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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2015 • Question 2

Question

Math 6010 • Linear Models Fall 2015 • Question 2

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ h(t;\theta )= \begin {cases} \dfrac {1}{\theta }, & \text {if } 0\leq t\leq \theta ,\\ 0, & \text {if } t\notin [0,\theta ]. \end {cases} \] We wish to test \(H_0:\theta =1\) against the alternative \(H_A:\theta <1\). We reject \(H_0\) if \[ X_{n,n}=\max _{1\leq i\leq n}X_i\leq 1-\frac {2}{n}. \] (a) Compute the significance level.

(b) Compute the power function.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed
random variables with density function
\[
h(t;\theta)=
\begin{cases}
\dfrac{1}{\theta}, & \text{if } 0\leq t\leq\theta,\\
0, & \text{if } t\notin[0,\theta].
\end{cases}
\]
We wish to test \(H_0:\theta=1\) against the alternative \(H_A:\theta<1\). We
reject \(H_0\) if
\[
X_{n,n}=\max_{1\leq i\leq n}X_i\leq 1-\frac{2}{n}.
\]
\textbf{(a)} Compute the significance level.

\textbf{(b)} Compute the power function.

Topic Tags

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