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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 10

Question

Math 6010 • Linear Models 1998 • Question 10

Let \(X_1,X_2,\ldots ,X_n\) be independent Poisson random variables with parameters \(\theta _1,\theta _2,\ldots ,\theta _n\). We wish to test \(H_o:\theta _1=\theta _2=\cdots =\theta _n\) against the alternative that \(H_o\) is not true.

(a) Find the likelihood ratio test. (5 points)

(b) Show that the likelihood ratio is asymptotically normal (after centralizing and normalizing) under \(H_o\). (5 points)

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent Poisson random variables
with parameters \(\theta_1,\theta_2,\ldots,\theta_n\). We wish to test
\(H_o:\theta_1=\theta_2=\cdots=\theta_n\) against the alternative that \(H_o\) is
not true.

(a) Find the likelihood ratio test. (5 points)

(b) Show that the likelihood ratio is asymptotically normal (after centralizing
and normalizing) under \(H_o\). (5 points)

Topic Tags

Covariance and Independence F-Tests and t-Tests Likelihood Ratio Tests Maximum Likelihood Estimation