GOODNESS-OF-FIT TESTS FOR THE CAUCHY DISTRIBUTION BASED ON THE EMPIRICAL CHARACTERISTIC FUNCTION

NORA GÜRTLER AND NORBERT HENZE

Institut für Mathematische Stochastik, Universität Karlsruhe, Englerstr. 2, 76128 Karlsruhe, Germany

(Received July 31, 1998; revised December 25, 1998)

Abstract. Let X_{1}, …, X_{n} be independent observations on a random variable X. This paper considers a class of omnibus procedures for testing the hypothesis that the unknown distribution of X belongs to the family of Cauchy laws. The test statistics are weighted integrals of the squared modulus of the difference between the empirical characteristic function of the suitably standardized data and the characteristic function of the standard Cauchy distribution. A large-scale simulation study shows that the new tests compare favorably with the classical goodness-of-fit tests for the Cauchy distribution, based on the empirical distribution function. For small sample sizes and short–tailed alternatives, the uniformly most powerful invariant test of Cauchy versus normal beats all other tests under discussion.

Key words and phrases: Goodness-of-fit test, Cauchy distribution, empirical characteristic function, kernel transformed empirical process, stable distribution, uniformly most powerful invariant test.

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