CIRJE-F-226 "Empirical Characteristic Function Approach to Goodness-of-Fit Tests for the Cauchy Distribution with Parameters Estimated by MLE or EISE"
Author Name Matsui, Muneya and Akimichi Takemura
Date June 2003
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Remarks Subsequently published in Annals of Institute of Statistical Mathematics, Institute of Statistical Mathematics.

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.