ON TESTS OF SYMMETRY AGAINST ONE-SIDED ALTERNATIVES

BHASKAR BHATTACHARYA

Department of Mathematics, Southern Illinois University, Carbondale, IL 62901-4408, U.S.A.

(Received May 31, 1995; revised September 13, 1996)

Abstract. To test that an experimental treatment is better than an existing one (or control), one can equivalently consider the difference in their response and test if the distribution of the difference is symmetric (about zero) versus it exhibits positive bias (skewness to the right). In this paper, we test the symmetry (about zero) of a discrete distribution against two particular classes of one sided alternatives. We obtain the maximum likelihood estimators under each alternative. The asymptotic null distributions of the likelihood ratio statistics are shown to have chi-bar square type distributions. A power study is performed to compare these one-sided alternatives with other one-sided tests. The theory developed is illustrated by an example.

Key words and phrases: Chi-bar square distribution, consistency, isotonic regression, likelihood ratio test, multinomial, positive bias, stochastic order relations.