Write a program in \(\mathrm{R}\) that simulates 1000 samples of size \(n\) from a Pois\(\operatorname{SON}(\theta)\) distribution, where \(n\) and \(\theta\) are specified below. For each sample compute the two estimators of \(P\left(X_{n}=0ight)=\exp (-\theta)\) given by

\[\hat{\theta}_{n}=n^{-1} \sum_{i=1}^{n} \delta\left\{X_{i} ;\{0\}ight\}\]

which is the proportion of values in the sample that are equal to zero, and \(\tilde{\theta}_{n}=\exp \left(-\bar{X}_{n}ight)\). Use the 1000 samples to estimate the bias, standard error, and the mean squared error for each case. Discuss the results of the simulations in terms of the theoretical findings of Exercise 11. Repeat the experiment for \(\theta=1,2\), and 5 with \(n=5,10,25,50\), and 100 .

Exercise 11.

Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables from a \(\operatorname{Poisson}(\theta)\) distribution. Consider two estimators of \(P\left(X_{n}=0ight)=\exp (-\theta)\) given by

\[\hat{\theta}_{n}=n^{-1} \sum_{i=1}^{n} \delta\left\{X_{i} ;\{0\}ight\}\]

which is the proportion of values in the sample that are equal to zero, and \(\tilde{\theta}_{n}=\exp \left(-\bar{X}_{n}ight)\). Compute the asymptotic relative efficiency of \(\hat{\theta}_{n}\) relative to \(\tilde{\theta}_{n}\) and comment on the results.