Suppose an SRSWOR of size n has been drawn from a population labelled 1,2,..,N, where the population size N is unknown.
(a)Find the maximum likelihood estimator N′ of N.
(b)Find the pmf of N′
(c)Show that E(n+1nN′−1)=N
We know that the selection probability is \frac{1}{{N \choose n}}.So, the MLE is the nth order statistic or the maximum X_{(n)}.It's pmf is also, P[X_{(n)}=m]=\frac{{m \choose n}-{m-1 \choose n}}{{N \choose n}}
But I cannot show the last part.Please help
Answer
Let's rewrite the distribution of N'=X_{(n)} a bit more precisely:
P(N'=j)=\begin{cases}\dfrac{\binom{j-1}{n-1}}{\binom{N}{n}}&,\text{ if }j=n,n+1,\ldots,N \\ \\\quad 0&,\text{ otherwise }\end{cases}
Then we have
\begin{align} E(N')&=\sum_{j=n}^N jP(N'=j) \\&=\frac{n}{\binom{N}{n}}\sum_{j=n}^N \frac{j}{n}\binom{j-1}{n-1} \\&=\frac{n}{\binom{N}{n}}\sum_{j=n}^N \binom{j}{n} \end{align}
If we can use the fact that \sum_{j=n}^N P(N'=j)=1\implies \sum_{j=n}^N \binom{j-1}{n-1}=\binom{N}{n}
, it follows that E(N')=\frac{n}{\binom{N}{n}}\binom{N+1}{n+1}=\frac{n(N+1)}{n+1}
Or you can prove this identity directly. See this, this and this.
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