Math 6010 • Linear Models

Let \(X_1,\ldots,X_n\) be independent, identically distributed random
variables with density function
\[
h(t,\theta)=
\begin{cases}
0, & \text{if } -\infty<t<0,\\
\theta(t+1)^{-\theta-1}, & \text{if } 0\leq t<\infty,
\end{cases}
\]
\(\theta>0\).

\textbf{(a)} Find the maximum likelihood estimator for \(\theta\).

\textbf{(b)} Is the estimator unbiased?

\textbf{(c)} Find the asymptotic variance of the maximum likelihood estimator
for \(\theta\).