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Non-parametric k-sample tests: Density functions vs distribution functions

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  • Martínez-Camblor, Pablo
  • de Uña-Álvarez, Jacobo

Abstract

Tests for the comparison of k samples based on kernel density estimators (KDE) are introduced. The Double Minimum method as a new and useful procedure for the crucial problem of bandwidth selection is developed. The statistical power of the proposed tests, as well as the impact of the smoothing degree and the performance of the Double Minimum algorithm, are studied via Monte Carlo simulations. Finally, the results of the tests based on the KDE are compared to those of the traditional k-sample tests based on empirical distribution functions (EDF), and to other tests based on the likelihood ratio introduced in the recent literature. Two main conclusions are obtained. First, the proposed bandwidth selection method attains quasi-optimal results. Second, the simulations suggest that KDE-based tests are the most powerful when the underlying populations are different in shape, and that the L1 distance among densities leads to optimal results in the considered situations.

Suggested Citation

  • Martínez-Camblor, Pablo & de Uña-Álvarez, Jacobo, 2009. "Non-parametric k-sample tests: Density functions vs distribution functions," Computational Statistics & Data Analysis, Elsevier, vol. 53(9), pages 3344-3357, July.
    • Handle: RePEc:eee:csdana:v:53:y:2009:i:9:p:3344-3357
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    References listed on IDEAS

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    1. Duc Devroye & J. Beirlant & R. Cao & R. Fraiman & P. Hall & M. Jones & Gábor Lugosi & E. Mammen & J. Marron & C. Sánchez-Sellero & J. Uña & F. Udina & L. Devroye, 1997. "Universal smoothing factor selection in density estimation: theory and practice," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 6(2), pages 223-320, December.
    2. 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.
    3. Alba Fernández, V. & Jiménez Gamero, M.D. & Muñoz Garcia, J., 2008. "A test for the two-sample problem based on empirical characteristic functions," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3730-3748, March.
    4. Zhang, Jin & Wu, Yuehua, 2007. "k-Sample tests based on the likelihood ratio," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4682-4691, May.
    5. Anderson, N. H. & Hall, P. & Titterington, D. M., 1994. "Two-Sample Test Statistics for Measuring Discrepancies Between Two Multivariate Probability Density Functions Using Kernel-Based Density Estimates," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 41-54, July.
    6. 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.
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    1. Martínez-Camblor, Pablo & Corral, Norberto, 2011. "Repeated measures analysis for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3244-3256, December.
    2. Chen, Yuhui & Hanson, Timothy E., 2014. "Bayesian nonparametric k-sample tests for censored and uncensored data," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 335-346.
    3. Álvarez-Esteban, P.C. & del Barrio, E. & Cuesta-Albertos, J.A. & Matrán, C., 2013. "Searching for a common pooling pattern among several samples," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 1-14.

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