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$X_1, X_2, X_3, \ldots{}, X_n$, i.i.d., follow $\mathcal{N}(0, \theta^2)$, $\theta > 0$.

What are the UMVUEs (Uniform Minimum Variance Unbiased Estimators) of $\theta$ as well as $\Phi(\tfrac{1}{\theta})$, where $\Phi$ is the cumulative distribution function of $\mathcal{N}(0,1)$?

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  • $\begingroup$ I have added the 'homework' tag. $\endgroup$ – ocram Feb 14 '12 at 8:12
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    $\begingroup$ Please explain why you cannot solve the homework and which level of help you require. $\endgroup$ – Xi'an Feb 14 '12 at 11:06
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Background:

  1. Find a sufficient statistic
  2. Show it is complete
  3. Find an arbitrary unbiased estimator
  4. Apply the Rao-Blackwell and the Lehmann-Scheffé theorems

Hint:

$$ \Phi(1/\theta) = \mathbb{P}(Z\le 1/\theta) = \mathbb{P}(\theta Z\le 1) = \mathbb{P}(X \le 1)\quad Z\sim\mathcal{N}(0,1),X\sim\mathcal{N}(0,\theta^2) $$

More hints:

  1. Write down the joint density of $(X_1,\ldots,X_n,S^2)$
  2. Derive the conditional distribution of $(X_1,\ldots,X_n)$ given $S^2$ as a uniform distribution on a sphere
  3. Compute $\mathbb{E}[\mathbb{I}(X_1\le 1)|S^2]$ as the surface of a truncated sphere.
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  • $\begingroup$ sigma(Xi^2) is sufficient and complete, because it is exponential family with full rank. But what is p(X1<=1|sigma(Xi^2)), I know that sigma(Xi^2) follows theta^2(chi square n). $\endgroup$ – br69 Feb 14 '12 at 20:37
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    $\begingroup$ Do you mean $\mathbb{E}[\mathbb{I}(X_1\le 1)|S^2]=\mathbb{P}(X_1\le 1|S^2)$? This is indeed the UMVUE. $\endgroup$ – Xi'an Feb 14 '12 at 21:15
  • $\begingroup$ but i want the explicit form of UMVUE. i.e given data values I can compute the value of S(x1,x2,.. xn), the UMVUE. $\endgroup$ – br69 Feb 15 '12 at 8:04
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    $\begingroup$ @br69 Please register your account and stop using answers for comments. $\endgroup$ – user88 Feb 15 '12 at 8:18

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