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Semiparametric Efficiency in Convexity Constrained Single-Index Model

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Version 2 2021-07-26, 18:20
Version 1 2021-05-17, 15:20
journal contribution
posted on 2021-07-26, 18:20 authored by Arun K. Kuchibhotla, Rohit K. Patra, Bodhisattva Sen

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 q2 moments and are allowed to depend on the covariates. When q5, we establish n1/2-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.

Funding

This work was supported by NSF (grants nos. DMS-17-12822 and AST-16-14743).

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