Goodness-of-fit tests for symmetric stable distributions—Empirical characteristic function approach

Abstract

We consider goodness-of-fit tests for symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard symmetric stable distribution with the characteristic exponent α estimated from the data. We treat α as an unknown parameter, but for theoretical simplicity we also consider the case that α is fixed. For estimation of parameters and the standardization of data we use the maximum likelihood estimator (MLE). We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distribution of the test statistic is obtained using complex integration. We find that if the sample size is large the calculated asymptotic critical values of test statistics coincide with the simulated finite sample critical values. Finite sample power of the proposed test is examined. We also present a formula of the asymptotic covariance function of the characteristic function process with parameters estimated by an efficient estimator for general distributions.