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Pointwise comparison of two multivariate density functions

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  • Martin L. Hazelton
  • Tilman M. Davies

Abstract

Testing the equality of two density functions based on independent samples is a classical problem in statistics. While the focus is often on global equality, it is also of interest to conduct local comparisons of density functions. Typically a type of Wald statistic is employed, where the local difference in densities is standardized by an estimate of the asymptotic standard error of that difference. We study the null distribution of this test statistic. The literature has suggested that this will be asymptotically standard normal, but we show that this is by no means always the case. In particular, when using bandwidth matrices of optimal order (for estimation), we prove that the asymptotic mean of this null distribution is nonzero when either the sample sizes differ, or when the Hessian matrices of the densities differ at the point where the densities are equal. In numerical studies we find the erroneous use of the standard normal null distribution in such cases can severely corrupt the test size. We show that these problems can be managed effectively by using common bandwidths when the Hessian matrices are equal, and applying adjusted undersmoothing bandwidth matrices when they are not.

Suggested Citation

  • Martin L. Hazelton & Tilman M. Davies, 2022. "Pointwise comparison of two multivariate density functions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(4), pages 1791-1810, December.
    • Handle: RePEc:bla:scjsta:v:49:y:2022:i:4:p:1791-1810
    DOI: 10.1111/sjos.12565
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    References listed on IDEAS

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    2. Peter J. Diggle, 1990. "A Point Process Modelling Approach to Raised Incidence of a Rare Phenomenon in the Vicinity of a Prespecified Point," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 153(3), pages 349-362, May.
    3. Signorini, D.F. & Jones, M.C., 2004. "Kernel Estimators for Univariate Binary Regression," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 119-126, January.
    4. Tarn Duong, 2013. "Local significant differences from nonparametric two-sample tests," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(3), pages 635-645, September.
    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.
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