The studentized empirical characteristic function and its application to test for the shape of distribution

Abstract

The empirical characteristic function is found to be effectively applied to test for the shape of distribution. The squared modulus of the studentized empirical characteristic function is suggested for testing the composite hypothesis that μ + σ X is subject to a known distribution, for unknown constants μ and σ. It is shown that the studentized empirical characteristic function, if properly normalized, converges weakly to a complex Gaussian process. Asymptotic considerations, as well as computer simulation, reveal that the proposed statistic, when applied to test normality, is more efficient than, or as efficient as, the test by the sample kurtosis, for certain types of alternatives.