Let X1, …, Xn be iid with pdf f(x ; θ)=1 /(3 θ),-θ 0. (a) Find th

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Let X1, …, Xn be iid with pdf f(x ; θ)=1 /(3 θ),-θ<x<2 θ,...

Let $X_{1}, \ldots, X_{n}$ be iid with pdf $f(x ; \theta)=1 /(3 \theta),-\theta<x<2 \theta$, zero elsewhere, where $\theta>0$. (a) Find the mle $\hat{\theta}$ of $\theta$. (b) Is $\widehat{\theta}$ a sufficient statistic for $\theta$ ? Why? (c) Is $(n+1) \widehat{\theta} / n$ the unique MVUE of $\theta$ ? Why?

Let $X_{1}, \ldots, X_{n}$ be iid with pdf $f(x ; \theta)=1 /(3 \theta),-\theta<x<2 \theta$, zero elsewhere, where $\theta>0$.
(a) Find the mle $\hat{\theta}$ of $\theta$.
(b) Is $\widehat{\theta}$ a sufficient statistic for $\theta$ ? Why?
(c) Is $(n+1) \widehat{\theta} / n$ the unique MVUE of $\theta$ ? Why?

Key Concepts

Maximum Likelihood Estimator (MLE)

The MLE is a method of estimating parameters by finding the parameter values that maximize the likelihood function, which represents the probability of observing the given data under the model. In problems where the support of the distribution depends on the parameter, the maximization often involves considering the boundaries (or order statistics) of the sample, since these boundaries constrain the parameter estimates.

Order Statistics

Order statistics are derived by arranging the sample values in order, and they become particularly significant when the support of a distribution depends on the parameter. In such cases, the extremes of the sample (such as the minimum or maximum) provide critical information for determining valid parameter estimates, often appearing naturally in the derivation of the MLE.

Sufficient Statistic

A sufficient statistic is a function of the sample data that encapsulates all the information needed to estimate a parameter effectively. According to the factorization theorem, if the joint probability density function can be expressed as a product where one factor depends solely on the parameter and the statistic, then this statistic is sufficient. Sufficient statistics simplify the estimation process and are central to finding optimal estimators.

Minimum Variance Unbiased Estimator (MVUE) and the Lehmann–Scheffé Theorem

The MVUE is an unbiased estimator that has the smallest possible variance among all unbiased estimators for a parameter. The Lehmann–Scheffé theorem states that if a complete and sufficient statistic exists, then any unbiased estimator that is a function of this statistic is the unique MVUE. This is a powerful result in finding estimators that are both unbiased and efficient.

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