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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 10

Question

Math 6010 • Linear Models 2000 • Question 10

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential \((\lambda )\) random variables. Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential \((\mu )\) random variables. The two samples are independent. We wish to test \(H_0:\theta =\mu \) against the alternative that \(H_0\) is not true.

(a) Find the likelihood ratio test.

(b) Find the asymptotic distribution of the likelihood ratio under \(H_0\).

(c) Find an equal-tail confidence interval for \(\tau =\lambda /\mu \).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed exponential \((\lambda)\) random variables. Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed exponential \((\mu)\) random variables. The two samples are independent. We wish to test \(H_0:\theta=\mu\) against the alternative that \(H_0\) is not true.

(a) Find the likelihood ratio test.

(b) Find the asymptotic distribution of the likelihood ratio under \(H_0\).

(c) Find an equal-tail confidence interval for \(\tau=\lambda/\mu\).

Topic Tags

Likelihood Ratio Tests Maximum Likelihood Estimation Simultaneous Confidence Intervals Uniform Integrability and Truncation Methods