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ON TESTS OF SYMMETRY AGAINST

ONE-SIDED ALTERNATIVES

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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.

**Source**
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