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On automatic kernel density estimate-based tests for goodness-of-fit

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  • Carlos Tenreiro

    (University of Coimbra)

Abstract

Although estimation and testing are different statistical problems, if we want to use a test statistic based on the Parzen–Rosenblatt estimator to test the hypothesis that the underlying density function f is a member of a location-scale family of probability density functions, it may be found reasonable to choose the smoothing parameter in such a way that the kernel density estimator is an effective estimator of f irrespective of which of the null or the alternative hypothesis is true. In this paper we address this question by considering the well-known Bickel–Rosenblatt test statistics which are based on the quadratic distance between the nonparametric kernel estimator and two parametric estimators of f under the null hypothesis. For each one of these test statistics we describe their asymptotic behaviours for a general data-dependent smoothing parameter, and we state their limiting Gaussian null distribution and the consistency of the associated goodness-of-fit test procedures for location-scale families. In order to compare the finite sample power performance of the Bickel–Rosenblatt tests based on a null hypothesis-based bandwidth selector with other bandwidth selector methods existing in the literature, a simulation study for the normal, logistic and Gumbel null location-scale models is included in this work.

Suggested Citation

  • Carlos Tenreiro, 2022. "On automatic kernel density estimate-based tests for goodness-of-fit," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 717-748, September.
  • Handle: RePEc:spr:testjl:v:31:y:2022:i:3:d:10.1007_s11749-021-00799-3
    DOI: 10.1007/s11749-021-00799-3
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    References listed on IDEAS

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    1. Ricardo Cao & Gábor Lugosi, 2005. "Goodness‐of‐fit Tests Based on the Kernel Density Estimator," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 599-616, December.
    2. C. Tenreiro, 2017. "A weighted least-squares cross-validation bandwidth selector for kernel density estimation," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(7), pages 3438-3458, April.
    3. Tenreiro, Carlos, 2001. "On the asymptotic behaviour of the integrated square error of kernel density estimators with data-dependent bandwidth," Statistics & Probability Letters, Elsevier, vol. 53(3), pages 283-292, June.
    4. Fan, Yanqin, 1998. "Goodness-Of-Fit Tests Based On Kernel Density Estimators With Fixed Smoothing Parameters," Econometric Theory, Cambridge University Press, vol. 14(5), pages 604-621, October.
    5. Hall, Peter, 1984. "Central limit theorem for integrated square error of multivariate nonparametric density estimators," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 1-16, February.
    6. Fan, Yanqin, 1994. "Testing the Goodness of Fit of a Parametric Density Function by Kernel Method," Econometric Theory, Cambridge University Press, vol. 10(2), pages 316-356, June.
    7. Norbert Henze, 2002. "Invariant tests for multivariate normality: a critical review," Statistical Papers, Springer, vol. 43(4), pages 467-506, October.
    8. Gouriéroux, Christian & Tenreiro, Carlos, 2001. "Local Power Properties of Kernel Based Goodness of Fit Tests," Journal of Multivariate Analysis, Elsevier, vol. 78(2), pages 161-190, August.
    9. T.W. Epps, 2005. "Tests for location-scale families based on the empirical characteristic function," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 62(1), pages 99-114, September.
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