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Goodness-of-Fit Tests for Symmetric Stable Distributions - Empirical Characteristic Function Approach


  • Muneya Matsui

    (Graduate School of Economics, University of Tokyo)

  • Akimichi Takemura

    (Graduate School of Information Science and Technology, University of Tokyo)


We consider goodness-of fit tests of symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard symmetric stable distribution with the characteristic exponentƒ¿ estimated from the data. We treat ƒ¿ as an unknown parameter, but for theoretical simplicity we also consider the case that ƒ¿ is fixed. For estimation of parameters and the standardization of data we use maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE) which minimizes the weighted integral. We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE and EISE. For the case of MLE, the eigenvalues of the covariance function are numerically evaluated and asymptotic distribution of the test statistic is obtained using complex integration. Simulation studies show that the asymptotic distribution of the test statistics is very accurate. We also present a formula of the asymptotic covariance function of the characteristic function process with parameters estimated by an efficient estimator for general distributions.

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  • Muneya Matsui & Akimichi Takemura, 2005. "Goodness-of-Fit Tests for Symmetric Stable Distributions - Empirical Characteristic Function Approach," CIRJE F-Series CIRJE-F-384, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2005cf384

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    References listed on IDEAS

    1. Besbeas, Panagiotis & Morgan, Byron J. T., 2001. "Integrated squared error estimation of Cauchy parameters," Statistics & Probability Letters, Elsevier, vol. 55(4), pages 397-401, December.
    2. Henze, Norbert & Wagner, Thorsten, 1997. "A New Approach to the BHEP Tests for Multivariate Normality," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 1-23, July.
    3. Nora Gürtler & Norbert Henze, 2000. "Goodness-of-Fit Tests for the Cauchy Distribution Based on the Empirical Characteristic Function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(2), pages 267-286, June.
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    1. Muneya Matsui & Akimichi Takemura, 2008. "Goodness-of-fit tests for symmetric stable distributions—Empirical characteristic function approach," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(3), pages 546-566, November.

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