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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: \[ \begin {alignedat}{2} \ell (0,1)&=0,\qquad & \ell (0,0)&=1,\\ \ell (1,0)&=0,\qquad & \ell (1,1)&=1. \end {alignedat} \] 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:
\[
\begin{alignedat}{2}
\ell(0,1)&=0,\qquad & \ell(0,0)&=1,\\
\ell(1,0)&=0,\qquad & \ell(1,1)&=1.
\end{alignedat}
\]
Is this procedure minimax?

Topic Tags

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