Consider estimation of the tail probability \(\mu=\mathbb{P}[X \geqslant \gamma]\) of some random variable \(X\), where \(\gamma\) is large. The crude Monte Carlo estimator of \(\mu\) is

\[ \begin{equation*} \widehat{\mu}=\frac{1}{N} \sum_{i=1}^{N} Z_{i} \tag{3.35} \end{equation*} \]

where \(X_{1}, \ldots, X_{N}\) are iid copies of \(X\) and \(Z_{i}=1\left\{X_{i} \geqslant \gamma\right\}, i=1, \ldots, N\).

(a) Show that \(\widehat{\mu}\) is unbiased; that is, \(\mathbb{E} \widehat{\mu}=\mu\).

(b) Express the relative error of \(\widehat{\mu}\), i.e.,

\[ \mathrm{RE}=\frac{\sqrt{\overline{\operatorname{Var} \widehat{\mu}}}}{\mathbb{E} \widehat{\mu}} \]

in terms of \(N\) and \(\mu\).

(c) Explain how to estimate the relative error of \(\widehat{\mu}\) from outcomes \(x_{1}, \ldots, x_{N}\) of \(X_{1}, \ldots, X_{N}\), and how to construct a \(95 \%\) confidence interval for \(\mu\).

(d) An unbiased estimator \(Z\) of \(\mu\) is said to be logarithmically efficient if

\[ \begin{equation*} \lim _{y \rightarrow \infty} \frac{\ln \mathbb{E} Z^{2}}{\ln \mu^{2}}=1 \tag{3.36} \end{equation*} \]

Show that the CMC estimator (3.35) with \(N=1\) is not logarithmically efficient.