Let X1, X2, …, Xn be a random sample from a distribution...

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a distribution with pmf $p(x ; \theta)=\theta^{x}(1-\theta), x=0,1,2, \ldots$, zero elsewhere, where $0 \leq \theta \leq 1$. (a) Find the mle, $\hat{\theta}$, of $\theta$. (b) Show that $\sum_{1}^{n} X_{i}$ is a complete sufficient statistic for $\theta$. (c) Determine the MVUE of $\theta$.

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a distribution with pmf $p(x ; \theta)=\theta^{x}(1-\theta), x=0,1,2, \ldots$, zero elsewhere, where $0 \leq \theta \leq 1$.
(a) Find the mle, $\hat{\theta}$, of $\theta$.
(b) Show that $\sum_{1}^{n} X_{i}$ is a complete sufficient statistic for $\theta$.
(c) Determine the MVUE of $\theta$.

Key Concepts

Complete Statistic

A statistic is complete if no nontrivial function of it has an expected value of zero for all possible values of the parameter. Completeness is important because it guarantees the uniqueness of unbiased estimators derived from it, ensuring that any unbiased estimator based on a complete statistic is the best possible in the sense of having minimum variance.

Unbiased Estimation and MVUE

Unbiased estimation seeks estimators whose expected value exactly equals the parameter being estimated. The Minimum Variance Unbiased Estimator (MVUE) is the unbiased estimator that, among all unbiased estimators, has the smallest variance. The MVUE is typically obtained by applying the Lehmann-Scheffé theorem to an unbiased estimator that has been conditioned on a complete sufficient statistic.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method for estimating the parameters of a probability distribution by maximizing the likelihood function, which measures how probable the observed data is for different parameter values. The MLE is derived by setting the derivative of the log?likelihood function to zero and solving for the parameter, resulting in estimators that often enjoy desirable large-sample properties.

Sufficient Statistic

A sufficient statistic is a function of the sample data that summarizes all the information needed to estimate a parameter of a distribution. According to the factorization theorem, if the joint distribution of the data can be factored into a product where one factor depends on the data only through this statistic and the other does not depend on the parameter, then the statistic is sufficient.

Lehmann-Scheffé Theorem

The Lehmann-Scheffé theorem establishes that if a statistic is both complete and sufficient for a parameter, then any unbiased estimator that is a function of this statistic is the unique Uniformly Minimum Variance Unbiased Estimator (UMVUE). This theorem is fundamental in the process of identifying the best unbiased estimator through conditioning on complete sufficient statistics.

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