# Finding uniform minimum variance unbiased estimators

$X_1, X_2, X_3, \ldots{}, X_n$, 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 $\mathcal{N}(0,1)$?

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

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.
• 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). – br69 Feb 14 '12 at 20:37
• 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. – Xi'an Feb 14 '12 at 21:15
• 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. – br69 Feb 15 '12 at 8:04