I observe a sample of the form $=[_1,_2,…,_]=[(_1,_1),(_2,_2),…,(_,_)]$ where each $_$ is an arrival time and is the amount of money spent by the th customer.
I assume that for each amount $$, the arrival times of the filtered sample $=[_:_>]$ follow a homogeneous Poisson process with the intensity $_x=^{−}$.
How do I estimate $$ and $$? Is it correct that $1/=\frac{1}{}∑^_{=1}_$?
Edit: It is easy to estimate and by counting the number of events in each $S_x$ for several values of $x$ and fitting a linear regression to the logarithms of these numbers.
Assuming that those are all arrivals within some time window of known duration $T$, you can try the maximal likelihood estimate. Under your assumptions, the probability density at the observed event will be proportional to $A^NT^Ne^{-AT}k^Ne^{-k\sum_j x_j}$, so, indeed, $\frac 1k=\frac 1N\sum_j x_j$ is a good choice. By the same logic, $A=N/T$ (which is not at all surprising). Of course, these estimates are meaningful only if the model itself is not too far from the truth, but checking that is a completely different issue.