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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 3

Question

Math 6010 • Linear Models 2005 • Question 3

Let \(X_1,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ g(t;\theta )= \begin {cases} 0, & \text {if } -\infty <t<\theta ,\\ e^{-(t-\theta )}, & \text {if } \theta \leq t<\infty . \end {cases} \] We wish to test \(H_0:\theta \leq \theta _0\) against \(H_A:\theta >\theta _0\).

(a) Find the uniformly most powerful test of size \(\alpha \).

(b) Compute the power function.

Raw LaTeX

Let \(X_1,\ldots,X_n\) be independent, identically distributed random variables with density function
\[
g(t;\theta)=
\begin{cases}
0, & \text{if } -\infty<t<\theta,\\
e^{-(t-\theta)}, & \text{if } \theta\leq t<\infty.
\end{cases}
\]
We wish to test \(H_0:\theta\leq\theta_0\) against \(H_A:\theta>\theta_0\).

(a) Find the uniformly most powerful test of size \(\alpha\).

(b) Compute the power function.

Topic Tags

Covariance and Independence F-Tests and t-Tests Measure-Theoretic Probability Uniform Integrability and Truncation Methods