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Exact Nonparametric Two-Sample Homogeneity Tests for Possibly Discrete Distributions

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  • Jean-Marie Dufour
  • Abdeljelil Farhat

Abstract

In this paper, we study several tests for the equality of two unknown distributions. Two are based on empirical distribution functions, three others on nonparametric probability density estimates, and the last ones on differences between sample moments. We suggest controlling the size of such tests (under nonparametric assumptions) by using permutational versions of the tests jointly with the method of Monte Carlo tests properly adjusted to deal with discrete distributions. We also propose a combined test procedure, whose level is again perfectly controlled through the Monte Carlo test technique and has better power properties than the individual tests which are combined. Finally, in a simulation experiment, we show that the technique suggested provides perfect control of test size and that the new tests proposed can yield sizeable power improvements. Dans ce texte, nous étudions plusieurs tests pour l'egalité de deux distributions inconnues. Deux de ces tests sont basés sur des fonctions de distribution empiriques, trois autres sur des estimateurs non-paramétriques de fonctions de densité, et les trois derniers sur des moments empiriques. Nous proposons de contrôler la taille des tests (sous des hypothèses non-paramétriques) en employant des versions permutationnelles de ces tests conjointement avec la méthode des tests de Monte Carlo ajustée pour tenir compte de la possibilité de distributions discontinues. Nous proposons aussi une méthode pour combiner plusieurs de ces tests, le niveau de ces procédures étant aussi contrôlé par la technique des tests de Monte Carlo, laquelle possède de meilleures propriétés de puissance que les tests individuels combinés. Finalement, nous montrons dans une étude de simulation que la technique suggérée contrôle parfaitement la taille des différents tests considérés et que les nouveaux tests proposés peuvent fournir de notables améliorations de puissance.

Suggested Citation

  • Jean-Marie Dufour & Abdeljelil Farhat, 2001. "Exact Nonparametric Two-Sample Homogeneity Tests for Possibly Discrete Distributions," CIRANO Working Papers 2001s-56, CIRANO.
  • Handle: RePEc:cir:cirwor:2001s-56
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    File URL: http://www.cirano.qc.ca/files/publications/2001s-56.pdf
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    References listed on IDEAS

    as
    1. Kiviet, Jan F. & Dufour, Jean-Marie, 1997. "Exact tests in single equation autoregressive distributed lag models," Journal of Econometrics, Elsevier, vol. 80(2), pages 325-353, October.
    2. Dufour, Jean-Marie, 1989. "Nonlinear Hypotheses, Inequality Restrictions, and Non-nested Hypotheses: Exact Simultaneous Tests in Linear Regressions," Econometrica, Econometric Society, vol. 57(2), pages 335-355, March.
    3. Dufour, Jean-Marie & Torres, Olivier, 2000. "Markovian processes, two-sided autoregressions and finite-sample inference for stationary and nonstationary autoregressive processes," Journal of Econometrics, Elsevier, vol. 99(2), pages 255-289, December.
    4. Jean-Marie Dufour & Jan F. Kiviet, 1998. "Exact Inference Methods for First-Order Autoregressive Distributed Lag Models," Econometrica, Econometric Society, vol. 66(1), pages 79-104, January.
    5. Dufour, Jean-Marie, 2006. "Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics," Journal of Econometrics, Elsevier, vol. 133(2), pages 443-477, August.
    6. G. Noether, 1963. "Note on the kolmogorov statistic in the discrete case," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 7(1), pages 115-116, December.
    7. Jean-Marie Dufour & Abdeljelil Farhat & Lucien Gardiol & Lynda Khalaf, 1998. "Simulation-based finite sample normality tests in linear regressions," Econometrics Journal, Royal Economic Society, vol. 1(Conferenc), pages 154-173.
    8. Dufour, Jean-Marie, 1990. "Exact Tests and Confidence Sets in Linear Regressions with Autocorrelated Errors," Econometrica, Econometric Society, vol. 58(2), pages 475-494, March.
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    Cited by:

    1. Kidd, Willis V. & Brorsen, B. Wade, 2004. "Why have the returns to technical analysis decreased?," Journal of Economics and Business, Elsevier, vol. 56(3), pages 159-176.

    More about this item

    Keywords

    Nonparametric methods; two-sample problem; discrete distribution; discontinuous distribution; goodness-of-fit test; Kolmogorov-Smirnov test; Cramér-von Mises; kernel density estimator; exact test; permutation test; Monte Carlo test; bootstrap; combined test procedure; induced test; Méthodes non-paramétriques; problème des deux échantillons; distribution discrète; distribution discontinue; test d'ajustement; test de Kolmogorov-Smirnov; estimateur à noyau pour une densité; test exact; test de permutations; test de Monte Carlo; bootstrap; test combiné; test induit;

    JEL classification:

    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General

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