How do I establish shape and scale parameters?

Question

I am trying to forecast customer LTV using an exponential gamma distribution suggested in a Journal of Forecasting article (Empirical Comparison of New Product Trial Forecasting Models; authored by Bruce Hardie, Peter Fader, and Michael Wisniewski).

The specific formula that I am looking to solve is $P(t)=1-(α/(α+t))^r$, where $t$ is the period, $P(t)$ is the probability of a customer still being a customer at time $t$, $α$ is the scale parameter, and $r$ is the shape parameter.

I have some initial data on the per period attrition / retention of customers over 12 periods but I don't know how to use this data to calculate $α$ or $r$ to enable me to forecast future periods attrition/retention and ultimately LTV.

Can anybody explain how I can use the data I have to calculate $α$ and $r$?

Answer 1

I'll get you started. You can estimate the parameters using maximum likelihood. The likelihood function for your distribution is

$\begin{equation} LF = \prod_{i=1}^{N} 1-\frac{\alpha}{(\alpha+t)^{r}} = 1 - \frac{\alpha^{N}}{(\alpha + t)^{Nr}} \end{equation}$,

where $N$ is your sample size. The log likelihood function is therefore

$\begin{equation} \ln LF = -N\ln \alpha + rN \ln(\alpha + t). \end{equation}$

You want to maximize the log likelihood function with respect to the two parameters - take partial derivatives and set them equal to zero. What software are you using? I could show you how to do this in R.