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Department of Mathematics

Question

Math 6010 • Linear Models

Summer 2020 • Question 8

Question

Math 6010 • Linear Models Summer 2020 • Question 8

Let \(X_1,\ldots ,X_n\) be independent and identically distributed with density function \[ h(t;\theta )=\theta t^{\theta -1}1\{0\leq t\leq 1\} \] for \(\theta >0\) unknown. Use the Neyman-Pearson Lemma to find the uniformly most powerful test of size \(\alpha \) for \(H_0:\theta =\theta _0\) versus \(H_a:\theta =\theta _1\), where \(\theta _1>\theta _0\).

Raw LaTeX

Let \(X_1,\ldots,X_n\) be independent and identically distributed with
density function
\[
  h(t;\theta)=\theta t^{\theta-1}1\{0\leq t\leq 1\}
\]
for \(\theta>0\) unknown. Use the Neyman-Pearson Lemma to find the uniformly most
powerful test of size \(\alpha\) for \(H_0:\theta=\theta_0\) versus
\(H_a:\theta=\theta_1\), where \(\theta_1>\theta_0\).

Topic Tags

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