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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1993 • Question 8

Question

Math 6010 • Linear Models 1993 • Question 8

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables, uniformly distributed on \([0,\theta ]\), \(\theta >\theta _0\). We want to test \(H_0:\theta =\theta _0\) against \(H_A:\theta >\theta _0\).

(a) Find the uniformly most powerful test. (You must prove your claim.)

(b) Show that the uniformly most powerful test and the likelihood ratio test are equivalent,

(c) Compute the power function of the most powerful test.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
random variables, uniformly distributed on \([0,\theta]\), \(\theta>\theta_0\). We
want to test \(H_0:\theta=\theta_0\) against \(H_A:\theta>\theta_0\).

(a) Find the uniformly most powerful test. (You must prove your claim.)

(b) Show that the uniformly most powerful test and the likelihood ratio test are
equivalent,

(c) Compute the power function of the most powerful test.

Topic Tags

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