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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Summer 2020 • Question 9

Question

Math 6010 • Linear Models Summer 2020 • Question 9

Let \(X_1,\ldots ,X_n\) be independent and identically distributed random variables. The density function of \(X_i\) is \[ g(t;\theta _i)=\theta _i t^{-\theta _i-1}1\{t\geq 1\}. \] Here \(\theta _i>0\) is an unknown parameter. We wish to test \[ H_0:\theta _1=\theta _2=\cdots =\theta _n \] against the alternative that \(H_0\) is not true. Find a test using the generalized likelihood ratio.

Raw LaTeX

Let \(X_1,\ldots,X_n\) be independent and identically distributed
random variables. The density function of \(X_i\) is
\[
  g(t;\theta_i)=\theta_i t^{-\theta_i-1}1\{t\geq 1\}.
\]
Here \(\theta_i>0\) is an unknown parameter. We wish to test
\[
  H_0:\theta_1=\theta_2=\cdots=\theta_n
\]
against the alternative that \(H_0\) is not true. Find a test using the
generalized likelihood ratio.

Topic Tags

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