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Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE. (English) Zbl 1083.62018

Summary: We consider goodness-of-fit tests of the Cauchy distribution based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard Cauchy distribution. For standardization of data N. Gürtler and N. Henze [Ann. Inst. Stat. Math. 52, 267–286 (2000; Zbl 0959.62041)] used the median and the interquartile range. We use the 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 the MLE or the EISE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distributions of the test statistics are obtained by the residue theorem. A simulation study shows that the proposed tests compare well to tests proposed by Gürtler and Henze and more traditional tests based on the empirical distribution function.

MSC:

62F05 Asymptotic properties of parametric tests
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics

Citations:

Zbl 0959.62041
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References:

[1] Anderson, T. W. and Darling, D. A. (1952). Asymptotic theory of certain ”goodness of fit” criteria based on stochastic processes.The Annals of Mathematical Statistics,23, 193–212. · Zbl 0048.11301 · doi:10.1214/aoms/1177729437
[2] Baker, C. T. H. (1977).The Numerical Treatment of Integral Equations, Clarendon Press, Oxford. · Zbl 0373.65060
[3] Besbeas, P. and Morgan, B. (2001). Integrated squared error estimation of Cauchy parameters,Statistics & Probability Letters,55, 397–401. · Zbl 0994.62014 · doi:10.1016/S0167-7152(01)00153-5
[4] Copas, J. B. (1975). On the unimodality of the likelihood for the Cauchy distribution,Biometrika,62, 701–704. · Zbl 0321.62037 · doi:10.1093/biomet/62.3.701
[5] Csörgo, S. (1983). Kernel-transformed empirical process,Journal of Multivariate Analysis,13, 517–533. · Zbl 0559.62012 · doi:10.1016/0047-259X(83)90037-4
[6] Durbin, J. (1973a).Distribution Theory for Tests Based on the Sample Distribution Function, SIAM, Philadelphia. · Zbl 0267.62002
[7] Durbin, J. (1973b). Weak convergence of the sample distribution function when parameters are estimated.The Annals of Statistics,1, 279–290. · Zbl 0256.62021 · doi:10.1214/aos/1176342365
[8] Gürtler, N. and Henze, N. (2000). Goodness-of-fit tests for the Cauchy distribution based on the empirical characteristic function.Annals of the Institute of Statistical Mathematics,52, 267–286. · Zbl 0959.62041 · doi:10.1023/A:1004113805623
[9] Hammerstein, A. (1927). Über Entwicklungen gegebener Funktionen nach Eigenfunktionen von Randwertaufgaben,Mathematische Zeitschrift,27, 269–311. · JFM 53.0466.01 · doi:10.1007/BF01171100
[10] Maesono, Y. (2001).Toukeitekisuisoku no Zenkinriron, Kyushu University Press, Fukuoka, Japan (in Japanese).
[11] Matsui, M. and Takemura, A. (2003). Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE, Discussion Paper CIEJE 2003-CF-226, Center for International Research on the Japanese Economy, Faculty of Economics, University of Tokyo, available from http://www.e.u-tokyo.ac.jp/cirje/research/dp/2003/list.htm.
[12] Serfling, R. J. (1980).Approximation Theorems of Mathematical Statistics, John Wiley, New York. · Zbl 0538.62002
[13] Slepian, D. (1957). Fluctuations of random noise power,Bell System Technical Journal,37, 163–184.
[14] Tanaka, K. (1996).Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, John Wiley, New York. · Zbl 0861.62062
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