Question

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?

Added by Paul S.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Andreas Papavassiliou

06/19/2025

Step 1

Step 1: **Identify the likelihood function** Given the pdf $ f(x; \theta) = \frac{1}{3\theta} $ for $ -\theta < x < 2\theta $, zero elsewhere, the likelihood function for the iid sample $ X_1, X_2, \ldots, X_n $ is: \[ L(\theta; X_1, X_2, \ldots, X_n) = Show more…

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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?