Question

For the likelihood

replace the binomial coefficients with the appropriate factorials. Find the log-likelihood, and then indicate which terms of it would not become zero if you took the derivative to find the MLE of n. (This should demonstrate that we really don't want to approach this problem in the usual way!)

Answer 1

$Consider~the~ratio:~R(n)=\frac{L_n}{L_{n-1}}=\frac{n-m}{n}\times \frac{n-t}{n-t-m&plus;r}\\ R(n)>1~according~as~(n-m)(n-t)>n(n-t-m&plus;r)~or,~n<\frac{mt}{r}\\ similarly~R(n)<1~according~as~n>\frac{mt}{r}\\ Hence~L_n~is~maximum~when~n\approx \frac{mt}{r}.\\ Thus~\hat{n}(r)=\left[ \frac{mt}{r}\right ],~where~[x]~denotes~the~largest~integer\leq x.$