New characterisation-based tests for symmetry

Abstract

We propose new tests for symmetry of a distribution around an unspecified centre. The first set of tests is based on a lesser-known characterisation involving symmetric integration of the distribution function around the estimated centre. The tests are straightforward to implement and, unlike many of the existing tests, do not require the practitioner to make a choice of a tuning parameter. This is an important consideration from a practical perspective, as an optimal choice of the tuning parameter usually depends on the unknown underlying distribution and it is known that the performance of tests can be quite sensitive, in terms of both size and power, to the choice of the tuning parameter. Numerical simulations indicate that the new tests have favourable finite-sample properties in the sense that they are level preserving and exhibit competitive power. The asymptotic null distribution of the Kolmogorov–Smirnov type test statistic is derived in the multiple linear regression setup where the goal is to test for symmetry of the error distribution, and the test is shown to be asymptotically consistent against general alternatives. The second set of tests is based on the idea of substituting smooth kernel-type distribution function estimators for the nonsmooth empirical counterparts appearing in classical Kolmogorov–Smirnov and Cramér–von Mises tests for symmetry. We demonstrate theoretically that, in finite samples, this leads to tests which potentially have significantly higher power. Numerical studies support this and also show that these tests have empirical size close to the nominal level. Even though kernel-type distribution function estimators have been studied extensively in the literature, its benefits in goodness-of-fit testing have only recently been recognised (see, e.g., Racine and van Keilegom, 2020). Recently, Allison and Pretorius (2017) proposed two new tests for symmetry around a specified centre. Although these tests were shown to compete very well with other modern tests for symmetry in a Monte Carlo study, the authors did not investigate the asymptotic properties of their tests. In this study we derive the asymptotic null distribution of one of these tests in the context of testing whether the observations have a distribution symmetric around an unknown centre. The study is concluded with an application of the new tests to real-world data.