"Goodness-of-Fit Tests for Symmetric Stable Distributions - Empirical Characteristic Function Approach"

We consider goodness-of fit tests of 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 exponentƒ¿ estimated from the data. We treat ƒ¿ as an unknown parameter, but for theoretical simplicity we also consider the case that ƒ¿ is fixed. For estimation of parameters and the standardization of data we use maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE) which minimizes the weighted integral. We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE and EISE. For the case of MLE, the eigenvalues of the covariance function are numerically evaluated and asymptotic distribution of the test statistic is obtained using complex integration. Simulation studies show that the asymptotic distribution of the test statistics is very accurate. 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.

"Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE"

We consider goodness-of-fit tests of Cauchy distribution based on weighted integrals of the squared distance of the difference between the empirical characteristic function of the standardized data and the characteristic function of the standard Cauchy distribution. For standardization of data Gurtler and Henze (2000) used the median and the interquartile range. In this paper we use maximum likelihood estimator (MLE)and an equivariant integrated squared error estimator (EISE), which minimizes the weighted integral. We derive an explicit form of the asymptotic covariance function of the characteristic function process with parameters estimated by MLE or EISE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distribution of the test statistics are obtained by the residue theorem. Simulation study shows that the proposed tests compare well to tests proposed by Gurtler and Henze (2000) and more traditional tests based on the empirical distribution function.

Skewness and kurtosis as locally best invariant tests of normality

Consider testing normality against a one-parameter family of univariate distributions containing the normal distribution as the boundary, e.g., the family of $t$-distributions or an infinitely divisible family with finite variance. We prove that under mild regularity conditions, the sample skewness is the locally best invariant (LBI) test of normality against a wide class of asymmetric families and the kurtosis is the LBI test against symmetric families. We also discuss non-regular cases such as testing normality against the stable family and some related results in the multivariate cases