I am having some trouble with an example from the book I am following.
Let $X_1,X_2,...,X_n$ for $n>2$ be an iid set of $N(\mu,\sigma^2)$ random variables with, $\mu\in\mathbb{R}$ and $\sigma^2\in\mathbb{R}^+$, both unknown. Under this circumtance $T=(\overline{X}_n,S_n^2)=(Y,W)$, the bi-variate statistic containing the sample mean and sample variance, is sufficient and complete for $(\mu,\sigma^2)$, and $\overline{X}_n$ and $\frac{nS_n^2}{\sigma^2}$ are independent, $N(\mu,\sigma^2/n)$ and $\chi^2_{n-1}$ random variables, respectively.
Because of the Lehmann-Scheffe lemma, an unbiased function $h(Y,W)$ for $\mu$ is the almost surely unique UMVUE of $\mu$. Therefore, such a function $h$ must satisfy \begin{equation} \int_0^\infty f(w)\left[\int_{-\infty}^\infty h(y,w)f(y)dy\right]dw=\mu\nonumber \end{equation} where $f(y)$ and $f(w)$ are the pdf of $Y$ and $W$ respectively. I have tried several of the techniques like differentiating several times both sides by $\mu$, and using completeness in order to find the condition that leads to \begin{equation} h(Y,W)=Y\text{ } a.s. P_{\mu,\sigma^2}, \forall (\mu,\sigma^2)\in\mathbb{R}\times\mathbb{R}^+ \end{equation}
I can not seem to find the conditions required for the last equation to hold.
Any suggestions? is there any general rule that should be applied?
Best regards,
JMJulio
It was quite simple. Thanks to all who took the trouble to look at it.
Since the joint cdf of $Y$ and $Z$ is of bounded variation, the derivative wrt $\mu$ of the left hand side of \begin{equation} E_{\mu,\sigma^2}\left[h(Y,Z)\right]=\mu \end{equation} interchanges with both integrals, leading to \begin{equation} E_{\mu,\sigma^2}\left[h(Y,Z)\frac{n}{\sigma^2}(y-\mu)\right]=1 \end{equation}
Therefore, \begin{equation} E_{\mu,\sigma^2}\left[h(Y,Z)Y\right]=E_{\mu,\sigma^2}\left[Y^2\right] \end{equation} and applying completeness to \begin{equation} E_{\mu,\sigma^2}\left[h(Y,Z)Y-Y^2\right]=0 \end{equation} this leads to $h(Y,Z)Y=Y^2$ a.s. $P_{\mu,\sigma^2}$ $\forall\mu,\sigma^2\in\mathbb{R}\times\mathbb{R}^+$, from where \begin{equation} h(Y,Z)=Y \text{ a.s. } P_{\mu,\sigma^2} \forall\mu,\sigma^2\in\mathbb{R}\times\mathbb{R}^+ \end{equation}