Unbiased estimator of $\mu\ 'S^{-1} \mu$

Question

$\vec X_1,\cdots ,\vec X_n$ are iid $N_2(\mu,\Sigma)$. Find an unbiased estimator of $\mu\ 'S^{-1} \mu$ as a linear function of its mle.

My work:

MLE is $\bar X\ 'S^{-1} \bar X$

Now

$$E(\bar X\ 'S^{-1} \bar X)=E(tr(\bar X\ 'S^{-1} \bar X))\\=tr(E(\bar X\ 'S^{-1} \bar X))\\=tr(E(S^{-1}\bar X\ '\bar X))\\=tr(E(S^{-1})E(\bar X\ '\bar X))$$

The last equation is because of the fact that $S$ and $\bar X\ ' \bar X$ are known to be independent.

I know $E(\bar X\ '\bar X)={\Sigma\over n}+\mu'\mu$

But I am not being able to find $E(S^{-1})$. Can someone help me proceed?

Answer 1

$S$, properly scaled, is a Wishart matrix. Inverse of a Wishart matrix can be easily computed by going through any standard reference on multivariate statistics.

Alternative easier solution: Recall that mle of $\mu'\Sigma^{-1}\mu$ is $\overline{X}'S^{-1}\overline{X}$ which, after proper scaling, has a $T^2$ distribution, which in turn is an $F$. Expectation of an $F$ s something you can know. Or if you want, write $F=cA/B$ where $A,B$ are independent $\chi$-squared, $c$ is a constant then find $E(F)=cE(A)E(1/B)$. $E(A)$ is straightforward, while $E(1/B)$ has to be computed by finding pdf of $B$, keeping in mind $B$ is chi squared.