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

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 investigat...

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Bibliographic Details
Published inScandinavian journal of statistics Vol. 47; no. 4; pp. 1192 - 1221
Main Authors Bagchi, Pramita, Sankar Dhar, Subhra
Format Journal Article
LanguageEnglish
Published Oxford Blackwell Publishing Ltd 01.12.2020
Subjects
consistency
convex function
cumulative sum diagram
Error analysis
Fixed points (mathematics)
Least squares
Multivariate analysis
nonstandard asymptotic distribution
rate of convergence
Regression models
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Summary: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.
ISSN:0303-6898
1467-9469
DOI:10.1111/sjos.12459