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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 4

Question

Math 6010 • Linear Models Spring 2010 • Question 4

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with density function \[ h(t,\theta )= \begin {cases} 0, & \text {if } t\notin [0,\theta ],\\ 2\theta ^{-2}t, & \text {if } 0\leq t\leq \theta . \end {cases} \] Find the uniformly minimum variance unbiased estimator for \(\theta \). Explain your answer. You need to prove directly that the sufficient statistic is also complete in this case.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
random variables with density function
\[
h(t,\theta)=
\begin{cases}
0, & \text{if } t\notin[0,\theta],\\
2\theta^{-2}t, & \text{if } 0\leq t\leq\theta.
\end{cases}
\]
Find the uniformly minimum variance unbiased estimator for \(\theta\). Explain
your answer. You need to prove directly that the sufficient statistic is also
complete in this case.

Topic Tags

Covariance and Independence Estimator Distributions Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods