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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 8

Question

Math 6010 • Linear Models 1998 • Question 8
Let \(X_1,\ldots,X_n\) be a random sample from the \(N(\mu,\sigma^2)\) distribution, with both \(\mu\) and \(\sigma^2\) unknown. **(a)** Derive the likelihood ratio test for \(H:\sigma^2=\sigma_0^2\) against all alternatives. Here \(\sigma_0^2\) is a known positive constant. (5 points) **(b)** Do the same for \(H:\sigma^2\geq\sigma_0^2\). (5 points)

Raw LaTeX

Let \(X_1,\ldots,X_n\) be a random sample from the \(N(\mu,\sigma^2)\) distribution, with both \(\mu\) and \(\sigma^2\) unknown.
\begin{parts}
\item Derive the likelihood ratio test for \(H:\sigma^2=\sigma_0^2\) against all alternatives. Here \(\sigma_0^2\) is a known positive constant. (5 points)
\item Do the same for \(H:\sigma^2\geq\sigma_0^2\). (5 points)
\end{parts}

Topic Tags

F-Tests and t-Tests Likelihood Ratio Tests Maximum Likelihood Estimation Uniform Integrability and Truncation Methods