LIKELIHOOD RATIO TESTS FOR SYMMETRY AGAINST ONE-SIDED ALTERNATIVES

RICHARD DYKSTRA^{1}, SUBHASH KOCHAR^{2} AND TIM ROBERTSON^{1}

1Department of Statistics and Actuarial Science, The University of Iowa, Iowa City, IA 52242, U.S.A.

2Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi - 110016, India

(Received May 6, 1994; revised January 25, 1995)

Abstract. A random variable X is said to have a symmetric distribution (about 0) if and only if X and −X are identically distributed. By considering various types of partial orderings between the distributions of X and −X, one obtains various notions of skewness or one-sided bias. In this paper we study likelihood ratio tests for testing the symmetry of a discrete distribution about zero against the alternatives, (i) X is stochastically greater than −X; and (ii) pr(X = j) ≥ pr(X = −j) for all j > 0. In the process, we obtain maximum likelihood estimators of the distribution function under the above alternatives. The asymptotic null distributions of the test statistics have been obtained and are of the chi-bar square type. A simulation study was performed to compare the powers of these tests with other tests.

Key words and phrases: Chi-bar square distribution, chi square test for goodness of fit, isotonic regression, positive biasedness, skewness, stochastic ordering.