# UMVUE of a function of sufficient statistic

Let $$X_1,\ldots, X_n$$ be a random sample from the following distribution: $$f(x\mid\theta)= \frac{4}{\theta}x^3e^{-\frac{x^4}{\theta}}$$

I know that $$S = \sum X_i^4$$ is a complete and sufficient statistic of $$\theta$$.

How can I find a UMVUE for $$\tau(\theta)=\frac{1}{\theta}$$?

• If this is for a homework or assignment, or a textbook question, please add the [self-study] tag and read its wiki as we tend to treat such questions a little differently here. Welcome to CV :-) – Andy Jun 3 '16 at 9:40
• What is the support of that density? Is there a sign missing in the exponent? – ekvall Jun 4 '16 at 15:50
• @Student001 You're right. There should be a minus into the e-power – arina Jun 6 '16 at 6:27

1. Define $Y_i = X_i^4$ and find the density for the $Y_i$
2. Compute $\mathbb E Y_i$
3. Use the result in 2. to construct an estimator based on $S = \sum_iY_i$ and appeal to a theorem about unbiased estimators that are functions of complete, sufficient statistics.