Semiparametric Efficiency in Convexity Constrained Single-Index Model

We consider estimation and inference in a single-index regression model with an unknown convex link function. We introduce a convex and Lipschitz constrained least-square estimator (CLSE) for both the parametric and the nonparametric components given independent and identically distributed observati...

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Bibliographic Details
Published in:Journal of the American Statistical Association 2023-01, Vol.118 (541), p.272-286
Main Authors: Kuchibhotla, Arun K., Patra, Rohit K., Sen, Bodhisattva
Format: Article
Language:English
Subjects:
Algorithms
Bundled parameter
Convergence
Convexity
Data analysis
Efficiency
Errors
Errors with finite moments
Geometric proof of semiparametric efficiency
Lipschitz constrained least squares
Normality
Regression analysis
Regression models
Shape restricted function estimation
Statistical methods
Statistics
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Summary:We consider estimation and inference in a single-index regression model with an unknown convex link function. We introduce a convex and Lipschitz constrained least-square estimator (CLSE) for both the parametric and the nonparametric components given independent and identically distributed observations. We prove the consistency and find the rates of convergence of the CLSE when the errors are assumed to have only moments and are allowed to depend on the covariates. When , we establish -rate of convergence and asymptotic normality of the estimator of the parametric component. Moreover, the CLSE is proved to be semiparametrically efficient if the errors happen to be homoscedastic. We develop and implement a numerically stable and computationally fast algorithm to compute our proposed estimator in the R package simest. We illustrate our methodology through extensive simulations and data analysis. Finally, our proof of efficiency is geometric and provides a general framework that can be used to prove efficiency of estimators in a wide variety of semiparametric models even when they do not satisfy the efficient score equation directly. Supplementary files for this article are available online.
ISSN:0162-1459
1537-274X
DOI:10.1080/01621459.2021.1927741