Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 5

Question

Math 6010 • Linear Models 2004 • Question 5

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t\notin [0,\theta ],\\ \dfrac {1}{\theta }, & \text {if } t\in [0,\theta ]. \end {cases} \] We use \(\hat {\theta }=cX_{1,n}\) and \(\hat {\eta }=\alpha \bar {X}\) to estimate \(\theta \), where \[ X_{1,n}=\min _{1\leq i\leq n}X_i\quad \text {and}\quad \bar {X}=\frac {1}{n}\sum _{1\leq i\leq n}X_i. \] (a) Find \(c\) and \(\alpha \) such that the estimators are unbiased.

(b) Compute the relative efficiency.

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed random variables with density function
\[
f(t,\theta)=
\begin{cases}
0, & \text{if } t\notin [0,\theta],\\
\dfrac{1}{\theta}, & \text{if } t\in [0,\theta].
\end{cases}
\]
We use \(\hat{\theta}=cX_{1,n}\) and \(\hat{\eta}=\alpha\bar{X}\) to estimate \(\theta\), where
\[
X_{1,n}=\min_{1\leq i\leq n}X_i\quad\text{and}\quad
\bar{X}=\frac{1}{n}\sum_{1\leq i\leq n}X_i.
\]
(a) Find \(c\) and \(\alpha\) such that the estimators are unbiased.

(b) Compute the relative efficiency.

Topic Tags

Covariance and Independence Estimator Distributions Laws of Large Numbers Uniform Integrability and Truncation Methods