UMVUE of $\theta^r$

Question

Given that $X_1,X_2,..X_n \sim f(x)=e^{\theta-x}$ , find the UMVUE of $\theta^r$.

I tried using Lehmann Scheffe knowing the fact that $X_{(1)}$ is sufficient and complete for $\theta$ but I cannot find the value of $E(X_{(1)})^r$ as it is a pretty tough integral.Any other approaches?

Answer 1

When support of distribution depends on parameter, one can use differentiation to solve integral equations.

It is clear than any unbiased estimator of $\theta^r$ based on $X_{(1)}$ will be the UMVUE of $\theta^r$.

Let $h(\cdot)$ be that (measurable) function of $X_{(1)}$.

Then

\begin{align} E_{\theta}\left[h(X_{(1)}\right]&=\theta^r\quad,\forall\,\theta\in\mathbb R \\\iff \int_{\theta}^\infty h(x)ne^{-n(x-\theta)}\,dx&=\theta^r\quad,\forall\,\theta \end{align}

That is, $$\int_{\theta}^\infty h(x)e^{-nx}\,dx=\frac{\theta^re^{-n\theta}}{n}\quad,\forall\,\theta$$

Differentiating both sides of the last equation with respect to $\theta$ you can find $h$.