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Testing equality of functions under monotonicity constraints

Author

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  • Cécile Durot
  • Piet Groeneboom
  • Hendrik P. Lopuhaä

Abstract

We consider the problem of testing equality of functions f j :[ a, b ]→ℝ for j =1, 2, ..., J on the basis of J independent samples from possibly different distributions under the assumption that the functions are monotone. We provide a uniform approach that covers testing equality of monotone regression curves, equality of monotone densities and equality of monotone hazards in the random censorship model. Two test statistics are proposed based on L 1 -distances. We show that both statistics are asymptotically normal and we provide bootstrap implementations, which are shown to have critical regions with asymptotic level α.

Suggested Citation

  • Cécile Durot & Piet Groeneboom & Hendrik P. Lopuhaä, 2013. "Testing equality of functions under monotonicity constraints," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(4), pages 939-970, December.
    • Handle: RePEc:taf:gnstxx:v:25:y:2013:i:4:p:939-970
    DOI: 10.1080/10485252.2013.826356
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    References listed on IDEAS

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    4. Piet Groeneboom, 2012. "Likelihood Ratio Type Two-Sample Tests for Current Status Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 39(4), pages 645-662, December.
    5. King, Eileen & Hart, Jeffrey D. & Wehrly, Thomas E., 1991. "Testing the equality of two regression curves using linear smoothers," Statistics & Probability Letters, Elsevier, vol. 12(3), pages 239-247, September.
    6. 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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    Cited by:

    1. Hendrik P. Lopuhaä & Eni Musta, 2017. "Smooth estimation of a monotone hazard and a monotone density under random censoring," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 71(1), pages 58-82, January.
    2. Holger Dette & Subhra Sankar Dhar & Weichi Wu, 2021. "Identifying shifts between two regression curves," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(5), pages 855-889, October.
    3. Jonas Peters & Peter Bühlmann & Nicolai Meinshausen, 2016. "Causal inference by using invariant prediction: identification and confidence intervals," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(5), pages 947-1012, November.
    4. Lopuhaä, Hendrik P. & Musta, Eni, 2018. "The distance between a naive cumulative estimator and its least concave majorant," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 119-128.

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