Virtual Bass Model and the Left-Hand Data-Truncation Bias

in Product Diffusion Studies

Zhengrui Jiang, Frank M. Bass,∗ and Portia Isaacson Bass

School of Management, University of Texas at Dallas, Richardson, Texas 75083-0688, USA

∗ Corresponding author. Tel.: +1-972-883-2744; fax: +1-972-883-2799 Mail address: School of Management, The University of Texas at Dallas, P.O. Box 830688, Richardson, Texas 75083-0688, USA E-mail address: mzjb@utdallas.edu (F. M. Bass)

2

Virtual Bass Model and the Left-Hand Data-Truncation Bias

in Product Diffusion Studies

Abstract

The Bass Model (Bass, 1969) has been widely employed in forecasting early sales of new

products and technologies prior to product launch using the “guessing by analogy” method. The

p and q parameter estimates of the Bass Model for a (previously introduced) analogous product

have been used to predict the diffusion pattern of the new product. In many cases, however, the

historical data for early year sales are not available. Estimates based on left-hand truncated data

will be biased. We demonstrate the prevalence of this bias in previous studies and present a

method for dealing with it by exploiting previously unknown mathematical properties of the

Bass model. We call this extension of the Bass Model the Virtual Bass Model.

Keywords: Bass Model, Diffusion of Innovations, New Product Forecasting, Left-Hand Trunca-

tion Bias

3

1. Introduction

From the outset of the publication of the Bass Model paper there was the notion that the

model provided a framework for “guessing without data.” In that article (Bass, 1969, p. 215) one

finds the following quotation: “Long-range forecasting of new product sales is a guessing game,

at best. Some things, however, may be easier to guess than others. The theoretical framework

presented here provides a rationale for long-range forecasting.” Soon after the publication of the

Bass Model paper in 1969 guessing algorithms built around the framework of the Bass Model

were developed. One of these algorithms was developed and used extensively at Eastman Kodak

and was also used by several other companies including RCA, IBM, Sears, and AT&T. This pro-

cedure is described by Lawrence (1981) and by Lawrence and Lawton (1981). Putsis and Srini-

vasan (2000) have reviewed various algorithms that have been employed in guessing parameter

values when little or no data are available.

Over time the idea of “guessing by analogy” has emerged as a basis for forecasting sales of

new products and technologies prior to launch. Extensive data bases of Bass Model parameter

estimates of previously introduced products have been developed. In the guessing-by-analogy

method the p and q parameter estimates for a (previously introduced) analogous product are used

to forecast the diffusion pattern for the new product. Lilien, Rao and Kalish (1981) used this

method to forecast sales of a new prescription drug and Bayus (1993) used a segmentation

scheme to group “similar” products and generate forecasts for high-definition television

(HDTV). An article by Bass et al. (2001) provides a case history of the successful use of the

guessing-by-analogy method in forecasting the diffusion of satellite television prior to launch.

Although the guessing-by-analogy method has proven to be useful, there is a potentially se-

rious problem with this method in some cases. The problem arises because in many instances the

4

historical data for early year sales are not available. Estimates based on left-hand truncated data

series will be biased. For products where there is a long left tail of the plot of sales over time the

bias will be substantial. Other authors have taken note of the importance of the left-hand trunca-

tion bias problem. These include Dekimpe, Parker, and Sarvary (1998, 2000) and Kohli, Leh-

mann, and Pae (1999). Meta analysis studies by Sultan, Farley, and Lehmann (1990) and Vanho-

nacker, Lehmann, and Sultan (1990) have compiled Bass Model parameter estimates for several

product categories. In addition, Bass Model parameter estimates for several new products have

been published in papers by Kohli et al. (1999), Bayus (1992), and by Lilien, Rangaswamy, and

Van Den Bulte (2000). Unfortunately, for many of the product categories included in these stud-

ies sales data are not available from the time of product introduction; and, therefore, the parame-

ter estimates for these products suffer from left-hand truncation bias.

In this paper we will demonstrate the prevalence of the left-hand truncation bias and present

a method for correcting the bias by exploiting previously unknown mathematical properties of

the Bass Model. We call this extension of the Bass Model the Virtual Bass Model. We show that

for any Bass diffusion process with a given start time there is an equivalent process for any other

start time. The Virtual Bass Model is of both theoretical and practical significance. In particular,

we will show how to obtain the correct parameter estimates when the data are left-truncated. This

permits us to develop a correct set of parameter values for guessing by analogy.

In the next sections we develop the theoretical bases for the Virtual Bass Model.

2. The Bass Model is Symmetric about T*

We begin with a proof that if q > p, the Bass diffusion curve over the interval between 0 and

2T* is symmetric with respect to T = T*, where T* is the time of peak sales. We use this result in

the theoretical developments to follow.

5

We use the essential Bass Model equations in the proof and thus we repeat their develop-

ment below. The Bass Model (Bass, 1969) is derived from a simple premise that the conditional

likelihood of adoption of a randomly chosen consumer at time T, given that adoption has not yet

occurred, is a linear function of the number of previous adopters. The mathematical form of the

assumption is shown in the hazard function of Eq. (1):

( ) /[1 ( )] ( / ) ( ),f T F T p q m Y T− = + (1)

where m, p, and q are constant parameters representing the total number of potential adopters, the

coefficient of innovation, and the coefficient of imitation, respectively; F(T) and f(T) are the cu-

mulative and non-cumulative proportion of adopters at time T; Y(T) is the total number of adopt-

ers by time T. The diffusion rate S(T) at time T can be expressed as (2) or, equivalently, as (3)

and the total number of initial adoptions by time T is shown in (4):

2( ) ( ) ( ) ( / )[ ( )] ,S T pm q p Y T q m Y T= + − − (2)

2 ( )

( ) 2

( )( ) ,[( / ) 1]

p q T

p q T

m p q eS Tp q p e

− +

− +

+=

+ (3)

( )

( )0

(1 )( ) ( ) .( / ) 1

p q TT

p q T

m eY T S t dtq p e

− +

− +

−= =

+∫ (4)

The shape of the Bass diffusion curve varies according to two different conditions. If q ≤ p,

the diffusion rate decreases monotonically with time T and if q > p, S(T) increases monotonically

before peak time, T*, and drops monotonically afterwards. The peak time can be calculated

based on the following equation:

* [1/( )] ( / )T p q Ln q p= + . (5)

Another important property of S(T) is stated in Theorem 1 and shown in Fig. 1.

6

Fig. 1. Symmetry of Bass Diffusion Curve

Theorem 1: If p < q, the Bass diffusion curve segment over [0, 2T*] is symmetric with respect

to T = T*1.

Proof: To prove that the Bass diffusion curve segment over [0, 2T*] is symmetric, we only need

to show that for any 0 ≤ t ≤ T*, S(T*-t)=S(T*+t) always holds. From Eq.(3), we have:

2 ( )( * ) 2 ( )

( )( * ) 2 ( ) 2

( ) *

( ) ( )( * ) ,[( / ) 1] [ 1]

where / .

p q T t p q t

p q T t p q t

p q T

m p q e m p q k eS T tp q p e p e

k e p q

− + ± +

− + ± +

− +

+ + ⋅± = =

+ +

= =

∓

∓

Noting that for any real x (thus in particular, for x=(p+q)t),

2

2 2 2 2[ 1] [ 1] [ 1]

x x x x

x x x x

e e e ee e e e

− − +

− − +

⋅= =

+ + ⋅ + , which completes the proof.

3. Virtual Bass Model

We show in this section that by a novel way of constructing a Virtual Bass Model (or

VBM), the symmetry property for the Bass model holds in (-∞, +∞). A Virtual Bass Model is

represented by a hypothetical Bass diffusion curve that starts from time negative infinity (-∞),

which can be obtained by extending a known Bass diffusion curve to time -∞. Before this can be

1 Mahajan and Wind (1986) indicated that the Bass Model assumed symmetry and Mahajan, Muller, and Srivastava (1990) noticed that the Bass Model was symmetric between 0 and 2T*. However, no proof of the symmetry property was offered in either of these papers. We believe that this is the first published formal proof of symmetry of the Bass Model.

S(T)

0 T* 2T*

Symmetric

7

done, we first introduce two ways of transforming the Bass model: forward transformation and

backward transformation. We then derive the Virtual Bass Model Equations (or VBE), which

make the computation of transformed parameters simpler without going through the forward and

backward transformations. Finally, we show how parameters for a backward transformed diffu-

sion process can be calculated based on VBE.

3.1. Forward Transformation

An in-progress Bass diffusion process D0 at time T = τ can be transformed into a new proc-

ess DF for the remaining portion of the potential adopters. The transformation, which we term

forward transformation, is illustrated in Fig. 2. We denote the three parameters of D0 by m, p,

and q and those of DF by m', p', and q'. Since the start times of the two processes are different, we

denote the timing relative to the start time of D0 by T and that for DF by T’, with T’= (T-τ). In

addition, we use S(T) and S′(T′) to represent the diffusion rate functions for D0 and DF, and Y(T)

and Y′(T′) to represent the cumulative number of adoptions for D0 and DF, respectively.

Fig 2. Diffusion Process Forward Transformation

From the forward transformation construction, the following must hold for any T ≥ τ or T' ≥ 0:

( ) '( ')S T S T≡ (6)

'( ') ( ) ( ),Y T Y T Y τ≡ − (7)

where Y(τ) is the number of adopters of D0 at T = τ and can be obtained based on (4).

S(T)

0 (0') T' = (T-τ)

S'(T')

8

Assuming that the three parameters for D0 are known, we can derive the parameters for DF.

By letting T (and thus T’) in (7) go to positive infinity, we get the total market potential for DF:

' ( )m m Y τ= − . (8)

By expanding both sides of equation (6) based on (2), we have:

2 2( ) ( ) ( / )[ ( )] ' ' ( ' ') '( ') ( '/ ')[ '( ')]pm q p Y T q m Y T p m q p Y T q m Y T+ − − = + − − . (9)

After plugging (7) into (9) and by algebraic rearrangement, we obtain the following:

2

2 2

( ) ( ) ( / )[ ( )]' ' ( ' ') ( ) ( '/ ')[ ( )] [( ' ') 2( '/ ') ( )] ( ) ( '/ ')[ ( )] .

pm q p Y T q m Y Tp m q p Y q m Y q p q m Y Y T q m Y Tτ τ τ

+ − −

= − − − + − + − (10)

Since (10) must hold for every T, the following three equations must also hold:

/ '/ ',q m q m= (11) ( ' ') 2( '/ ') ( )q p q p q m Y− = − + τ , (12) 2' ' ( ' ') ( ) ( '/ ')[ ( )]pm p m q p Y q m Y= − − τ − τ . (13)

From (11) and (8), we obtain q′:

' ( / ) ( )q q q m Y τ= − . (14)

Based on (11), (12), and (14), we derive the solution for p’:

' ( / ) ( )p p q m Y τ= + (15)

Substituting (8), (14) and (15) into (13), we find that the equality in (13) holds. Therefore,

(8), (14) and (15) combined constitute the unique solution for forward transformation. The solu-

tion guarantees ( ) '( ')S T S T= for any T. Based on (14) and (15), we further conclude that

' 'p q p q+ = + is always true for forward transformation.

As a special case of forward transformation, if p < q and τ = 2T*, the parameters for the

forward transformed diffusion DFs can be derived as follows:

9

(2 *) (1 / ),(2 *) ( / ) ,

( / ) (2 *) ,( / ) (2 *) .

Fs

Fs

Fs

Y T m p qm m Y T p q mq q q m Y T pp p q m Y T q

= −

= − = = − = = − =

We name DFs the forward transformed symmetric diffusion process of D0 since the starting

times for D0 and DFs are of equal distance to T*.

3.2. Backward Transformation

The reverse of forward transformation is backward transformation. In the forward transfor-

mation just discussed, if we assume the known diffusion process is DF, D0 can be considered the

backward transformed process of DF. In backward transformation, we refer to the original proc-

ess again as D0 and the transformed process as DB. We still denote the three parameters of the

original process by m, p, q and those of the transformed process by m', p' and q'. Since D0 can be

considered the forward transformed process of DB, based on the solution of forward transforma-

tion, we express the parameters for D0 in terms of the parameters for DB, as shown in Table 1.

After some algebraic rearrangements, we obtain the parameters for the backward transformed

process, as shown in Table 2. Since this solution is derived from forward transformation, the

equality ( ) '( ')S T S T= still holds for any T.

Table 1. Parameters in Backward Transformation

Backward Transformed Process (DB) Original Process (D0) 'T τ−= 'TT

)'(' TY )(')'(')( τYTYTY −= 'm )('' τYmm −= 'q )(')'/'(' τYmqqq −= 'p )(')'/'(' τYmqpp +=

)()'(' TSTS =

10

Table 2. Parameters Rearranged for Backward Transformation

Original Diffusion (D) Transformed Process (B) ∗ T τ+= TT '

Y(T) )(')()'(' τYTYTY += m )('' τYmm += q )(')/(' τYmqqq += p )(')/(' τYmqpp −=

)'(')( TSTS = ∗ In case of p ≤ q, *)2()*2()(' TYTYY −+= ττ .

Unlike in forward transformation, '( )Y τ in backward transformation is not readily avail-

able. The solution, therefore, cannot be directly used unless p ≤ q. As illustrated in Fig. 3, in the

case of p ≤ q, based on the symmetric property of the Bass Model, we have:

'( ) (2 * ) (2 *).Y Y T Y Tτ τ= + − (16)

We call the solution shown in Fig. 3 for the p ≤ q case direct backward transformation.

Fig. 3. Direct Backward Transformation

In the case of p > q, the problem can be solved through indirect backward transformation.

With indirect backward transformation, we first find the backward transformed symmetric diffu-

sion process DBs, and then use DBs to obtain DB, the requested backward transformed process of

D0. Analogous to general backward transformation, backward symmetric transformation can be

considered the reverse of forward symmetric transformation discussed in the previous subsec-

T = 0 T′ = τ

S'

T′ T = 2T* T = 2T*+τ

11

tion; therefore, D0 is the forward transformed symmetric diffusion process of DBs. The DBs pa-

rameters are derived (see Table 3) by following a similar algebraic transformation (see Tables 1

and 2). In both forward and backward symmetric transformation, the values of p and q for the

original process are switched in the transformed symmetric process.

Table 3. Parameters Rearranged for Backward Transformation

Original Diffusion (D0) Transformed Diffusion (DBs) m mqpmBs )/(= q pqBs = p qpBs =

The distance between the start times of DBs and D0, denoted by θ, can be calculated with:

)./ln()]/(2[)/ln()]/(2[)*(2 qpqppqqpT BsBsBsBsBs +=+==θ

To obtain the backward transformed process DB from DBs, we first compare τ with θ and then

follow the following rule:

(1) If τ = θ, DB is the same as DBs.

(2) If τ < θ, DB can be obtained by forward transforming DBs by (θ-τ) units of time.

(3) If τ > θ, DB can be obtained by backward transforming DBs by (τ-θ) units of time.

θ

Start time of D0

Start time of DB1

Start time of DBs Start

time of DBs

Start time of DB2

τ2 > θ τ1 < θ

Start time of D0

θ

τ2-θ τ2 τ 1 θ-τ 1

Fig. 4. Indirect Backward Transformation (p ≤ q)

Fig. 4 illustrates the indirect transformation with two examples where DB1 is obtained by forward

transforming DBs by (θ-τ1) units of time since τ1 is less than θ, and DB2 is obtained by backward

12

transforming DBs by (τ2-θ) units of time since τ2 is greater than θ. The indirectly obtained back-

ward transformed processes still satisfy the condition that S(T)=S’(T’) for T’≥τ or T ≥ 0, and the

symmetric property still holds if p’<q’.

It can be verified that regardless of the condition in the original process, the following equa-

tion still holds in backward transformation: ( p' + q' ) = ( p + q ).

3.3. The Virtual Bass Model

By backward transforming a Bass diffusion curve to negative infinity, we can imagine that

there exists a virtual diffusion curve that spans (-∞, +∞), as shown in Fig. 5. We call this a Vir-

tual Bass Model (or VBM). Along a VBM curve, once a starting point is fixed, a specific Bass

diffusion process is known and its three parameters m, p, and q are fixed. The VBM Curve is

bell-shaped, unimodal, and symmetric in (-∞, +∞) with respect to T = T*.

Fig. 5. The Virtual Bass Model

Another important property of the VBM is stated in the following theorem:

Theorem 2. Any segment of a Bass diffusion curve uniquely determines a VBM curve.

Proof: In Appendix A.

Corollary 1. The solutions for the forward and backward transformations are unique solutions.

Proof: We have shown that a forward or backward solution satisfies S(T)=S’(T’). For any curve

segment to the right of the start time of the rightmost process, DF in forward transformation or

S(T)

0 T* T

Symmetric Start time flexible

13

D0 in backward transformation, the transformed process matches the curve segment perfectly.

From Theorem 2, this must be the unique solution.

From the solutions for forward and backward transformation we obtain parameters for a

Bass diffusion process starting from -∞, +∞, or T*.

(I) In order to get the parameters for the diffusion process that starts from -∞, we let τ go to

∞ in the backward transformation. If p is less than q, from Table 2, we have:

'( ) ( ) (2 *) (1 / ) / ,' '( ) / (1 / ) ,' [ '( )] / , in case of .' ( / ) '( ) 0,

Y Y Y T m m p q mp qm m Y m mp q p q mq q m Y m p q p qp p q m Y

∞ = ∞ − = − − =

= + ∞ = + = + = + ∞ = + ≤ = − ∞ =

(17)

If q is less than p in the known diffusion process, we can get the following parameters indirectly

based on (17) and the parameters for the backward transformed symmetric process Bs.

' (1 / ) (1 / ) ,' , in case of .' 0,

Bs Bs Bs

Bs Bs

m p q m p q mq p q p q p qp

= + = + = + = + > =

(18)

The expressions for the three parameters are the same in (17) and (18).2 For convenience, we

introduce two more parameters, M and U, and let (1 / )M p q m≡ + and ( )U p q≡ + . Since the

parameter m for any Bass diffusion process is interpreted as the total number of potential adopt-

ers and equals the integral of S(T) from time zero to infinity, and the Bass diffusion curve that

starts from -∞ is the same as the VBM curve, M can be interpreted as the total number of virtual

adopters of a VBM and equals the integral of the VBM curve from -∞ to +∞. From Theorem 2,

2 Strictly speaking, the parameter values in (17), (18) and (19) should be expressed as limits. For example, p’ = 0 should be lim ' 0p

τ →∞= , implying that as the assumed start time of a diffusion

process is extended to negative infinity, the value of p’ becomes infinitely close to zero; how-ever, this result does not imply that we can start from time negative infinity with p = 0, q = U > 0, and m = M > 0 and then derive other results by forward transformation.

14

we know that regardless of the relative position of the known Bass diffusion process in the corre-

sponding VBM, the same VBM is obtained by backward transforming the known process to -∞;

thus, depending on the start time of the known process, the original parameters are different, but

the VBM’s (1 / )M p q m≡ + is a constant. In our forward and backward transformations, the

sum of p and q does not change in any transformation; thus, the VBM’s U is also a constant.

(II) We now check the diffusion process that starts from the other end, positive infinity. By

letting τ go to ∞ in the forward transformation, we have:

( ) ,' ( ) 0,' [ ( )] / 0,' ( / ) ( ) .

Y mm m Yq q m Y mp p q m Y p q

∞ =

= − ∞ = = − ∞ = = + ∞ = +

(19)

We see that for the diffusion process that starts from positive infinity, m' equals zero, and the

value of p' and q' are switched from the negative infinity case.

(III) For a diffusion process that starts from the VBM’s peak time, we obtain the three pa-

rameters by forward transformation if p ≤ q or by backward transformation if p > q:

' (1 / ) / 2 / 2,' ' ( ) / 2 / 2.

m m p q Mq p p q U

= + = = = + =

(20)

Now we have a clearer picture of how the parameters change as the start time of a VBM

process moves from -∞ to +∞: m decreases from M to 0, q decreases from U to 0, and p in-

creases from zero to U. As the process moves, the sum of p and q is constant, which implies that

some value is continuously shifting from q to p. For a start time at T*, m = M/2, and p=q =U/2.

3.4. Virtual Bass Model Equations

We have shown that two of the constant parameters M and U of a VBM can be obtained by

(V1) and (V2), defined below. We now examine the other parameters in the model. To permit

15

time to vary according to start times, we denote the start time relative to the time of peak of a

Bass diffusion process by z. We set z = 0 for the process that starts from the peak time point of

the VBM curve; therefore, a Bass diffusion process that starts before the peak point has a nega-

tive z value and a process that starts after the peak point has a positive z value. For a given proc-

ess, if p and q are given, z can be calculated by (V3). From (V3) we see that z takes a negative

value if p < q. We denote the Bass diffusion process that starts from time z by D(z) and its diffu-

sion rate function by S(z)(T).3 For example, S(0)(T) represents the diffusion rate of the diffusion

process with z = 0. For such a diffusion process the peak diffusion rate, H, is a constant along

with M and U and is given by (0) (0) ( / 2)( / 2) / 4H S M U MU= = = .

We have shown for any Bass diffusion process that although the parameters m, p, and q will

change with the change in the start time, the parameters M, U, and H are invariant to the start

time. In the equations below the VBM parameters for a process that starts τ periods away from

the original start time are indicated and the new start time, z’, is indicated as z + τ, where τ is

negative for backward transformation and positive for forward transformation.

(1 / ) (V1)

M p q mU p q

= += +

z'U

(V2) (VBE)[1/( )]ln( / ) (V3)

z' = z + q' = U/(1 + e ); p' = U - q'; m' = Mz p q p q= +

τ; /(1+ p'/q'); (VBM)

The Virtual Bass Model Equations (VBE) summarizes the relationships among the parame-

ters. By examining (VBE), we conclude: the parameters (z’, q’, p’, and m’) for a transformed dif-

fusion process that starts τ periods away from the start time of the original process is obtained

from the four equations (VBE). The transformed parameters provide us with a unique D(z’).

3 Although the values of m, p and q in equations (V1)-(V3) are different with different z, we have chosen not to denote them by p(z), q(z) m(z) for notational simplicity and clarity.

16

Eq. (VBE) enables us to compute the parameters for a new process from the parameters of the

known process without going through the forward or backward transformations. (VBE) may be

used for both forward and backward transformation and for the cases where p≤q and when p>q.

3.5. An Example of Backward Transformation

To illustrate the application of VBE we use the case of color television. We use the 1963 -

1970 data series from Mahajan, Mason and Srinivasan (1986). The parameters estimated by the

Nonlinear Least Squares method (Srinivasan & Mason, 1986) are: m = 39658.62, p = 0.018466,

and q = 0.615863. The first sales of color television began in 1954 and thus the parameter esti-

mates based on a start time of 1963 suffer from a left-hand data truncation bias.

We use the VBM transformations from the equations VBE with τ = -9. We first calculate

the two constants based on (V1) and (V2):

(1 / ) 40847.74,M p q m= + = and U p q= + = .634329.

The parameter z for the process starting from 1963 is calculated from (V3) to be z = -5.52882;

thus, z' for the process starting from 1954 is: -14.52882. Using the other VBE equation we have:

'q = U/(1 + ez’U) =.634265935, 'p = (U – 'q ) = 6.3065E-5 and 'm = M/(1 + 'p / 'q ) = 40843.68.

The predicted years to the peak is 5.5 for the first series and 14.5 (T’* = τ + T*) for the series

beginning in 1954 and the predicted sales rate at the peak is 6,477 (in thousands) in both cases.

The VBM procedure for obtaining adjusted parameter values just illustrated is an indirect

approach in that the parameters m, p, and q are estimated from the original data series and then

transformed by the VBM equations. There does exist, however, a direct approach using nonlin-

ear least squares estimation when time is the exogenous variable that will produce equivalent pa-

rameter estimates for m’, p’, and q’. In this approach the initial value for time is set equal to τ

17

rather than 0. For example, using the color television data if the initial value for time is set equal

to 9 (representing 1963) and the next value is set equal to 10 (representing 1964) and so on and

nonlinear least squares regression is employed, the resulting parameter values will be the same as

the VBM estimates for m’, p’, and q’. Fig. 6 below provides a graphical comparison of the two

methods. In estimation the dashed and solid lines will converge and be the same for the estima-

tion period. The direct approach and the VBM approach will produce the same curve with the

starting time as the time of product launch.

Fig. 6. Direct Approach and VBM Approach with Left Truncated Data

As indicated in Fig. 6, when the VBM approach is used, the start time of the estimation is

T1 and the estimation takes place over the interval from T1 to T2. The parameter estimates from

this estimation are then transformed on the basis of information about the true introduction time,

T0. In the direct approach the estimation is also over the interval between T1 and T2, but the initial

value for time will be τ rather than 0 used as the initial value in the indirect approach. Both esti-

mations will have the same fit to the data and both will produce the same fitted curve over the

interval covered by the data. From Theorem 2 we know that any segment of a Bass diffusion

curve uniquely determines the backward and forward transformations. Thus both curves will

match between time T0 and T1; therefore, when nonlinear least squares is used with time as the

T0 - Product Launch Time

T2– Last Observation

Indirect Backward VBM

Direct Appr.

T1 - Data First Available

18

exogenous variable the direct approach and the VBM approach to estimation for the true intro-

duction time are equivalent. This conclusion has been confirmed by substantial experimentation.

In the paper we have provided a mathematical justification for the use of the (previously in-

tuition-based) direct approach to dealing with left-hand truncation bias. Our theoretical devel-

opments here provide the flexibility of adjusting for the left-hand truncation bias when the data

are not available but when the parameters have been estimated and reported. These parameter

estimates may be transformed by VBM to be consistent with different introduction times.

4. Correcting Left-Hand Truncation Bias for Use in Guessing by Analogy

We examine here the extent of the left-hand truncation bias issue for new products, services,

and technologies. To do this we determine the introduction time for a substantial number of new

products from historical sources and collect the available sales data for these products. We can

then measure the extent of the problem and we use the VBM methodology to correct the parame-

ter estimates where the data start time is different from the product introduction time. In this way

we will create a database of appropriate p’s and q’s for use in guessing by analogy.

4.1. The Introduction Time Issue and Data Collection

Considered in the context of guessing by analogy, the determination of the introduction time

to be used is not a simple matter. For a manager attempting to make a forecast for a new product

prior to product launch and seeking an appropriate analogy, judgments must be made about the

state of development of the new product, the state of available infrastructure support, the level of

marketing support, and other considerations that will influence whether or not the new product

sales will takeoff early or whether sales will initially grow slowly for a period of time before

takeoff. The manager will want the introduction time to be analogous to that provided by histori-

cal information for new products. In some cases the introduction time will be defined as the time

19

in which the product was first made available for sale, but under other circumstances the intro-

duction time may be judged to be the year in which significant sales begin. The matter is further

complicated by the fact that the commercialization date indicated by historical sources may refer

to the time when a business unit was created or when a patent was granted and will not be the

time of first product sales. In some cases historical sources will differ in introduction dates. We

have applied our judgment to resolve uncertain or conflicting information in deciding on the in-

troduction years used in this study. To accommodate possible differences in managerial interpre-

tation of the beginning year when guessing by analogy, we have used two different definitions of

the introduction year for the new products in our study. The first of these is our best judgment of

the introduction year as indicated by historical sources and the second is the year in which “sig-

nificant sales” begin. We define the start year of significant sales as the first year in which sales

equal or exceed 1.8% of m, the estimated Bass Model market potential in which case the start

year for the product will be equal to or later than the first year of the data series.

We consulted numerous historical sources for information about the introduction year for

the products in this study. These historical sources as well as sources used for the products sales

or subscriptions are listed in Appendix B. We excluded products for which we could not obtain

the introduction date. In total we have constructed a database of 39 products and services in five

different categories: home appliances (10 products), house-wares (7 products), consumer elec-

tronics (11 products), products bought by both business and consumers (7 products), and sub-

scription services (4 products). Products in the first three categories have been included in previ-

ous studies of comparative diffusion rates, while the last two categories have not.

20

4.2. Previous Studies

Previous studies have published parameter estimates for the Bass Model p and q parameters

for several product categories. Kohli et al. (1999) studied the relationship between incubation

time, defined as the time that elapses been essential completion of product development and ma-

jor market introduction (defined as the time when “noticeable” sales occur), and the diffusion

pattern for new products using data for 32 consumer durables in three categories: home appli-

ances (10 products), house-wares (11 products), and consumer electronics (11 products). Be-

cause of the differing definitions of introduction year, the introduction years will not all be the

same even if a product is in both studies. Their estimation procedure, however, is similar to ours

in that it took into account the left censoring of data and appears to have been the direct method

that we have previously discussed in section 3.5. Therefore, we expect the estimates of the p pa-

rameter from this study to be closer to the estimates that we develop for this parameter with

VBM than in studies that are based on the start of the data series.

Bayus (1992) explored whether or not diffusion rates have been accelerating over time. His

study included 30 consumer durables: home appliances (6 products), house-wares (16 products),

and consumer electronics (8 products). For most of the products Bayus analyzed the estimates

were made starting with the first year of data availability; thus, we expect the estimates of the p

parameter from his study to be substantially greater than our estimates using the VBM method.

Lilien et al. (2000) estimated the Bass Model parameters for products on the basis of mostly

penetration data (data collected by surveys on ownership) and mostly for U.S. diffusion. In their

estimates for short data series there were 54 products including: agricultural (4 products), medi-

cal equipment (4 products), production technology (8 products), electrical appliances and house-

wares (25 products), and consumer electronics (13 products). The estimates were made over pe-

21

riods for which the data were available and hence we expect the estimates of the p parameter to

be substantially larger than our estimates using the VBM method. Sultan et al. (1990) conducted

a meta-analysis study of 213 applications with many of the same products included in the differ-

ent studies. The estimates from these studies were based on available data as were the estimates

for 11 consumer durables in the original Bass (1969) study. We will also include here a compari-

son of the estimates for 8 products estimated with nonlinear least squares with the Srinivasan-

Mason method and reported by Mahajan et al. (1986). Included in this analysis were home appli-

ances (3 products), medical technology (2 products) and educational innovations (2 products).

4.3. Parameter Estimates for Introduction and Start Time of Data

Shown in Table 4 (below) are the nonlinear least squares estimates for the Bass Model p and

q parameters for the 39 products in our analysis (as indicated by p data and q data) over the time

periods for which data are available for each product (indicated under the column Data Used).4

Also shown are the VBM estimates based on the year each product was introduced (as indicated

by p VBM and q VBM). The actual years from introduction to peak sales are indicated by t* Act

(actual) and the predicted years to peak sales based on the VBM estimates of p’ and q’ and the

predicted years to peak sales based on the parameter estimates of p data and q data. In all but 6

cases the product introduction year is different from the first year of available data; thus, we con-

clude that the extent of the left-hand truncation bias is substantial. In most cases there is a large

difference between the actual years to peak sales and the predicted years to peak sales based on

parameter estimates derived from the data available. Thus reliance on the estimates of p’s and q’s

from previously published databases for guessing by analogy can be strongly questioned. The

prevalence of long left tails in the diffusion process is greater than would be suggested by previ-

4 For the data series for each product data were included that extend just slightly past the peak in sales in order to minimize the extent of repeat sales in the data.

22

ous studies and we believe that managers would be well advised to take this into account when

forecasting prior to product launch. The data in Table 4 should be helpful for this purpose.

Table 4. Actual Time from Introduction to Peak Sales and Comparison of Peak Time and Parameters of VBM Estimate and Estimate Using Data

Categories Year

Introduced t* Act t*VBM t*data p VBM q VBM p data q data Data UsedHome Appliances

Clothes Dryers 1930 25 26 6 2.13E-06 0.4717 0.02523 0.4464 1950-1958Clothes Washers 1910 20 18 6 0.00148 0.2899 0.04194 0.2494 1922-1932

Electric Range 1919 12 10 4 0.00188 0.5804 0.05616 0.5261 1925-1933Freezers 1929 24 24 7 4.8E-05 0.3803 0.02830 0.3520 1946-1954

Microwave Ovens 1955 32 32 17 4E-06 0.3535 0.00080 0.3527 1970-1989Power Lawnmowers 1926 34 33 11 6.7E-06 0.3231 0.00804 0.3151 1948-1961

Refrigerators 1913 25 27 20 0.00042 0.2305 0.00211 0.2288 1920-1940Room Air Conditioners 1928 29 30 12 3.5E-06 0.3869 0.00366 0.3833 1946-1961

Trash Compactors 1964 15 12 5 0.0072 0.3017 0.05310 0.2558 1971-1981Vacuum Cleaners 1908 22 19 5 0.00116 0.2981 0.06114 0.2382 1922-1932

HousewaresBlenders 1946 24 23 14 4.6E-07 0.6028 0.00010 0.6027 1955-1971Broilers 1937 18 17 8 8.2E-08 0.9451 0.00041 0.9447 1946-1956

Coffee Makers 1934 23 24 10 0.0002 0.2993 0.01284 0.2867 1948-1961Heating Pads 1918 12 12 5 0.0013 0.5127 0.04341 0.4706 1925-1933

Electric Blankets 1930 32 33 14 6.5E-05 0.2487 0.00714 0.2417 1949-1961Electric Shavers 1931 26 27 10 7.8E-05 0.3045 0.01317 0.2914 1948-1961

Steam Irons 1936 22 20 7 7.6E-05 0.4362 0.02111 0.4151 1949-1958Consumer Electronics

B&W TV 1939 17 17 10 0.00228 0.2894 0.01671 0.2749 1946-1961Camcorders 1973 19 18 6 0.00015 0.4468 0.02981 0.4172 1985-1992

Cassette Decks 1964 24 21 11 0.00204 0.2159 0.01679 0.2012 1974-1988CD Players 1983 16 15 0.00249 0.3118 1983-1998Color TV 1954 15 14 0.00009 0.6272 1954-1970

Digital Watches 1971 13 11 8 0.00678 0.3627 0.01981 0.3497 1974-1984Laser Disc Players 1980 14 13 8 0.00081 0.5015 0.00977 0.4925 1985-1998

Projection TV 1984 16 17 0.00569 0.2056 1984-2001Radio 1922 8 7 0.01216 0.4905 1922-1932

Record Players 1906 52 53 7 2.6E-11 0.4495 0.02340 0.4261 1952-1961VCR's 1975 11 11 6 0.00049 0.6475 0.01217 0.6358 1980-1989

Bus. & Consumer ProductsAnswering Machines 1960 30 30 8 7.7E-07 0.4476 0.01411 0.4335 1982-1993

ATM Machines 1971 14 13 0.00070 0.4935 1971-1986Copying Machines 1949 31 31 16 0.00106 0.1626 0.01346 0.1502 1965-1982

Fax Machines 1980 18 17 10 0.00214 0.2757 0.01430 0.2636 1987-2001Handheld Calculators 1967 14 13 10 0.00194 0.4116 0.00663 0.4069 1970-1982

PC Printers 1976 9 8 4 0.00116 0.7912 0.02669 0.7657 1980-1986Portable Dictation Machines 1955 25 26 16 0.00087 0.2139 0.00723 0.2075 1965-1981

Subscription ServicesAOL Change in Subs. 1989 7 7 3 0.00043 1.1408 0.03955 1.1017 1993-1997

Cable TV Change in Subs. 1948 27 26 21 0.00014 0.2916 0.00061 0.2912 1953-1976Cell Phones (Analog) Change 1983 13 16 15 0.00061 0.4137 0.00092 0.4134 1984-1995Satellite TV Change in Subs. 1994 5 5 0.05547 0.3246 1994-1998

Shown in Table 5 is the distribution of the number of years to peak sales for the 39 products

in our study. Only a single product reached peak sales within 5 years and only 4 of the 39 prod-

23

ucts had sales peak in 10 years or less. On average it took 20 years from the time of introduction

for the sales to reach peak sales.

Table 5. Distribution Time to Peak from introduction

Time to Peak From

IntroductionNo. of

ProductsPercent of Products

More than 35 1 2.5631-35 4 10.2626-30 4 10.2621-25 9 23.0816-20 7 17.9511-15 10 25.646-10 3 7.69

5 or Less 1 2.56

In Table 6 the averages for each of the measures in Table 4 are shown by each of the

product categories. These averages do not vary a great deal over the categories although con-

sumer electronics has the shortest mean time to peak sales at 19 years and home appliances has

the longest mean time to peak at 24 years. Overall, the mean time to peak sales is 20 years.

Table 6. Averages by Category of Actual Time from Introduction to Peak Sales and Comparison of Peak Time and Parameters of VBM Estimate and Estimate Using Data

t* Act t*VBM t*data p VBM q VBM p data q dataHome Appliances 23.8 23.1 9.3 0.0012 0.3616 0.0280 0.3347

Housewares 22.4 22.3 9.7 0.0002 0.4785 0.014027 0.4647Consumer Electronics 18.6 20.6 9.9 0.0018 0.4162 0.0135 0.4030

Bus. & Consumer Products 20.1 20.8 11.0 0.0012 0.3838 0.0119 0.3887Subscription Services 21.3 22.0 10.0 0.0011 0.4188 0.0169 0.3978

Overall Average 20.3 21.4 10.0 0.0011 0.4228 0.0181 0.4073

4.4. Comparisons with Previous Studies

In order to further show the extent of the left-hand truncation bias we compare the mean parame-

ter estimates for the p’s and q’s across the different studies. In Table 7 these averages are shown

with the ordering based on the reverse ranking of the estimates of the p’s. For the current study

based on the available data the average of the estimates of p is more than 16 times the average

24

based on the introduction year. Other than the study by Kohli et al. (1999) and the VBM esti-

mates for the current study, all of the other averages in Table 7 are based on the available data.

For many of the products included in previous studies the available data did not include the data

points between the product introduction time and the first observation; thus the estimates for

these products suffer from the left-hand truncation bias.

Table 7. Average p’s and q’s for Various Studies

Study Average p Average qNo. of

ProductsLilien et al. (2000) 0.0400 0.3980 54

Bayus (1992) 0.0373 0.3590 30Sultan et al. 0.0300 0.3800 213

Current, Data 0.0181 0.4073 39Bass (1969) 0.0163 0.3248 11

Mahajan et al. (1986) 0.0121 0.6319 7Kohli et al. (1999) 0.0076 0.2759 32

Current, VBM 0.0011 0.4228 33

4.5. Estimates Based on Starting Point When Significant Sales Begin

As indicated in Section 4.1, there are circumstances when a manager will conclude that the

appropriate analogy of the introduction year of his/her new product is when significant sales

started for a previously introduced product. To accommodate such judgments we have estimated

the p’s and q’s for the 39 products in our study with the starting time in the estimation as the first

year in which sales equaled or exceeded 1.8% of the estimated m. For 24 of the 39 products this

definition resulted in a starting year that was later than the start of the data series. Shown in Ta-

ble 8 for each product are: the year in which significant sales began, the actual number of years

to the peak of sales from the beginning of significant sales, t* Act., the predicted number of years

based on the parameter estimates. t*s, the estimates of the p’s and q’s, and the data interval used

for estimation. For those products where the start of significant sales was later than the start year

of the data series, the estimate of p increased, but the estimate of q changed very little.

25

Table 8. Statistics for Time to Peak with Starting Point when Significant Sales Begin

Categories

Year of Start of Significant Sales t* Act. t*s p est. q est. Data Used

Home Appliances Clothes Dryers 1950 7 6 0.0252 0.4464 1950-1958

Clothes Washers 1922 8 6 0.0419 0.2494 1922-1932Electric Range 1925 6 4 0.0562 0.5261 1925-1933

Freezers 1946 7 7 0.0283 0.3520 1946-1954Microwave Ovens 1978 10 9 0.0120 0.3631 1978-1989

Power Lawnmowers 1950 10 9 0.0149 0.3097 1950-1961Refrigerators 1929 9 16 0.0114 0.1513 1929-1940

Room Air Conditioners 1952 5 6 0.0377 0.3055 1952-1961Trash Compactors 1971 8 5 0.0531 0.2558 1971-1981Vacuum Cleaners 1922 8 5 0.0611 0.2382 1922-1932

Housewares Blenders 1964 6 5 0.0220 0.5944 1964-1971Broilers 1946 10 8 0.0004 0.9447 1946-1956

Coffee Makers 1951 10 8 0.0312 0.2353 1951-1961Heating Pads 1925 5 5 0.0434 0.4706 1925-1933

Electric Blankets 1955 7 14 0.0151 0.1531 1955-1961Electric Shavers 1948 9 10 0.0132 0.2914 1948-1961

Steam Irons 1949 0 7 0.0211 0.4151 1949-1958Consumer Electronics

B&W TV 1949 7 6 0.0444 0.1597 1949-1961Camcorders 1985 6 6 0.0298 0.4172 1985-1992

Cassette Decks 1975 13 10 0.0204 0.2011 1975-1988CD Players 1990 9 9 0.0202 0.2668 1990-1998Color TV 1963 6 5 0.0244 0.5977 1963-1970

Digital Watches 1976 8 6 0.0422 0.2460 1976-1984Laser Disc Players 1985 9 8 0.0098 0.4925 1985-1998

Projection TV 1991 9 8 0.0229 0.2877 1991-2001Radio 1924 6 5 0.0338 0.4491 1924-1932

Record Players 1953 5 5 0.0377 0.3896 1953-1961VCR's 1981 5 5 0.0229 0.6249 1981-1989

Bus. & Consumer Products Answering Machines 1983 7 7 0.0222 0.4194 1983-1993

ATM Machines 1978 7 6 0.0213 0.4729 1978-1986Copying Machines 1968 12 12 0.0205 0.1248 1968-1982

Fax Machines 1989 9 9 0.0246 0.2449 1989-2001Handheld Calculators 1974 8 6 0.0361 0.3132 1974-1982

PC Printers 1980 5 4 0.0267 0.7657 1980-1986Portable Dictation Machines 1970 10 11 0.0195 0.1872 1970-1981

Subscription ServicesAOL Change in Subs. 1994 2 2 0.1228 0.9420 1994-1997

Cable TV Change in Subs. 1966 9 8 0.0257 0.2454 1966-1976Cell Phones (Analog) Change 1992 4 3 0.0662 0.7974 1992-1995Satellite TV Change in Subs. 1994 5 5 0.0555 0.3246 1994-1998

The averages of the statistics in Table 8 are shown in Table 9 by category and overall. As

expected, the average number of years from the product introduction time to the sales peak of

26

20.3 years is much greater than the average years from substantial sales of 7.3 years. The time

from the start of significant sales to the peak was also a good bit lower than the 10.0 years when

the starting point of the data was used as the beginning year. Equally significant is that the aver-

age estimate of p increased by a factor of 28.8 compared with the estimate based on the year of

introduction and by a factor of 1.75 when compared with the estimate based on the first year of

the data series. The change in the average estimate of q was, in contrast, rather small.

Table 9. Averages Statistics with Starting Point when Significant Sales Begin

t*Act. t*s p est. q est.Home Appliances 7.8 7.3 0.0342 0.3198

Housewares 6.7 8.1 0.0209 0.4435Consumer Electronics 7.5 6.6 0.0280 0.3757

Bus. & Consumer Products 8.3 7.9 0.0244 0.3612Subscription Services 5.0 4.5 0.0675 0.5774

Overall Average 7.3 7.1 0.0317 0.3916

Taken as a whole, these results indicate that managerial interpretation of product introduc-

tion time can have an important impact on the question of the appropriate p’s and q’s to use in

guessing by analogy. Whatever the manager’s preference, the Virtual Bass Model provides a

method of transforming existing Bass Model parameters to conform with his/or her definition.

5. Conclusion

The use of guessing by analogy in conjunction with the Bass Model for forecasting the dif-

fusion of new products and technologies prior to product launch has brought to the fore the im-

portance of the left-hand truncation bias. We have demonstrated here that for most of the prod-

ucts for which there is historical information about the time of introduction the data series for

sales of these products do not extend back to the introduction date. In these cases the parameter

estimates of the Bass Model will be biased and will imply a shorter length of time for sales to

reach the peak than the actual length. Managers relying on databases of Bass Model parameter

27

estimates in which the estimates have not been adjusted for the left-hand truncation bias can be

badly mislead. We have shown that the Virtual Bass Model may be used to transform biased pa-

rameter estimates so that the truncation bias is removed.

In creating the Virtual Bass Model we have uncovered previously unknown mathematical

properties of the Bass Model. We have proven that for any Bass diffusion process with a given

start time there is an equivalent diffusion process for any other start time. We have also proved

that the transformation of parameters in moving from one start time to another is unique. The

Virtual Bass Model has both theoretical and practical significance. On the practical side it can be

used to transform parameter estimates based on left-hand truncated data into unbiased parameter

estimates. We also believe that through increased understanding of the properties of the Bass

Model the theoretical development we have presented here stands a good chance of finding other

practical applications of significance.

Acknowledgement

The authors would like to thank an anonymous reviewer for constructive suggestions for

simplifying the mathematics in our paper.

28

Appendix A Proof of Theorem 2

Suppose we have a Bass diffusion curve segment between time A and B. To prove that the

curve segment uniquely determines a VBM curve, we only need to show that for any diffusion

start time z < A, there is only one unique diffusion curve that begins from z and matches the

known curve segment completely.

S(T)

S'(T)

z

T2 T1

A B Figure A1. If two VBM Curves Both Fit a Bass Diffusion Curve Segment …

Suppose that we have found such a diffusion curve S(T). If S(T) is not the unique solution,

there must exist another S'(T) that also starts from z and matches the same curve segment. Then

the two curves S(T) and S'(T) must overlap between A and B. The scenario is shown in Fig. A1.

We denote the three parameters for S(T) by m, p, q, and those for S'(T) by I', p', and q'. Since

S(T) and S'(T) are different, at least one of their parameters must be different. We denote the dis-

tance between z and A by T1 and the distance between z and B by T2.

Equations (A4), (A5), and (A6) hold since S(T) and S'(T) overlap between time A and B.

(τ satisfies 2 10 ( )T Tτ≤ ≤ − throughout the proof.)

1 1( ) '( )S T S T= (A4)

1 1( ) '( )S T S Tτ τ+ = + (A5)

1 1 1 1( ) ( ) '( ) '( ) ( )Y T Y T Y T Y T Iτ τ τ+ − = + − = (A6)

I(τ) in (A6) represents the number of new adoptions from time A to (A + τ). From (A6), we have:

29

1 12 2

1 1 1 12 2

1 1 1 12

1

( ) ( )

( ) ( ) ( / )[ ( )] ( ( ) ( ) ( / )[ ( )] )

( )[ ( ) ( )] ( / )[ ( ) ( )] ( ) ( ) ( / )[ ( )] )

[( ) 2( / ) ( )] ( ) ( / )[ ( )] .

S T S T

pm q p Y T q m Y T pm q p Y T q m Y T

q p Y T I q m Y T I q p Y T q m Y T

q p q m Y T I q m I

τ

τ τ

τ τ

τ τ

+ −

= + − + − + − + − −

= − + − + − − +

= − − −

(A7)

Similarly, we have (A8):

2

1 1 1'( ) '( ) [( ' ') 2( '/ ') '( )] ( ) ( '/ ')[ ( )] .S T S T q p q m Y T I q m Iτ τ τ+ − = − − − (A8)

Since )(')(')()( 1111 TSTSTSTS −+=−+ ττ holds for every τ, both (A9) and (A10) must also hold:

/ '/ ' q m q m= (A9)

1 1( ) 2( / ) ( ) ( ' ') 2( '/ ') '( ). q p q m Y T q p q m Y T− − = − − (A10)

From (A9) and (A10), we get (A11) and (A12).

)()/(2 )(')/(2)()''( 11 TYmqTYmqpqpq −+−=− (A11)

)()/(2

)()''()(' 11 TYmq

pqpqTY +−−−

= (A12)

Plug (A11) first and then (A12) into S'(T1), and after algebraic transformation, we get (A13).

2 2

21 1 1

( ' ') ( )'( ) ' ' ( ) ( ) ( / )[ ( )] .4( / )

q p q pS T p m q p Y T q m Y Tq m

− − −= + − − + (A13)

From , 1 1'( ) ( )S T S T= we get:

2 2

2 21 1 1 1

( ' ') ( )' ' ( ) ( ) ( / )[ ( )] ( ) ( ) ( / )[ ( )]4( / )

q p q pp m q p Y T q m Y T pm q p Y T q m Y Tq m

− − −+ − − + = + − −

pmmq

pqpqmp =−−−

+⇒)/(4

)()''(''22

)/(4)()''()'/'(4'' 22 mqpmpqpqmqmp =−−−+⇒

2 24 ' ' ( ' ') 4 ( )

' 'p q q p pq q p

q p q p⇒ + − = + −⇒ + = +

(A14)

From (A14) and )(')( 11 ττ +=+ TSTS , we have the following:

30

2 21 1

2 21 1

2 21 1

( ) [ ( )( )] '( ' ') [ ( ' ')( )][( / ) [ ( )( )] 1] ' [( '/ ') [ ( ' ')( )] 1]

'[( / ) [ ( )( )] 1] '[( '/ ') [ ( ' ')( )] 1]

( '/ ') [ ( )(

m p q Exp p q T m p q Exp p q Tp q p Exp p q T p q p Exp p q T

m mp q p Exp p q T p q p Exp p q Tq p Exp p q

τ ττ τ

τ τ

+ − + + + − + +=

− + + + − + + +

⇒ =− + + + − + + +

− +⇒ 1/ 21

1

)] 1 '( ) is a constant.( / ) [ ( )( )] 1 '( '/ ') ( / )

T m pq p Exp p q T mpq p q p

ττ

+ +=

− + + +⇒ =

(A15)

(A9), (A14) and (A15) lead to the (A16).

.' ,' ,' qqppmm === (A16)

(A16) contradicts the original assumption that at least one of the parameters is different.

Therefore S(T) has to be the unique solution. The theorem is therefore proved.

Appendix B Historical Sources and Data Sources

Historical Sources:

Bayus, B. L. (1992). Have diffusion rates been accelerating over time? Marketing Letters

3(3), 215-226.

Bud, R., Niziol, S., Boon, T, & Nahum A. (2000). Inventing the modern world: technology

since 1750 New York: Dorling Kindersley.

Bunch, B. & Hellemans, A. (1993). The timetables of technology: a chronology of the most

important people and events in the history of technology New York: Simon & Schuster.

Desmond, K. (1987). The Harwin chronology of inventions, innovations, discoveries: from

pre-historic to present day London: Constable.

Dummer, G.W.A. (1983). Electronic inventions and discoveries: electronics from its earliest

beginnings to the present New York: Pergamon Press.

DuVall, N. (1988). Domestic technology: a chronology of developments Boston: GK Hall.

31

Fisher, D.E. & Fisher, M.J. (1996). Tube: the invention of television Washington, D.C.:

COUNTERPOINT.

Giscard D’Estaing, V. & Young, M (1993). Inventions and discoveries: what’s happened,

what’s coming, what’s that? Ann Arbor, MI: Books on Demand.

Giscard D’Estaing, V. (1985). The world almanac book of inventions Mahwah, NJ: The

World Almanac.

Kohli, R., Lehmann, D. R. & Pae, J. (1999). Extent and impact of incubation time in new

product diffusion. Journal of Product Innovation Management, 16, 134-144.

Lifshey, E. (1973). The housewares story: a history of the American housewares industry

Chicago: National Housewares Manufacturers Association.

McNeil, I. (Ed.) (1990). An encyclopedia of the history of technology London: Routledge.

Merchandising Week (1972). Special 50-Year Summary Issue, February 28, 1972.

Smith, A. (Ed.) (1998). Television: an international history New York: Oxford University

Press.

Travers, B & Muhr, J. (Eds.) (1994). World of invention Detroit: Gale Research.

Van Dulken, S. (2000). Inventing the 20th century: 100 inventions that shaped the world

New York: New York University Press.

Williams, I. T. (2000). A history of invention: from stone axes to silicon chips New York:

Check-mark Books.

Yu, J. (2002). Factors affecting diffusion patterns for new products: a hierarchical Bayesian

mixture model, Ph.D. Dissertation, The University of Texas at Dallas.

Data Sources:

Dealerscope

32

Dealerscope Marketplace

Economic Almanac

Electrical Merchandising

Electrical Merchandising Week

Electronic Market Data Book (2003). Arlington, VA: Consumer Electronics Assn.

Historical Statistics of the United States

In-Stat

Kagan Media Services

Merchandising Week

Statistical Abstract of the U.S.

Strategis

The Information Technology Industry Databook

Yankee Group

33

References

Bass, F. M. (1969). A new product growth model for consumer durables. Management Science,

15(5), 215-227.

Bass, F. M., Gordon, K., Ferguson, T. L., & Githens, M. L. (2001). DIRECTV: Forecasting dif-

fusion of a new technology prior to product launch. Interfaces, 31(3), S82-S93.

Bayus, B. L. (1992). Have diffusion rates been accelerating over time? Marketing Letters 3(3),

215-226.

Bayus, B. L. (1993). High-definition television: assessing demand forecasts for a next generation

consumer durable. Management Science, 39(11), 1569-1585.

Dekimpe, M. G., Parker, P. M., & Sarvary, M. (2000). Multimarket and global diffusion. In V.

Mahajan, E. Muller, & Y. Wind (Eds.), New product diffusion models (pp. 49-73). Bos-

ton: Kluwer.

Kohli, R., Lehmann, D. R. & Pae, J. (1999). Extent and impact of incubation time in new product

diffusion. Journal of Product Innovation Management, 16, 134-144.

Lawrence, K. D. (1981). New product forecasting diffusion modeling with scenario sensitivity

analysis. In R. Leone (ed.), Proceedings, Market Measurement and Analysis Conference

(pp. 194-199). Austin: University of Texas at Austin.

Lawrence, K. D. & Lawton, W. H. (1981). Application of diffusion models: some empirical re-

sults. In Y. Wind, V. Mahajan and R. N. Cardozo, (eds.), New-Product Forecasting:

Models and Applications (pp. 529-541). Lexington, MA: Lexington Books.

Lilien, G., Rangaswamy, A., & Van Den Bulte, C. (2000). Diffusion models: managerial applica-

tions and software. In V. Mahajan, E. Muller, & Y. Wind (Eds.), New product diffusion

models (pp. 295-311). Boston: Kluwer.

34

Lilien, G. L, Rao, A. G. & Kalish, S. (1981). Bayesian estimation and control of detailing effort

in a repeat purchase diffusion environment. Management Science, 27(5), 493-506.

Mahajan, V., Mason, C. H., & Srinivasan, V. (1986). An evaluation of estimation procedures for

new product diffusion models. In V. Mahajan and Y. Wind (Eds.), Innovation diffusion

models of new product acceptance (pp. 203-232). Cambridge, MA: Ballinger.

Mahajan, V., Muller, E., & Srivastava, R. (1990). Determination of adopter categories by using

innovation diffusion models. Journal of Marketing Research, 27(1), 37-50.

Mahajan, V., & Wind, Y. (1986). Innovation diffusion models of new product acceptance: a re-

examination. In V. Mahajan and Y. Wind (Eds.), Innovation diffusion models of new

product acceptance (pp. 3-25). Cambridge, MA: Ballinger Publishing

Putsis, W. P., & Srinivasan, V. (2000). Estimation techniques for micro diffusion models. In V.

Mahajan, E. Muller, & Y. Wind (Eds.), New product diffusion models (pp. 263-291).

Boston: Kluwer.

Srinivasan, V., & Mason, C. H. (1986). Nonlinear least squares estimation of new-product diffu-

sion models. Marketing Science, 5(2), 169-178.

Sultan F., Farley, J. U., & Lehmann, D. R. (1990). A Meta-analysis of applications of diffusion

models. Journal of Marketing Research, 27, 70-77.

Vanhonacker, W. R., Lehmann, D. R., & Sultan, F. (1990). Combining related and sparse data in

linear regression models. Journal of Business and Economic Statistics, 8, 327–335.