Efficiency of Point Estimates

The sample corresponds to the distribution $\operatorname {Bin}\left(N, \theta \right), \theta \in (0 ; 1)$. Verify the regularity conditions, find $i(\theta )$, and prove the efficiency of the ML estimator of the parameter $\theta$.

Show that the Poisson distribution $\operatorname {Pois}\left(\theta \right)$ is regular. Find $i(\theta )$. Prove the efficiency of the ML estimator $\widehat{\theta }_{n}$ of the parameter $\theta$.

For the distribution $\mathcal{N}\left(\mu ; \theta^{2}\right), \theta > 0$, where $\mu$ is known, find the Fisher information $i(\theta )$.

Verify the regularity of the distribution $\operatorname {Exp}\left(\theta \right), \theta > 0$, and compute $I_{n}(\theta )$. Prove that the estimator $\widetilde{\theta }_{n} = \frac{n-1}{n} \widehat{\theta }_{n}$ is asymptotically efficient if $\widehat{\theta }_{n}$ is the ML estimator of $\theta$.

Show that the information $I_{n}(\theta )$ contained in a sample $Z_{n}$ corresponding to the Laplace distribution (4.10) equals $\frac{n}{\lambda^{2}}$.

Compare in accuracy the estimator $\theta_{n}^{*}$ of the parameter $\theta$ of the distribution $R[0 ; \theta ]$, $\theta > 0$, obtained by the method of moments, with the estimator $\tilde{\theta }_{n}$ considered in Example 4.5.

Let the sample correspond to the distribution $\mathcal{N}(\mu ; \theta ), \theta > 0, \mu$ is known. Prove that the ML estimator of the variance $\theta$ is efficient.

The sample $Z_{n}$ corresponds to the distribution $\mathcal{N}\left(\theta_{1} ; \theta_{2}^{2}\right)$. Find the Fisher information matrix $I\left(\theta_{1}, \theta_{2}\right)$.

For the estimators constructed in Problems 1, 2 and 7, find their representation (4.6) in terms of the sample score.