A new test for symmetry against right skewness

Abstract

Let X be a continuous random variable with distribution function F. If F is symmetric about 0, then for 0 < p < 1, η p = −η 1−p , where η p is the pth quantile of X. Using this well-known observation, based on a random sample from F, a new test of symmetry against right skewness is proposed and its exact null distribution is obtained. It is shown that the proposed test is relatively more powerful than some other tests in the literature including a test given by Corzo and Babativa [A modified runs test for symmetry. J Statist Comput Simul. 2013; 83: 984-991].

Comparison of efficiencies of some symmetry tests around an unknown centre

“…sample from an absolutely continuous symmetric distribution, with distribution function F . Let U n (µ) with kernel Φ(X; µ) be a U -statistic of order m from the family U; and let µ(α), 0 ≤ α ≤ 1/2, be the α-trimmed sample mean (2). Then √ nU n ( µ(α)) converges in distribution to a zero mean normal random variable with the following variance:…”

“…sample from an absolutely continuous symmetric distribution, with function F . Let {U n (µ; t)} be a nondegenerate family of U-statistics of order m with kernel Φ(·; t), that belong to the family U; and let µ(α), 0 ≤ α ≤ 0.5, be the α-trimmed sample mean (2). Then the family {U n ( µ(α); t)} is also non-degenerate with the variance function…”

Characterization of symmetric distributions based on concomitants of ordered variables from FGMs family of bivariate distributions

“…Here, d = means that the two random variables have the same distribution. Testing of symmetry based on characterizations have been also considered by several authors, see, for example, Baringhaus and Henze (1992), Nikitin and Ahsanullah (2015), Amiri and Khaledi (2016), Milošević and Obradović (2016) and Božin et al (2018) and references therein. The aim of this paper is to provide some new characterization results for symmetric distributions by using some properties of concomitants of ordered variables from the FGM family of bivariate distributions.…”