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

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from the uniform distribution with pdf $f\left(x ; \theta_{1}, \theta_{2}\right)=1 /\left(2 \theta_{2}\right), \theta_{1}-\theta_{2}<x<\theta_{1}+\theta_{2}$, where $-\infty<\theta_{1}<\infty$ and $\theta_{2}>0$ and the pdf is equal to zero elsewhere. (a) Show that $Y_{1}=\min \left(X_{i}\right)$ and $Y_{n}=\max \left(X_{i}\right)$, the joint sufficient statistics for $\theta_{1}$ and $\theta_{2}$, are complete. (b) Find the MVUEs of $\theta_{1}$ and $\theta_{2}$.

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from the uniform distribution with pdf $f\left(x ; \theta_{1}, \theta_{2}\right)=1 /\left(2 \theta_{2}\right), \theta_{1}-\theta_{2}<x<\theta_{1}+\theta_{2}$, where $-\infty<\theta_{1}<\infty$ and $\theta_{2}>0$
and the pdf is equal to zero elsewhere.
(a) Show that $Y_{1}=\min \left(X_{i}\right)$ and $Y_{n}=\max \left(X_{i}\right)$, the joint sufficient statistics for $\theta_{1}$ and $\theta_{2}$, are complete.
(b) Find the MVUEs of $\theta_{1}$ and $\theta_{2}$.

Key Concepts

Sufficient Statistics

A statistic is sufficient for a parameter if it captures all the information in the sample about that parameter. In the context of parametric inference, the Neyman–Fisher factorization theorem is used to determine whether a statistic is sufficient by expressing the joint probability density function as a product where one factor depends on the sample only through this statistic and the parameter, while the other factor does not depend on the parameter.

Completeness

A statistic is complete if no non-zero function of it has an expected value of zero for all parameter values, unless the function is zero almost surely. Completeness is a crucial property when deriving optimal estimators because it ensures the uniqueness of unbiased estimators derived as functions of the complete sufficient statistic.

Order Statistics in Uniform Distribution

When sampling from a uniform distribution, the minimum and maximum order statistics often serve as natural and informative summaries of the sample. These order statistics tend to contain all the necessary information regarding the bounds of the distribution, making them ideal candidates for sufficient statistics in the estimation of location and scale parameters.

Lehmann–Scheffé Theorem

The Lehmann–Scheffé theorem states that if a statistic is complete and sufficient for a parameter, then any unbiased estimator which is a function of that statistic is the unique minimum variance unbiased estimator (MVUE) for that parameter. This theorem provides a systematic way to derive optimal estimators in parametric inference problems.

Minimum Variance Unbiased Estimator (MVUE)

The MVUE is an estimator that is unbiased and has the smallest variance among all unbiased estimators for a parameter. The process of finding an MVUE often involves identifying a complete sufficient statistic and then constructing an unbiased estimator as a function of that statistic, ensuring minimal estimation variance due to the uniqueness provided by the Lehmann–Scheffé theorem.

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