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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1994 • Question 10

Question

Math 6010 • Linear Models 1994 • Question 10

Suppose \(\theta \in \{\theta _0,\theta _1\}\) is an unknown (\(\theta _0\) and \(\theta _1\) are, however, known.) Put some prior, \(\pi \), on \(\theta \). We are to test \(H_0:\theta =\theta _0\) versus \(H_1:\theta =\theta _1\). Our actions are 0 (accept) and 1 (reject). Find the Bayes’ procedure for doing this test, if the loss function, \(\ell (\theta ,a)\), is the 0–1 loss given by: \[ \ell (0,1)=0,\qquad \ell (0,0)=1, \] \[ \ell (1,0)=0,\qquad \ell (1,1)=1. \] Is this procedure minimax?

Raw LaTeX

Suppose \(\theta\in\{\theta_0,\theta_1\}\) is an unknown (\(\theta_0\) and \(\theta_1\) are, however, known.) Put some prior, \(\pi\), on \(\theta\). We are to test \(H_0:\theta=\theta_0\) versus \(H_1:\theta=\theta_1\). Our actions are 0 (accept) and 1 (reject). Find the Bayes' procedure for doing this test, if the loss function, \(\ell(\theta,a)\), is the 0--1 loss given by:
\[
\ell(0,1)=0,\qquad \ell(0,0)=1,
\]
\[
\ell(1,0)=0,\qquad \ell(1,1)=1.
\]
Is this procedure minimax?

Topic Tags

Bayesian Testing and Decision Theory F-Tests and t-Tests Uniform Integrability and Truncation Methods