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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 \(H:\sigma ^2\ge \sigma _0^2\). (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.

(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 \(H:\sigma^2\ge \sigma_0^2\). (5 points)

Topic Tags

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