UMVUE of $\mu^4$ when $(X_{i})_{i=1 \ldots n} \sim N(\mu, \sigma^2)$

Question

I am trying to find UMVUE of $\mu^4$ when $(X_{i})_{i=1 \ldots n} \sim N(\mu, \sigma^2)$, both $\mu, \sigma^2$ are unknown.

I know that the complete sufficient statistics for $\mu, \sigma^2$ and $\bar{X}_n, S_n^2$. And I want to find an unbiased estimator for $\mu^4$.

I know that: $$ E[(\bar{X}_n)^4] = \mu^4 + 3 \frac{\sigma^4}{n^2} + 6\mu^2 \frac{\sigma^2}{n} \\ E[S_n^2] = \frac{n+1}{n-1} \sigma^4 $$

Note that here: \begin{align*} \bar{X}_n & := \frac{1}{n}\sum_i X_i \\ S_n^2 &:= \frac{1}{n-1} \sum_i (X_i - \bar{X}_n)^2 \end{align*}

But I am stuck on finding an estimator contains $\mu^2 \sigma^2$.

Answer 1

The idea is to construct UMVUEs for all quartic order estimators $(\mu^4, \mu^2\sigma^2, \sigma^4)$ by considering all possible monomial functions of the complete and sufficient statistic $$\left(T=\frac{1}{n}\sum_i X_i, U=\frac{1}{n}\sum_iX_i^2\right)$$

whose expectation values contain the requisite parameters to be estimated. It turns out there are exactly $3$ such monomials, $(T^4,T^2 U,U^2)$ and their expectation values read, in matrix form:

$$\begin{pmatrix}E(T^4)\\E(T^2 U)\\E(U^2)\end{pmatrix}=\begin{pmatrix}3&6/n&3/n^2\\3&1+5/n&1/n+2/n^2\\3&2+4/n&1+2/n\end{pmatrix}\begin{pmatrix}{\mu}^4\\\mu^2\sigma^2\\\sigma^4\end{pmatrix}$$

After inversion of the above matrix, we conclude that the estimator

$$\hat{\mu^4}=\frac{1}{n^2-1}\left(\frac{(n+2)(n+4)}{3}T^4-2(n+2)T^2 U+U^2\right)$$

is unbiased and hence by the Lehmann-Scheffé theorem, the unique UMVUE for this quantity.

Note: the OP's set of complete and sufficient statistics is related to this one by

$$\bar{X}_n=T,~~ S_n^2=\frac{n}{n-1}(U-T^2)$$

In terms of these variables the UMVUE estimator reads

$$\hat{\mu^4}= \frac{\bar{X}_n^4}{3}- \frac{2}{n} S_n^2 \bar{X}_n^2+\frac{n-1}{(n+1)n^2} S_n^4$$