Student Seminar Series - November 15, 2005
University of Minnesota
School of Statistics
College of Liberal Arts
Hypothesis
Tests with Multiple Linear Inequality Constraints
Yumin Huang
Tuesday, November 15, 2005
2:00 PM, 300
Ford Hall
Minneapolis, East Bank Campus
Refreshments at 1:30 PM
300 Ford Hall
Abstract
We
consider the problem of testing the null hypothesis that a
multivariate normal mean vector is constrained to lie in a set
satisfying multiple inequality constraints, which can also be
described as testing multivariate one-sided null hypotheses. Lehmann
(1952) showed unbiasedness does not hold for such tests.
Related but different problems of testing one-sided hypotheses have
been investigated by Liu and Berger (1995), Perlman and Wu (2004).
Our null hypothesis is a polyhedral convex subset of a Euclidean space.
Our alternative hypothesis is the complement of the null. We give
examples showing the biasedness of several traditional tests and
present a new test having better properties. Our new test consists of
two stages: pretest and posttest. In the pretest, a face of the null
hypothesis is selected according to the value of PIC (pretest
information
criterion). In the posttest, the null hypothesis is rejected if the
likelihood ratio statistic is greater than a critical value that
depends on the face chosen in the pretest. The new test is constructed
such to be pointwise asymptotic unbiased. We also discuss computational
methods for finding the critical values of the test.