Friday, 4 November 2016

probability - Expected value of MLE of SRSWOR(n)



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+1nN1)=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.


No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...