

ol.

1,

No.

a

departure.

Customers

may

also become unsatisfied

due

to

changes

in their

habits,

age,

income, etc.

Many studies still assume that

interpurchase

imes

follow

an

exponential distribution.

When

the

customer's

buying

incidence is

more regular than a Poisson

process,

using

NBD-type models could cause bias.

Chatfield and

Good-

hardt14

provided some

empirical evidence of

purchasing

behaviour

which

followed a

more

regular

distribution

han

a

Poisson

process.

They argued

that an

Erlang-2

distribu-

tion is more

appropriate

or

interpurchase

imes,

yielding

a

'condensed'

negative

binomial model

(CNBD).

Some

studies, such as

those

by Lawrence15

and

Gupta,16

upport

an

Erlang-2distribution.

However, special

care is

necessary

since

customer

purchase behaviour can

be more

regular.

For

example, there are

some consumer

goods

for

which

habitual

usage

behaviour can

be

easily formed,

that

would

cause their

interpurchase

time

to be

more

regular

than

Erlang-2.

In

order

to take

into account

such

behaviour,

the

interpurchase ime

distribution

should

be

extended

to

Erlang-c,

c

>o

1.

The

Erlang distribution s an

important

generalisation

of

the

exponential

distribution,

which is

flexible and can

effectively capture

the

spirit

of

regular

or

irregular

nterpurchase

imes.

One alternativemodel

was

developed

by

Jeulandet al.8

Under

the

assumption of

independence

between

the

zero-

order choice

process

and

the

Erlang purchase

timing

process,

the

output of

the

developed

model

includes

analy-

tical

expressionsfor

market

hareand

penetration.

However,

a

disadvantage

of

this model

is that it

lacks informationon

learning and

departures.

Another

alternativemodel

was

developed

by Schmittlein

et

al,6

which examined consumer purchase

patterns

by

considering

'death

rates.' In

their

study,

for an

individual

customer, besides the

Poisson

purchases,

the

exponential

lifetime

and

gamma

heterogeneity or

death

rateswere

also

taken

into

account. The

socalled

NBD/Pareto

model

they

developed

allows

the

company

to

determine he

numberof

'active' and

'inactive'

customers over

time.

They

proved

that

the model

provided

answersto the

following

questions

that

are often

asked by

marketing

practitioners:

1.

How

many retail

customersdoes the

firm now

have?

2.

How has

the

customer

base grown over

the past

year?

3. Whichindividualson the list are most likelyto represent

active

customers?

Inactive

customers?

4.

What

level

of

transactions

hould be

expected

next

year

by

those

on

the

list,

both

individuallyand

collectively?

However,

the

assumptionof an

exponential

lifetime of

each

individual

and

then

gamma

heterogeneity

across the

population

s a

restriction.In

practiceit is

hardto

evaluate

the

lifetime

distribution

f each

individualsince

we observe

only one

'death'

occurrence of

an active

customer who

becomes an

inactive

customer.

Also, the

change of

an individual's

purchase

behaviour

often arises

from

past

experience.

A customer's

departure

is not

merely

determined

by

the duration of

his

usage. Major determi-

nants

are past

experienceand

purchasebehaviour.

Usually,

the

last purchase

occasion or

use

experience will

signifi-

cantly

affect

whether

a customer will

make another

purchase or not.

Therefore

the

learning

behaviour

should

be

taken

into

account and should be

involved in the

models.

The

objective

of this

article,

therefore,is to

extend the

NBD-type

models, while

incorporatingconsumer's

learn-

ing

and

departure behaviour

and

Erlang interpurchase

times, and their

unobserved

heterogeneity.By these

exten-

sions, the

model

allows us to

determine he

probability

hat

a

customer with

a

given patternof

purchasing

behaviour

still

remains,

or

has

departed,

at

any

time

after

k

?

1

purchases are

made. The

model also can

be used

to

determine

how

manypurchasesare

made

by

an

experienced

or an

inexperiencedcustomer

during

a

given

period.

Our

model

promotes

the influence

of

learningon consumers'

purchasing

behaviour,

which

would be more

useful in the

market

since

customers'

purchase

decisions

significantly

depend on

past

experience.

Using the consumer

purchase

data

for

tea,

the

empiricalresults

substantially

ndicate

that

learningand

departure

ehavioursarethe

important

actors

while

predicting he

purchase

frequencies

of

inexperienced

customers.

In

the

following

sections;

Firstly

the

interpurchase

ime

model

with

gamma

heterogeneity

and

learning

and

depar-

ture

behaviour

s

specified; Secondly,

an

integrated

model

is

developed;Thirdly, he estimation

procedureand

empiri-

cal

results are

reported, and

finally,

we

conclude with

a

discussion of

implications

of our

findings

and

suggestions

for

future research.

The

model

Imagine that

you

arethe

marketingmanager

of a

company

selling product

Alpha. You

have a list

of

customers who

have

ever

done

business with

the

company in

the past,

as

well as

informationon

the

frequency

and

timing

of

each

customer's

transactions.You

are

interested n

understand-

ing the

individual's

purchasingbehaviour

across

the popu-

lation of

customers

and

predicting

the

growth of

the

company's customer

base,

which

would be

helpful in

planning

marketing

trategies.To

address

hese

managerial

issues, let us startwith

a brief

review

of

the

interpurchase

time

models.

Interpurchase

ime

model

To

generalise

the

NBD

model, we

first

suppose the

custo-

mer's

interpurchase

imes

follow

an Erlang

distribution

(c,

u):

ucXc1

e-uX

f(XIc,

u)

=

(cXc)

0

o

O.

(7)

F(y)

The

coefficient

of variation

of u is

1/,

so thehigherthe y

value, the more

homogeneous arethe

customers.

If

the gamma

distribution

exactly governs an

individual

customer's

purchasepattern,

we can add up all

the custo-

mers to find

the

average

probability of

purchase for a

random

population

member."7'9"6With

gamma hetero-

geneity

of

u,

and

normalisingthe

repurchaseprobabilities,

(4) becomes:

p

Eck

+

-j-

I

( t

\CkJ

V

P-j=1

\.

Yl

I

t

+

o/\t

+

for

(8)

This

gives the

probabilityof

the number

of purchases

made

in

time

period

(0, t] while a

customer is

still active. The

formula is

the

NBD model

of

Ehrenberg' for c

=

1

and

Pk-I

=

1-

The

combined

model we call

the integrated

model. To

compute the

distribution of

purchases by

the inactive

customer,(5) and (6) still hold underconditionsof normal-

isationof

repurchase

probabilityand

gamma

heterogeneity.

Empirical

Analysis

The

model

is fully

determinedwhen

the

following types of

parameters

are

known: the

repeat buying

probability,

Pk,

k

>

1,

the

order of

the

Erlang

timing process,

c, and

two

parameters

which describe

the heterogeneity

over

the

population

of the

purchase

rate,a shape

parameter, , and

a

scale parameter,

c.

Our

approach s

illustratedwith

consumer

purchase

data

for tea

provided

by

Ten

Ren Tea

Co.,

Ltd.,

the

largest

company

in

the Oriental

tea market.

The datasetcovers

a

panelof

customersat one selected store.

There are

901

new

customers

made

purchase

during

July

1994 to

May

1996

(96

weeks).

The data

contains records

of the

complete

purchase

history

of these

customers

(sex, age, purchasing

duration,

etc.).

To

examine the

efficiency

of the

integrated

model

andexplorethe

importance

of

learning

and

departure

factors, we divided

the observation

duration into

two

periods of

48 weeks

each

(verify

the

model

twice).

In

the

first

period,

there are

363 new

customers and 538

new

customers in

the second

period.Fromthe

summary

of

the

customers'

characteristics,

or both

periods,

frequent

buyers

are older

than

light

buyers

and

have

higher

incomes.

Frequent

buyers who are

older

may

have

more

purchases

due to their

higher ncome

(less

financial

constrain),

and the

fact that

they are more

likely

to form a

habit of

drinking

tea.

In

addition,males aremore

likely

to

be

frequent

buyers

than

females.

While

there is not

enough

information

to

reliably

esti-

mate

Pk

on

an

individualevel

therewill

generally

be

enough

to

estimate across

customers.

The mean

probabilityPk

can

be

obtained

by

the

following

analytical

expression:

(no.

f

people

who

buy less than

k

times

anddepart

J

Pk

-

no. of total

population

no.

of

people

who

buy less

than

k-I

times

and

departJ

Due to

the

one-time

trials

may have

different

purchase

behaviour with

the

customers who

have

made

purchase

more than

once,

to

find

the

leaming

model,

we

do

not

include

Pl.

Excluding

Pl,

the

regression

analysis

shows

the

linear

learning

model, Pk

=

a,

Pk-l

+

ao,

has

a

signi-

ficant

power

of

predictionwith a

high

R2

for

each

period

(the

R2

for the first

and

second

periods

are

0.9354

and

0.9432,

respectively).

We

therefore

employ

the

learning

model to

estimate the

series of

Pk.

From the

calculation

we

found

that

ao

is

0.201

and

0.210

for

periods

1

and 2,

respectively;

a,

is

0.772

and

0.780 for

periods

1

and 2,

respectively.

Learning

behaviour

does

exist

and it

causes the

high

a,

value.

In

order

to

determine

the

regularity of

a

customer's

purchase,

he

individual's

distributions f

the

time

intervals

between

consecutive

purchases

are

examined. In

the data

available for

this

study,

when a

purchase

was made,

it was

recorded;

therefore, we

could

directly

calculate

the mean

and

standard

deviation

of

an

individual's

interpurchase

times. For

any

random

variable,the

coefficient of

variation

(CV)

is

determined

by

the

ratio

of

the

standard

deviation to

the

mean.

The

order

of

Erlang

interpurchase

time,

c,

happens to

be

the

inverse

of

square CV,

that

is,

c

=

I/CV2.

It

is

clear

that

there

is

some

heterogeneityof

8/11/2019 254189

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CWu

ndH-L hen-A

onsumer

urchasing

odel ith

earning

nd

eparture

ehaviour

87

Table 1

Actual vs

predicted

number

of

purchases

k

equals no. Observed

Integrated

of purchases

frequency

model

NBD CNBD

Pareto/NBD

Period 1

k=

1 198 197.79

67.93

50.90

86.83

c=4

k=2

40

29.09 59.46

57.32

60.95

y

=

.13365

k=3

11 6.13 49.54

53.80

45.38

oc 0.1968 k=4 15 9.34 40.24 46.25 34.79

ocO

0.201

k=5

9

11.14

32.18

37.75

27.12

oc,

=

0.772

k=6

9

11.84

25.46

29.78

21.36

k=7 1

11.75 19.99

22.92

16.95

k=8

8 10.16

15.60

17.33

13.52

k=9

2

10.26

12.13 12.91

10.82

k= 10 6

9.22

9.39

9.52

8.69

k=

11 8

8.15

7.25

6.95

6.99

k=

12 9

7.11

5.59

5.04

5.63

k=

13 3

6.15

4.29

3.62

4.55

k=

14

4

5.28

3.29

2.60 3.68

k=

15 2

4.50 2.52

1.85

2.97

k=

16

1

3.82

1.93

1.31

2.41

k=

17 8

3.23

1.47

0.93

1.95

k=

18

4

2.73

1.12

0.65 1.58

k= 19 6 2.30 0.86 0.46 1.28

k=20+

19

9.85

2.06

0.74

4.50

Thiel's

U

-

0.0526

0.4424

0.5050

0.3718

Period 2 k=

1

315

314.99

110.81

88.98 139.81

c=4

k=2

59

52.59

96.48

95.67

96.87

y=

1.2305 k=3

12

14.19

78.73

85.63

70.70

oc=0.3201 k=4

18

14.71

62.09

70.18

52.90

o%o=0.78

k=5

10

14.48

47.95

54.61

40.15

oc,

=

0.210

k=6

19

13.80

36.50

41.06

30.75

k=7

15

12.86

27.50

30.12

23.68

k=8

2

11.79

20.56

21.70

18.32

k=9

2

10.68

15.28

15.42

14.21

k=

10

8

9.59

11.30

10.83

11.05

k= 11 9 8.54 8.31 7.54 8.61

k=

12

13

7.57

6.10 5.21

6.72

k=

13

7

6.68

4.47

3.57

5.25

k= 14

8

5.88

3.26

2.43

4.10

k=

15

0

5.16

2.38

1.66

3.21

k= 16

3

4.51

1.73

1.12

2.51

k=

17

4

3.94 1.25

0.75

1.97

k=

18

0

3.44

0.91

0.51

1.54

k=

19

9

2.99

0.66

0.34

1.21

k=20+

25

16.97 1.23

0.45

3.47

Thiel's U

-

0.0307

0.4429

0.4939

0.3735

Note:

1.

Predictednumber

of customers

s based on

predicted

probabilityof

numberof

customers.

2.

Traditional

models can't

capture ight

buyers.

According to

the formula

of Theil's U,

it causes

high

value of

Theil's

U.

the

population

with

respectto

the

order

of

the

interpurchase

time

process.

Therefore,

Jeuland

et a18

suggested that

a first

step

would

be

to

assume the

population is

homogeneous

with

respect

to

the

order,and

heterogeneous

with

respect to

the

second

parameter f

the

Erlang

model, that

is,

u.

Using

this

idea,

each

customer's

c

is

then

averaged

to

yield an

overall

population

c.

16

We

obtained

he

mode

of

the

order

of

the

Erlang

process

and it

approximates

our

in

both

periods.

Erlang-4

shows

the

customer's

nterpurchase

imes

are

much

more

regular

han

suggested

by the

exponential

distribution.

The

final

step

is to

allow the

parameter

u to

vary

over

customers.

The

CV

of

u is

1/,

so

y

is

a

measurementof

the

heterogeneityacross

the

populationof

customers.

From

our

data

base,

we find

that

y

is

1.3365

and

1.2305 for

periods

1

and

2,

respectively;

the

other

parameterof

the

gamma

distribution,

c,

is

0.1968

and

0.3201

for

periods

1

and

2,

respectively.

We

composed

a

computer

program

in

C-language

to

calculate

the

probability of

customers'

purchases

on

the

integrated

model,

Pareto/NBD,

CNBD,

and

NBD

models.

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588

Journalf he

Operational

esearch

ociety

ol.

1,

No.

Table

2 Actual vs

predicted

numberof

purchases

k equals no. Observed

Integrated

of purchases frequency model

NBD CNBD

Pareto/NBD

Period

1

k=2 40 33.89 15.47 12.29 24.01

c=4

k=3 11 5.92

15.48 13.86

19.14

7=.3365

k=4 15 9.01 14.84

14.32 15.78

x=0.1968 k=5 9 10.75 13.88 14.08 13.27

io

=

0.201

k=6

9 11.42 12.74 13.86

11.30

a,

=

0.772

k=7

1

11.34 11.54 12.39

9.73

k=8

8 10.76 10.35 11.28

8.43

k=9 2 9.90 9.22 10.12

7.34

k= 10

6

8.90 8.16

8.98 6.42

k=

11

8 7.86

7.18 7.90 5.63

k=

12 9

6.86

6.30

6.89 4.95

k= 13 3 5.93

5.50 5.98

4.36

k=

14 4

5.09 4.79

5.16 3.85

k= 15 2 4.34

4.16 4.42 3.41

k= 16 1

3.69

3.60 3.78 2.02

k=

17 8 3.12

3.11 3.22 2.67

k= 18

4

2.63 2.69 2.74 2.37

k= 19

6 2.22 2.31 2.31 2.11

k=20+

19 9.50 11.68 9.90 15.35

Thiel's U

-

0.1967 0.3314

0.3727

0.2472

Period2

k=2 59

52.59 23.56 18.03

31.54

c=4

k=3 12

14.19

22.76 20.10 26.01

y=1.2305

k=4

18

14.70 21.24 20.54

21.89

x=0.3201 k=5 10

14.48 19.42 19.94

18.64

xo

=0.78 k=6 19

13.80 17.50 18.72

16.00

a,

=0.210

k=7 15

12.86 15.61

17.14

13.80

k= 8 2

11.79

13.81 15.42

11.95

k= 9

2

10.68

12.15 13.68

10.37

k=10

8

9.58

10.64 12.00

9.03

k= 11

9 8.54

9.27

10.43 7.87

k=

12

13

7.57 8.06 8.99

6.87

k= 13 7 6.68 6.98 7.71 6.00

k= 14

8

5.87 6.03

6.57

5.25

k=15

0

5.16 5.20 5.58

4.60

k=

16 3

4.51

4.48 4.70

4.03

k=

17 4

3.94

3.85 3.96

3.53

k= 18 0

3.43 3.30

3.33 3.09

k=

19

9

2.99 2.83

2.78 2.71

k=20+

25

16.69 13.89

10.99 18.11

Thiel's U

-

0.1433 0.3368

0.3777 0.2853

Note:

Predictednumberof customers s

based on predictedprobabilityof

numberof customers.

The predictedresults are reportedin

Tables

1

and

2

and

Figures

1

and

2.

The

predictive

quality of

the

model is

assessed

using Theil's U

inequality coefficient. The

U

ranges

from

0

to 1,

where

smaller

values

indicate

better

predictions.

We

see from

Tables

1

and

2

thatthe

integrated

model

performs

better than

the

Pareto/NBD, CNBD

and

NBD

models,

as

indicated

by the

relatively

small U

forboth

periods.

ComparingTable

1

with

Table 2,

when

the

one-time

purchasers

are

included,

the

NBD-type

models

would

have

much

higher

Theil's

U, but

the

integrated

model

does not.

This

is one

of the

advantagesof

the

integrated

model,

which

is not

provided

by the

NBD-type

models.

In case

that

the

first

repurchase

probability s much

smaller than

the

other

repurchase

probabilities, he

integrated

model still

provides

the

functionto

evaluatethe

consumer

purchase

behaviour

well.

Summaryof

substantial

indings and

managerial

implications

According

to

Tables

1

and 2,

the

NBD/Pareto

model

performsbetter

than the NBD

and CNBD

models

in

both

periods, which

implies

that

when a firm

observes

a new

customer

and

wishes to

predict his

purchase

pattem, the

8/11/2019 254189

8/10

C

Wu

nd

H-L

Chew-A

onsumerurchasingodel

ith

earning

nd epartureehaviour 89

45--- - Actual

---- El-----

ntegrated

Model

40-

-_---

NBD

-A-CNBD

35-l

........x

Pareto/NBD

Y30-

, 25-

020-

z ~~~~~~~t

10

8 zn-

A3*>-

2

3

4

5 6 7 8

9 10

11

12

13 14 15 16

17 18 19

20

Number f

Purchases

Figure

1

Actual s

predicted

umber

f

purchases

Period

).

70-

-40

Actual

60-

----E3-

Integrated

Model

-~-e-- NBD

-

A-

CNBD

50-i

--.

.

Pareto.NBD

?

40-

0 0

-

30-

z

20

10

2 3

4 5

6

7

8

9

10 11

12

13

14 15 16 17

18

19

20

Number of

Purchases

Figure

2

Actual

vs

predicted

umber

f

purchases

Period

).

'defection

effect'

should be

utilised.

Furthernore,

apart

from the

'defection

effect',

the

learning

effect should

also

be

utilised.

This

is

because

before

the

customer

gets

experience,

there

always

exists

learning

effect.

This

is

evidenced

from the

result

that

the

integrated model

performsmuch betterthanthe NBD/Pareto model.

It

is

interesting

o

find that when the

one-time

purchasers

are

included,

the R2

of

the

linear

learning model is

much

smaller than

the

R2

of

the

case

that

one-time

purchasers

are

excluded.

This has

another

important

managerial

mplica-

tion.

The

one-time

trial

really

exhibits

different

purchase

behaviour rom

the

customers

who

repurchase t

least

once.

Whilethere

s

lower

sensitivity

o

the

marketingmix,

once a

customer

repurchases,

here

is a

higher

possiblility to

follow

the

regular

leaming

model and

purchase

again.

Also,

according

to

the

learning

model, the

probability is

that

the

repurchasingwill

go

steady

once the

customer

gets

experience. This model

explains

why

state

dependence

wears out

as

a

customer

gains

experiences

and

the

tradi-

tional

NBD-type models

assume

thatthe

experiencedconsu-

mers

are in a

steady state.

These

findings

have

substantial

managerial

mplications.

For

example,

a retailer

may

induce

high

inertial

households

to

buy

its

store

brands in

various

categories

such as

using

samplingprograms

on

'bundles'

of

store brands.As

long

as

the

store brandsare

in the

consumers'choice

set,

thereis

a

high

probability

hat the household

keeps

purchasing

n the

future.

Next,

we would like

to discuss

the

regularity

assumption

of

the

interpurchaseimes.

According

to Tables

1

and

2,

we

observe that

Erlang-2

does not

really

improve

the fit.

Interestingly,

if we

assume

the

interpurchase

imes are

exponentially

distributed,

nstead of

Erlang-4

distribution,

in

most of

the

cases,

the Theil's U

is almost the

same

as

before.

For

example,

when the one-time

purchasers

are

included,the Theil's U

is

0.0498 and

0.0308 for

periods

1

and

2,

respectively; when

the

one-time

purchasers

are

excluded,the Theil's

U

is 0.2190

and 0.1424

for

periods

I

and 2,

respectively.

Due to this

finding,

if the

majority

of a

firm's

customers

just purchase

once, to

predict

the firm's

sales,

we can infer that the

regularityassumption

of the

interpurchasetimes

may

not

be

important.

Also,

it

is

difficult to

predict

these

customers'

purchase behaviours.

However, n some

cases,

the

regularityassumption

may

be

substantial,

uch as in

period

1,

when the

one-time

purcha-

sers are

excluded,

the

difference of the

Theil's

U

would be

significant.To avoid the

prediction

bias,

a

model

shouldbe

flexible

and take

most of

the

substantial

actors

ntoaccount.

With

these

concerns,

the

developed model would

be more

useful in

most

of

the

cases.

Conclusions

We

have

attempted

o

provide a

more general

framework o

analyse

the

customer's

interpurchase ime

by considering

the

regularity of

interpurchase

ime, adding learning and

the

departurefactors

and

including the

heterogeneity of

customers. We

provided these

extensions by replacing

some

NBD

assumptions.

Firstly, as regular

purchases

exist,

we

adopted

Erlang interpurchase

times in

our

model.

We

found that

the

customer's

interpurchase ime

can

be

extended

to Erlang-c

and still can

be easy to

estimate.

Secondly,

considerationof

the customer's

earning

and

departure

s

shown to

be

necessarywhen we treat

the

buying

population

as

havingeasy exit

and entry.

Combining

these

elements with

gamma

heterogeneity

provides

many good

tools and is

useful to the

marketing

manager

in

solving

managerial

issues. Our

model can be

used

to

analyse

the company's

annual sales

and customer

base for

a

product.

For

example, the model can

monitor he

ratio

of

customersretained

and customers

leaving,

specify

the

value and

satisfaction of

core

customersand

measure

8/11/2019 254189

9/10

590

Journalf heOperational

esearchociety

ol.

1,

No.

business

performance.By not eliminating ight buyers, as is

often

done in similar studies, the integratedmodel we have

developed

achieves more precise

results,

which

can be seen

by using Theil's

U.

Further

research in

this area could try to take into

account

the interrelationship

between repeat buying prob-

ability and

interpurchasetime, and at the same time

incorporating

marketing

variables.

Appendix

The derviation

of (5)

and (6) can

be expressed

diagrama-

tically

below:

P(T,

t,

Tk-

I

t and

repurchase

k

-

1

times) is

the

probability

hat a customer

who arrives at time 0,

is still active

at time t

and makes

k

purchases, and

P(Tk

t)

=1

(c

-j)

(8)

t

P(T1

t)

=

{P(T2

>

t

-yIT,

=y)P(T1

=y)dy

0

(9)

t

c

~~(u(t

y))C-I

FeU(t

)

E

_(c_j) _]

Y:ud;y

;(t)c1u

(10)

F(c)

j=E (2c-j)

and

inductively,

P(Tk-l

t)

-

X

e-u(t-Y)

E

[(

(

y)]J

Jo

j=i

(c

-

)

Uc(k-l)yc(k-1)-I eu

y

F(c(k

-

1))

Y

j=i

(ck-j)

k=3

,***(1

Therefore,

we are able to

obtain

the

closed form

of

P(Tk-

I

t,

Tk

>

OPfk-1,

Vk.

The

probability

hat a customer s not active

at time

t

but

has

made k

purchases

would be obtained

recursively.

The

probability

hat

the customer

purchases

wice and abandons

the

product,

s

P(T2

< t

repurchases nce andthendeparts)

=

P(T2

?

t) *

p

Iq2

=[P(T1

t)(I

-ql)-P(T1