On Testing the Symmetry of Innovation Distribution in Autoregression Schemes

Abstract

We consider a stationary linear \(AR(p)\) model with zero mean. The autoregression parameters, as well as the distribution function (d.f.) \(G(x)\) of innovations, are unknown. We test symmetry of innovations with respect to zero in two situations. In the first case the observations are a sample from a stationary solution of \(AR(p)\). We estimate parameters and find residuals. Based on them we construct a kind of empirical d.f. and the omega-square type test statistic. Its asymptotic d.f. under the hypothesis and the local alternatives are found. In the second situation the observations are subject to gross errors (outliers). For testing the symmetry of innovations we again construct the Pearson’s type statistic and find its asymptotic d.f. under the hypothesis and the local alternatives. We establish the asymptotic robustness of Pearson’s test as well.