How do I establish shape and scale parameters? 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 $r$?
 A: 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.
A: Actually, for a nice convex objective function such as the one  that Wally provides, you could solve the optimization problem using the Excel "Solver" function. The default option is basically a Newton-Raphson search for a local maximum (or minimum, depending on the sign of the objective function). Note that Solver is an add-in -- it comes with Excel, but you have to install it. Excel help shows how.
Surprisingly powerful little gadgets, these general-purpose hill-climbing algorithms. 
