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On mode testing and empirical approximations to distributions

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  • Cheng, Ming-Yen
  • Hall, Peter

Abstract

There are close connections between the theory of statistical inference under order restrictions and the theory of tests for unimodality. In particular, a result of Kiefer and Wolfowitz, on the error of convex approximations to empirical distribution functions, is basic to limit theory for the dip test for unimodality. We develop a version of Kiefer and Wolfowitz' result in the context of distributions that are strongly unimodal, and apply it and related limit theory to compare the powers, against local alternatives, of three different tests of unimodality. In this context it is shown that the dip, excess mass and bandwidth tests are all able to detect departures of size n-3/5 (measured in terms of the distribution function) from the null hypothesis, where n denotes sample size; but are not able to detect departures of smaller order. Thus, they have similar powers.

Suggested Citation

  • Cheng, Ming-Yen & Hall, Peter, 1998. "On mode testing and empirical approximations to distributions," Statistics & Probability Letters, Elsevier, vol. 39(3), pages 245-254, August.
    • Handle: RePEc:eee:stapro:v:39:y:1998:i:3:p:245-254
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    References listed on IDEAS

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    1. Hall, Peter & Titterington, D. M., 1988. "On confidence bands in nonparametric density estimation and regression," Journal of Multivariate Analysis, Elsevier, vol. 27(1), pages 228-254, October.
    2. P. M. Hartigan, 1985. "Computation of the Dip Statistic to Test for Unimodality," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 34(3), pages 320-325, November.
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    1. Hsu, Chih-Yuan & Wu, Tiee-Jian, 2013. "Efficient estimation of the mode of continuous multivariate data," Computational Statistics & Data Analysis, Elsevier, vol. 63(C), pages 148-159.
    2. Polonik, Wolfgang & Wang, Zailong, 2005. "Estimation of regression contour clusters--an application of the excess mass approach to regression," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 227-249, June.

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