AISM 52, 267-286

## 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.

**Source**
( TeX , DVI )