Let X1, X2, …, Xn be a random sample from each of the...

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from each of the following distributions involving the parameter $\theta .$ In each case find the mle of $\theta$ and show that it is a sufficient statistic for $\theta$ and hence a minimal sufficient statistic. (a) $b(1, \theta)$, where $0 \leq \theta \leq 1$. (b) Poisson with mean $\theta>0$. (c) Gamma with $\alpha=3$ and $\beta=\theta>0$. (d) $N(\theta, 1)$, where $-\infty<\theta \leq \infty$. (e) $N(0, \theta)$, where $0<\theta<\infty$.

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from each of the following distributions involving the parameter $\theta .$ In each case find the mle of $\theta$ and show that it is a sufficient statistic for $\theta$ and hence a minimal sufficient statistic.
(a) $b(1, \theta)$, where $0 \leq \theta \leq 1$.
(b) Poisson with mean $\theta>0$.
(c) Gamma with $\alpha=3$ and $\beta=\theta>0$.
(d) $N(\theta, 1)$, where $-\infty<\theta \leq \infty$.
(e) $N(0, \theta)$, where $0<\theta<\infty$.

Key Concepts

Maximum Likelihood Estimation (MLE)

MLE is a fundamental method in statistical inference for estimating the parameters of a statistical model. It involves selecting the parameter values that maximize the likelihood function, which measures how well the model explains the observed data. In the context of the problem, finding the MLE for different distributions means finding the parameter that makes the observed sample most probable under each specific model.

Likelihood Function

The likelihood function is a function of the parameter(s) given the observed data. It is constructed as the product of probability density functions or probability mass functions for independent observations. Analyzing and maximizing this function is central to obtaining the maximum likelihood estimator. The proper construction of the likelihood is essential for deriving both the estimator and the subsequent sufficiency arguments.

Sufficient Statistic

A sufficient statistic for a parameter is a function of the sample that captures all the information needed to compute the likelihood of the parameter, given the sample. Once the value of a sufficient statistic is known, no additional data can increase the information about the parameter. This concept is crucial in summarizing data effectively without loss of information for parameter estimation.

Factorization Theorem

The factorization theorem provides a method to verify whether a statistic is sufficient for a parameter. It states that a statistic is sufficient if and only if the joint probability function of the sample can be factored into two parts: one that depends on the data only through this statistic and the parameter, and another that depends solely on the data. This theorem underlies the process of proving sufficiency in the given problems.

Minimal Sufficient Statistic

A minimal sufficient statistic is the most reduced form of a statistic that still retains full information about the parameter. It is sufficient and is a function of every other sufficient statistic. Establishing that an estimator is not only sufficient but also minimal implies that it is the simplest possible summary of the data with respect to the parameter estimation problem, thus achieving parsimony in statistical inference.

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