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Testing equality of a large number of densities

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  • D. Zhan
  • J. D. Hart

Abstract

The problem of testing equality of a large number of densities is considered. The classical k-sample problem compares a small, fixed number of distributions and allows the sample size from each distribution to increase without bound. In our asymptotic analysis the number of distributions tends to infinity but the size of individual samples remains fixed. The proposed test statistic is motivated by the simple idea of comparing kernel density estimators from the various samples to the average of all density estimators. However, a novel interpretation of this familiar type of statistic arises upon centring it. The asymptotic distribution of the statistic under the null hypothesis of equal densities is derived, and power against local alternatives is considered. It is shown that a consistent test is attainable in many situations where all but a vanishingly small proportion of densities are equal to each other. The test is studied via simulation, and an illustration involving microarray data is provided.

Suggested Citation

  • D. Zhan & J. D. Hart, 2014. "Testing equality of a large number of densities," Biometrika, Biometrika Trust, vol. 101(2), pages 449-464.
    • Handle: RePEc:oup:biomet:v:101:y:2014:i:2:p:449-464.
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    File URL: http://hdl.handle.net/10.1093/biomet/asu002
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    Citations

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    Cited by:

    1. Cousido-Rocha, Marta & de Uña-Álvarez, Jacobo & Hart, Jeffrey D., 2019. "A two-sample test for the equality of univariate marginal distributions for high-dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    2. Marta Cousido-Rocha & Jacobo Uña-Álvarez & Jeffrey D. Hart, 2019. "Testing equality of a large number of densities under mixing conditions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(4), pages 1203-1228, December.
    3. Hart, Jeffrey D., 2016. "A nonparametric test of stationarity for independent data," Statistics & Probability Letters, Elsevier, vol. 108(C), pages 40-44.
    4. M. D. Jiménez-Gamero & M. Cousido-Rocha & M. V. Alba-Fernández & F. Jiménez-Jiménez, 2022. "Testing the equality of a large number of populations," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(1), pages 1-21, March.
    5. Jiménez-Gamero, M. Dolores & Franco-Pereira, Alba M., 2021. "Testing the equality of a large number of means of functional data," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    6. Reza Modarres, 2020. "Graphical Comparison of High‐Dimensional Distributions," International Statistical Review, International Statistical Institute, vol. 88(3), pages 698-714, December.

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