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Confidence intervals in monotone regression

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  • Piet Groeneboom
  • Geurt Jongbloed

Abstract

We construct bootstrap confidence intervals for a monotone regression function. It has been shown that the ordinary nonparametric bootstrap, based on the nonparametric least squares estimator (LSE) f^n, is inconsistent in this situation. We show that an n2/5‐consistent bootstrap can be based on the smoothed f^n, to be called the SLSE (Smoothed Least Squares Estimator). The asymptotic pointwise distribution of the SLSE is derived. The confidence intervals, based on the smoothed bootstrap, are compared to intervals based on the (not necessarily monotone) Nadaraya Watson estimator and the effect of Studentization is investigated. We also give a method for automatic bandwidth choice, correcting work in Sen and Xu (2015). Analogous methods for constructing confidence intervals in the current status model are discussed, improving on work in Groeneboom and Hendrickx (2018).

Suggested Citation

  • Piet Groeneboom & Geurt Jongbloed, 2024. "Confidence intervals in monotone regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 51(4), pages 1749-1781, December.
    • Handle: RePEc:bla:scjsta:v:51:y:2024:i:4:p:1749-1781
    DOI: 10.1111/sjos.12730
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    References listed on IDEAS

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    1. Moulinath Banerjee & Jon A. Wellner, 2005. "Confidence Intervals for Current Status Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(3), pages 405-424, September.
    2. Groeneboom,Piet & Jongbloed,Geurt, 2014. "Nonparametric Estimation under Shape Constraints," Cambridge Books, Cambridge University Press, number 9780521864015, November.
    3. Piet Groeneboom & Kim Hendrickx, 2018. "Confidence Intervals for the Current Status Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 45(1), pages 135-163, March.
    4. Manski, Charles F., 1975. "Maximum score estimation of the stochastic utility model of choice," Journal of Econometrics, Elsevier, vol. 3(3), pages 205-228, August.
    5. Timothy B. Armstrong, 2014. "Adaptive Testing on a Regression Function at a Point," Cowles Foundation Discussion Papers 1957R, Cowles Foundation for Research in Economics, Yale University, revised Feb 2015.
    6. Jason Abrevaya & Jian Huang, 2005. "On the Bootstrap of the Maximum Score Estimator," Econometrica, Econometric Society, vol. 73(4), pages 1175-1204, July.
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