Methods for Constructing Point Estimates of Parameters

Prove that the parameter estimators constructed in Examples 3.1 and 3.3 are asymptotically unbiased and strongly consistent.

A sample of size $n$ is generated by the random variable $X \sim \operatorname {Exp}\left(\theta \right), \theta > 0$. Find the ML estimator of the parameter $\theta$ and prove its strong consistency.

A sample of size $n$ corresponds to the Poisson distribution $\operatorname {Pois}\left(\theta \right)$, $\theta > 0$. Find the ML estimator of $\theta$ and prove its unbiasedness, strong consistency, and asymptotic normality.

The sample $\left\{ X_{k}, k = 1, \ldots , n\right\}$ is generated by the random variable $X \sim E(\theta_{1}, \theta_{2}), \theta_{2} > 0$, i.e.

Find estimators of the parameters $\theta_{1}$ and $\theta_{2}$ by the method of moments. Prove the strong consistency of the obtained estimators.

The sample $Z_{n}$ corresponds to the Rayleigh distribution with distribution function $F(x ; \theta ) = 1-\exp \left\{ -\frac{x^{2}}{\theta }\right\} , x \geqslant 0, \theta > 0$. Find the ML estimator of the parameter $\theta$.

The sample $Z_{n} = \left\{ X_{k}, k = 1, \ldots , n\right\}$ of size $n = 2 m+1$ ($m$ is a natural number) corresponds to the Laplace distribution with density $p(x ; \theta ) = \frac{1}{2} \exp \left\{ -\left|x-\theta \right|\right\}$. Find the ML estimator of the parameter $\theta$.

Under the conditions of Problem 6, for the case $n = 2 m$, show that any statistic of the form $\widehat{\theta }_{n} = (1-\lambda ) X_{(m)}+\lambda X_{(m+1)}, \lambda \in [0 ; 1]$ is an ML estimator of $\theta$.

Let $\widehat{\theta }_{n}$ be the ML estimator of the parameter $\theta$ of the Bernoulli distribution $\operatorname {Bin}\left(1, \theta \right)$. Show that the sequence $\sqrt{n}\left(\widehat{\theta }_{n}-\theta \right)$ is asymptotically normal with parameters ($0 ; \theta (1-\theta )$).

The sample $Z_{n}$ corresponds to the normal distribution with parameters $(\sqrt{\theta } ; 2), \theta \geqslant 0$. Find the ML estimator of $\theta$.

The sample $Z_{n} = \left\{ X_{k}, k = 1, \ldots , n\right\}$ corresponds to the lognormal distribution with parameters $\left(\theta_{1} ; \theta_{2}\right)$, i.e. $\ln X_{k} \sim \mathcal{N}\left(\theta_{1} ; \theta_{2}\right)$. Find the ML estimator of the parameter $\theta = \mathbb {E}\left[X_{k}\right]$ and prove its strong consistency.

Find the ML estimator of the parameter $\theta = \mathbb {E}\left[X_{k}\right]$ from the sample $\left\{ X_{k}, k = 1, \ldots , n\right\}$ corresponding to the distribution $R\left[\theta_{1}, \theta_{2}\right]$.