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A study on the least squares estimator of multivariate isotonic regression function

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  • Pramita Bagchi
  • Subhra Sankar Dhar

Abstract

Consider the problem of pointwise estimation of f in a multivariate isotonic regression model Z=f(X1,…,Xd)+ϵ, where Z is the response variable, f is an unknown nonparametric regression function, which is isotonic with respect to each component, and ϵ is the error term. In this article, we investigate the behavior of the least squares estimator of f. We generalize the greatest convex minorant characterization of isotonic regression estimator for the multivariate case and use it to establish the asymptotic distribution of properly normalized version of the estimator. Moreover, we test whether the multivariate isotonic regression function at a fixed point is larger (or smaller) than a specified value or not based on this estimator, and the consistency of the test is established. The practicability of the estimator and the test are shown on simulated and real data as well.

Suggested Citation

  • Pramita Bagchi & Subhra Sankar Dhar, 2020. "A study on the least squares estimator of multivariate isotonic regression function," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(4), pages 1192-1221, December.
    • Handle: RePEc:bla:scjsta:v:47:y:2020:i:4:p:1192-1221
    DOI: 10.1111/sjos.12459
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    References listed on IDEAS

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    2. Pramita Bagchi & Moulinath Banerjee & Stilian A. Stoev, 2016. "Inference for Monotone Functions Under Short- and Long-Range Dependence: Confidence Intervals and New Universal Limits," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(516), pages 1634-1647, October.
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    4. Paranjape, S. R. & Park, C., 1973. "Laws of iterated logarithm of multiparameter wiener processes," Journal of Multivariate Analysis, Elsevier, vol. 3(1), pages 132-136, March.
    5. Subhra sankar Dhar, 2016. "Trimmed Mean Isotonic Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(1), pages 202-212, March.
    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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