Math 6010 • Linear Models

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
random variables with density function
\[
  f(t,\theta)=
  \begin{cases}
    0, & \text{if } t<0,\\
    \dfrac{1}{\theta}e^{-t/\theta}, & \text{if } t\geq 0.
  \end{cases}
\]
Let \(\alpha>0\) and
\[
  \tau=\frac{1}{\theta^\alpha}.
\]

\textbf{(a)} How big \(n\) should be (as a function of \(\alpha\)) so that \(\tau\)
has uniformly minimum variance unbiased estimator?

\textbf{(b)} Find the uniformly minimum variance unbiased estimator for \(\tau\)
(you need to justify that your estimator has the required property).