Deriving an Expression for P (X(t) = x) Under the Pareto/NBD Model

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1 Deriving an Expression for P Xt x Under the Pareto/NBD Model Peter S. Fader Bruce G. S. Hardie September 26 Introduction The Pareto/NBD Schmittlein et al. 987, hereafter SMC is a model for customer-base analysis in a noncontractual setting. One result presented in SMC is an expression for P Xt x, where the random variable Xt denotes the number of transactions observed in the time interval,t]. This note derives an alternative expression for this quantity, one that is simpler to evaluate. In Section 2 we review the assumptions underlying the Pareto/NBD model. In Section 3, we derive an expression for P Xt x conditional on the unobserved latent characteristics λ and μ; we remove this conditioning in Section 4. For the sake of completeness, SMC s derivation is replicated in the Appendix. 2 Model Assumptions The Pareto/NBD model is based on the following assumptions: i. Customers go through two stages in their lifetime with a specific firm: they are alive for some period of time, then become permanently inactive. ii. While alive, the number of transactions made by a customer follows a Poisson process with transaction rate λ. Denoting the number of transactions in the time interval,t]bythe random variable Xt, it follows that P Xt x λ, alive at t λtx e λt, x,, 2,... iii. A customer s unobserved lifetime of length ω after which he is viewed as being inactive is exponentially distributed with dropout rate μ: fω μ μe μω. iv. Heterogeneity in transaction rates across customers follows a gamma distribution with shape parameter r and scale parameter : gλ r, r λ r e λ. Γr c 26 Peter S. Fader and Bruce G. S. Hardie. Document source: <

2 v. Heterogeneity in dropout rates across customers follows a gamma distribution with shape parameter s and scale parameter. gμ s, s μ s e μ Γs. 2 vi. The transaction rate λ and the dropout rate μ vary independently across customers. 3 P Xt x Conditional on λ and μ Suppose we know an individual s unobserved latent characteristics λ and μ. Assuming the customer was alive at time, there are two ways x purchases could have occurred in the interval,t]: i. The individual remained alive through the whole interval; this occurs with probability e μt. The probability of the individual making x purchases, given she was alive during the whole interval, is λt x e λt /. Therefore, the probability of remaining alive through the interval,t] and making x purchases is λt x e λ+μt. ii. The individual died at some point ω < t and made x purchases in the interval,ω]. The probability of this occurring is λω x e λω μe μω dω λ x ω x e λ+μω μ dω λ x μ t λ + μ x+ ω x e λ+μω λ + μ x+ dω which, noting that the integrand is an Erlang-x + pdf, λ x [ μ λ + μ x+ e λ+μt i [ λ + μt ] i Combining these two scenarios gives us the following expression for the probability of observing x purchases in the interval,t], conditional on λ and μ: P Xt x λ, μ λtx e λ+μt λ x [ μ + λ + μ x+ e λ+μt i 4 Removing the Conditioning on λ and μ i! [ λ + μt ] i i! ]. ]. 3 In reality, we never know an individual s latent characteristics; we therefore remove the conditioning on λ and μ by taking the expectation of 3 over the distributions of Λ and M: P Xt x r,, s, Substituting 3 in 4 give us P Xt x r,, s, A + A 2 P Xt x λ, μgλ r, gμ s, dλ dμ. 4 i t i i! A 3 5 2

3 where A A 2 A 3 i. Solving 6 is trivial: A λt x e λ+μt gλ r, gμ s, dλ dμ 6 λ x μ gλ r, gμ s, dλ dμ λ + μ x+ 7 λ x μe λ+μt gλ r, gμ s, dλ dμ 8 λ + μ x i+ Γr + x r t x s 9 Γr + t + t + t ii. To solve 7, consider the transformation Y M/Λ + M and Z Λ+M. Using the transformation technique Casella and Berger 22, Section 4.3, pp ; Mood et al. 974, Section 6.2, p. 24ff, it follows that the oint distribution of Y and Z is gy, z,, r, s r s ΓrΓs ys y r z r+s e z y. Noting that the inverse of this transformation is λ yz and μ yz, it follows that A 2 r s ΓrΓs r s ΓrΓs r s Γr + s ΓrΓs r s Br, s y y x gy, z,, r, s dy r+s y s y r+x z r+s e z y dy y s y r+x { z r+s e z y dy y s y r+x y r+s dy y s y r+x [ y ] r+s dy which, recalling Euler s integral for the Gaussian hypergeometric function, r s Br + x, s + r+s 2F r + s, s +;r + s + x +; Br, s. Looking closely at, we see that the argument of the Gaussian hypergeometric function,, is guaranteed to be bounded between and when since >, thus ensuring convergence of the series representation of the function. However, when <we can be faced with the situation where <, in which case the series is divergent. Applying the linear transformation Abramowitz and Stegun 972, equation F a, b; c; z z a 2F a, c b; c; z z, 2 gives us A 2 r s Br + x, s + r+s 2F r + s, r + x; r + s + x +; Br, s. 3 We see that the argument of the above Gaussian hypergeometric function is bounded between and when. We therefore present and 3 as solutions to 7: we use when and 3 when. 2 F a, b; c; z Bb,c b tb t c b zt a dt, c>b. 3

4 iii. To solve 8, we also make use of the transformation Y M/Λ + M and Z Λ+M. Given, it follows that A 3 r s ΓrΓs r s ΓrΓs r s Γr + s + i ΓrΓs r s Br, s y s y r+x z r+s+i e z+t y dy y s y r+x { Γr + s + i Γr + s z r+s+i e z+t y dy y s y r+x + t y r+s+i dy + t r+s+i y s y r+x [ +t y ] r+s+i dy which, recalling Euler s integral for the Gaussian hypergeometric function, r s Br + x, s + + t r+s+i Br, s Γr + s + i Γr + s 2 F r + s + i, s +;r + s + x +; +t. 4 Noting that the argument of the Gaussian hypergeometric function is only guaranteed to be bounded between and when, we apply the linear transformation 2, which gives us r s Br + x, s + A 3 + t r+s+i Br, s Γr + s + i Γr + s. 5 2 F r + s + i, r + x; r + s + x +; +t The argument of the above Gaussian hypergeometric function is bounded between and when. We therefore present 4 and 5 as solutions to 8, using 4 when and 5 when. Substituting 9,, 3, 4, and 5 in 5, and noting that Γr + s + i Γr + s i! ibr + s, i, yields the following expression for the distribution of the number of transactions in the interval,t] for a randomly-chosen individual under the Pareto/NBD model: where and P Xt x r,, s, B B 2 Γr + x r t Γr + t + r s Br + x, s + Br, s x + t + t { B 2F r + s, s +;r + s + x +; r+s 2F r + s, r + x; r + s + x +; r+s i s t i ibr + s, i B 2 if if 2F r + s + i, s +;r + s + x +; +t + t r+s+i if 2F r + s + i, r + x; r + s + x +; +t + t r+s+i if. 6 4

5 We note that this expression requires x+2 evaluations of the Gaussian hypergeometric function; in contrast, SMC s expression see the attached appendix requires 2x + evaluations of the Gaussian hypergeometric function. The equivalence of 6 and A, A3, A4 is not immediately obvious. Purely from a logical perspective, they must be equivalent. Furthermore, equivalence is observed in numerical investigations. However, we have yet to demonstrate direct equivalence of these two expressions for P Xt x r,, s,. Stay tuned. References Abramowitz, Milton and Irene A. Stegun eds. 972, Handbook of Mathematical Functions, New York: Dover Publications. Casella, George, and Roger L. Berger 22, Statistical Inference, 2nd edition, Pacific Grove, CA: Duxbury. Mood, Alexander M., Franklin A. Graybill, and Duane C. Boes 974, Introduction to the Theory of Statistics, 3rd edition, New York: McGraw-Hill Publishing Company. Schmittlein, David C., Donald G. Morrison, and Richard Colombo 987, Counting Your Customers: Who Are They and What Will They Do Next? Management Science, 33 January, 24. 5

6 Appendix: SMC s Derivation of P Xt x r,, s, SMC derive their expression for P Xt x by first integrating over λ and μ and then removing the conditioning on ω, which is the reverse of the approach used in Sections 3 and 4 above. This gives us Γr + x r t x s P Xt x r,, s, where C Γr + t + t {{ + NBD P Xtx + t {{ P Ω>t x Γr + x r ω Γr + ω + ω {{ NBD P Xωx Γr + x Γr + t Making the change of variable y + ω, C +t y x y r+x y + s+ dy r t + t x + t ω x + ω r+x + ω s+ dω. which, recalling the binomial theorem, 2 +t x y x y r+x y + s+ dy s s+ + ω {{ fω dω s Γr + x + s r s C Γr x +t y r+ y + s+ dy { x y r+ y + s+ dy y r+ y + s+ dy +t letting z /y in the first integral which implies dy z 2 and z + t/y in the second integral which implies dy + tz 2, x { r+s+ + t r+s+ z r+s+ z s+ z r+s+ +t z s+ which, recalling Euler s integral for the Gaussian hypergeometric function, { x 2F s +,r+ s + ; r + s + +; r + s + r+s+ 2 F s +,r+ s + ; r + s + +; +t + t r+s+. A3 A A2 2 x + y r r k x k y r k for integer r. 6

7 We note that the arguments of the above Gaussian hypergeometric functions are only guaranteed to be bounded between and when. We therefore revisit A2, applying the change of variable y + ω: C +t y x y s+ y + r+x dy which, recalling the binomial theorem, +t x y x y s+ y + r+x dy x +t y s++ x y + r+x dy { x y s++ x y + r+x dy y s++ x y + r+x dy +t letting z /y in the first integral which implies dy z 2 and z + t/y in the second integral which implies dy + tz 2, x { r+s+ + t r+s+ z r+s+ z r+x z r+s+ +t z r+x which, recalling Euler s integral for the Gaussian hypergeometric function, { x 2 F r + x, r + s + ; r + s + +; r + s + r+s+ 2 F r + x, r + s + ; r + s + +; +t + t r+s+. A4 We note that A3 and A4 each require 2x + evaluations of the Gaussian hypergeometric function. 7