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LIKELIHOOD RATIO TESTS FOR SYMMETRY

AGAINST ONE-SIDED ALTERNATIVES

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

^{1} *Department of Statistics and Actuarial Science, The
University of Iowa,*

Iowa City, IA 52242, U.S.A.

^{2} *Indian 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.

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