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 $\tau(\theta)=\frac{1}{\theta}$?

  • 3
    $\begingroup$ 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 :-) $\endgroup$ – Andy Jun 3 '16 at 9:40
  • $\begingroup$ What is the support of that density? Is there a sign missing in the exponent? $\endgroup$ – ekvall Jun 4 '16 at 15:50
  • $\begingroup$ @Student001 You're right. There should be a minus into the e-power $\endgroup$ – arina Jun 6 '16 at 6:27

I leave the details for you to fill in, and do not provide proofs, because this is self-study.

  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.
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