Question

Refer to exercise 9.109{

Suppose that n integers are drawn at random and with replacement from the integers 1, 2, ..., N. That is, each sampled integer has probability of 1/N of taking on any of the values 1, 2, ..., N, and the sampled values are independent.

(a) Find the method-of-moments estimator of N.

Ni

$p(x) =\begin{Bmatrix} \frac{1}{N}, x= 1,2,...,n\\ 0, elsewhere \end{Bmatrix}$

$E(X)=\sum_{i=1}^{N}xf(x) = \frac{1}{N}\sum_{i=1}^{N}x = \frac{1}{N}\frac{N(N+1)}{2} = \frac{N+1}{2}$

$\bar{X} = \frac{N+1}{2} => \widehat{N_{1}}= 2\bar{X}-1$

(b) Find $E( \widehat{N_{1}}) and V(\widehat{N_{1}})$

$E( \widehat{N_{1}}) = E(2\bar{X}-1)=2E(\bar{X})-1=...=N \\ E(X_{i}^{2}) = .\sum_{i=1}^{N}x_{i}^{2}f(x) = ... = \frac{(N+1)(2N+1)}{6} \\ V(X_{i})= E(X_{i}^{2}) - E(X_{i})^{2} = \frac{N^{2}-1}{12}$

$V\widehat{N_{1}} = V(2\bar{X}-1) = 4V(\bar{X}) = 4V(\frac{1}{n}\sum_{i=1}^{n}x_{i}) = 4\cdot \frac{1}{n^{2}}\sum_{i=1}^{n}V(x_{i})=4\cdot \frac{1}{n^{2}}\cdot n(\frac{N^{2}-1}{12}) = \frac{4}{n}\cdot \frac{N^{2}-1}{12} = \frac{N^{2}-1}{3n}$

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(a) Find the MLE, $\widehat{N_{2}}$ of N.

Answer 1

$The~likelihood~function~is\\ L(N;x_1,x_2,...,x_n)=\left\{\begin{matrix} \frac{1}{N^n} &1\leq \max(x_1,x_2,...,x_n)\leq N \\ 0 & otherwise~~~~~~~~~~~~~~~~~~~~~~~ \end{matrix}\right.$

Clearly the MLE of N is given by

$\hat{N}(X_1,X_2,...,X_n)=\max(X_1,X_2,...,X_n),\\ for ~if~we~take~any~\hat{\alpha}<\hat{N}~as~the~MLE,~then~P_{\hat{\alpha}} (x_1,x_2,...,x_n)=0;~and~if~we~take~any ~\hat{\beta}>\hat{N}~as~the~MLE,~then ~P_{\hat{\beta}}(x_1,x_2,...,x_n)=\frac{1}{\hat{\beta}^n}<\frac{1}{\hat{N}^n}=P_{\hat{N}(x_1,x_2,...,x_n).}$