Mixtures of equispaced normal distributions and their use for testing symmetry with univariate data

Given a random sample of observations, mixtures of normal densities are often used to estimate the unknown continuous distribution from which the data come. The use of this semi-parametric framework is proposed for testing symmetry about an unknown value. More precisely, it is shown how the null hypothesis of symmetry may be formulated in terms of a normal mixture model, with weights about the center of symmetry constrained to be equal one another. The resulting model is nested in a more general unconstrained one, with the same number of mixture components and free weights. Therefore, after having maximized the constrained and unconstrained log-likelihoods, by means of the Expectation–Maximization algorithm, symmetry is tested against skewness through a likelihood ratio statistic with p-value computed by using a parametric bootstrap method. The behavior of this mixture-based test is studied through a Monte Carlo simulation, where the proposed test is compared with the traditional one, based on the third standardized moment, and with the non-parametric triples test. An illustrative example is also given which is based on real data.