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.
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References
Baringhaus, L. and Henze, N. (1991). A class of consistent tests for exponentiality based on the empirical Laplace transform, Ann. Inst. Statist. Math., 43, 551–564.
Baringhaus, L., Gürtler, N. and Henze, N. (2000). Weighted and components of smooth tests of fit, Australian and New Zealand Journal of Statistics, 42 (to appear).
Burke, M., Csörgő, M., Csörgő, S. and Révész, P. (1978). Approximations of the empirical process when parameters are estimated, Ann. Probab., 7(5), 790–810.
Chernoff, H., Gastwirth, J. and Johns, M. (1967). Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation, Ann. Math. Statist., 38, 52–72.
Csörgő, S. (1981). The empirical characteristic process when parameters are estimated, Contributions to probability (eds. J. Gani and V. Rohatgi), 215–230, Academic Press, New York.
Csörgő, S. (1983). Kernel-transformed empirical processes, J. Multivariate. Anal., 13, 517–533.
D'Agostino, R. and Stephens, M. (1986). Goodness-of-Fit Techniques, Marcel Dekker, New York.
Durbin, J. (1973). Weak convergence of the sample distribution function when parameters are estimated, Ann. Statist., 1(2), 279–290.
Epps, T. and Pulley, L. (1983). A test for normality based on the empirical characteristic function, Biometrika, 70(3), 723–726.
Epps, T. and Pulley, L. (1986). A test of exponentiality vs. monotone-hazard alternatives derived from the empirical characteristic function, J. Roy. Statist. Soc. Ser. B, 48(2), 206–213.
Franck, W. (1981). The most powerful invariant test of normal versus Cauchy with applications to stable alternatives, J. Amer. Statist. Assoc., 76(376), 1002–1005.
Hájek, J. and Šidák, Z. (1967). Theory of rank tests, Academic Press, New York.
Henze, N. (1993). A new flexible class of omnibus tests for exponentiality, Comm. Statist. Theory Methods., 22, 115–133.
Henze, N. (1997). Extreme smoothing and testing for multivariate normality, Statist. Probab. Lett., 35, 203–213.
Henze, N. and Wagner, T. (1997). A new approach to the BHEP tests for multivariate normality, J. Multivariate Anal., 62(1), 1–23.
Henze, N. and Zirkler, B. (1990). A class of invariant consistent tests for multivariate normality, Comm. Statist. Theory Methods., 19(10), 3595–3617.
Johnson, N., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, 2nd ed., Wiley, New York.
Jörgens, K. (1970). Lineare Integraloperatoren, Teubner, Stuttgart.
Karatzas, I. and Shreve, S. (1988). Brownian Motion and Stochastic Calculus, Springer, New York.
Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent rv's, and the sample df, Z. W. theorie Verw. Geb., 32, 111–131.
Meintanis, S. (1997). Goodness-of-Fit tests for Cauchy Distributions Derived from the Empirical Characteristic Function (Preprint). Greece: University of Patras.
Serfling, R. (1980). Approximation Theorems of Mathernatical Statistics, Wiley, New York.
Stephens, M. (1991). Tests of fit for the Cauchy distribution based on the empirical distribution function, Teck. Report, No. 449, Department of Statistics, Stanford, California.
Stigler, S. (1974). Studies in the history of probability and statistics. XXXIII. Cauchy and the witch of Agnesi: An historical note on the Cauchy distribution, Biometrika, 61, 375–379.
Widder, D. (1959). The Laplace Transform, 5th ed, Princeton University Press, Princeton.
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Gürtler, N., Henze, N. Goodness-of-Fit Tests for the Cauchy Distribution Based on the Empirical Characteristic Function. Annals of the Institute of Statistical Mathematics 52, 267–286 (2000). https://doi.org/10.1023/A:1004113805623
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DOI: https://doi.org/10.1023/A:1004113805623