Consider the family of probability mass functions given by
f(x;k) = 3(4^(k-x)) x = k + 1, k + 2,....
and indexed by parameter k E Z. For a random sample of size n, derive with justification:
a) the method of moments estimator for k.
b) the maximum likelihood estimator for k.
The MME isn't too bad, after a bit of algebra I get the sample mean - 4/3. The MLE is a bit more difficult. I don't think any calculus is useful here? In the likelihood function I need to maximise (nk -(sumXi)) which means that the biggest value of k will do this. But what is confusing me is that the parameter, k, is in the support of X. I dont know if this affects the summation or what. So pretty stuck at this point if anybody could help? Thanks