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.