11 D X where X ~ unif{1, . . . ,0), θ e N. max(X1,-..,An) 2....

Question

Question

11 D X where X ~ unif{1, . . . ,0), θ e N. max(X1,-..,An) 2. Suppose that X1,... ,Xn Let T = X(n) =

Answer 1

Answer #1

here,

since, the sufficient and complete statistics $X(n)_{}$ has the lebesgue pdf ,

n $n\theta ^{-n}x^{n-1} I_{(0,\theta )}(x)$

$E(X_{n})=n\theta ^{-n}\int_{0}^{\infty }x^{n}dx=(n/n+1)\theta$

hence , an unbiased estimator of $\theta$ is $(n+1)X_{(n)}/n$

Which is UMVUE.

Suppose, that $\gamma =g(\theta )$ ,where g is differentiable function on $(0,\infty )$

An unbiased esrimator $h(X_{(n)})$ of $\gamma$ must satisfy ,

$\theta ^{n}g(\theta )=n\int_{0}^{\infty }h(x)x^{n-1}dx$ for all $\theta > 0$

differentiating both size of previous equations and apply result of different of an integral lead to

$n\theta ^{n-1}g(\theta )+\theta ^{n}g'(\theta )=nh(\theta )\theta ^{n-1}$

Hence , the UMVUE of $\gamma$ is

$h(X_{n})=g(X_{(n)})+n^{-1}X_{(n)}g'(X_{(n)})$

where, $\gamma$ = real valued parameter related to population .

In particular , if $\gamma =\theta$ ,then the UMVUE of $\theta$ is $(1+n^{-1})X_{(n)}$

Answer 2

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