(i) Let $X_{1}, \ldots, X_{n}$ be independent, identically distributed random variables, with the exponential density $f(x ; \theta)=\theta e^{-\theta x}, x>0$.

Obtain the maximum likelihood estimator $\hat{\theta}$ of $\theta$. What is the asymptotic distribution of $\sqrt{n}(\hat{\theta}-\theta)$ ?

What is the minimum variance unbiased estimator of $\theta ?$ Justify your answer carefully.

(ii) Explain briefly what is meant by the profile log-likelihood for a scalar parameter of interest $\gamma$, in the presence of a nuisance parameter $\xi$. Describe how you would test a null hypothesis of the form $H_{0}: \gamma=\gamma_{0}$ using the profile log-likelihood ratio statistic.

In a reliability study, lifetimes $T_{1}, \ldots, T_{n}$ are independent and exponentially distributed, with means of the form $E\left(T_{i}\right)=\exp \left(\beta+\xi z_{i}\right)$ where $\beta, \xi$ are unknown and $z_{1}, \ldots, z_{n}$ are known constants. Inference is required for the mean lifetime, $\exp \left(\beta+\xi z_{0}\right)$, for covariate value $z_{0}$.

Find, as explicitly as possible, the profile log-likelihood for $\gamma \equiv \beta+\xi z_{0}$, with nuisance parameter $\xi$.

Show that, under $H_{0}: \gamma=\gamma_{0}$, the profile $\log -$ likelihood ratio statistic has a distribution which does not depend on the value of $\xi$. How might the parametric bootstrap be used to obtain a test of $H_{0}$ of exact size $\alpha$ ?

[Hint: if $Y$ is exponentially distributed with mean 1 , then $\mu Y$ is exponentially distributed with mean $\mu$.]