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Abstract:
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We consider estimation and inference in a single index regression model with an unknown convex link function. We propose two estimators for the unknown link function: (1) a shape-constrained smoothing spline estimator and (2) a Lipschitz constrained least squares estimator. Moreover, both these procedures lead to estimators for the unknown finite dimensional parameter. We develop methods to compute both the penalized least squares estimator (PLSE) and the Lipschitz constrained least squares estimator (LLSE) of the parametric and the nonparametric components given independent and identically distributed data. We prove the consistency and find the rates of convergence for both the PLSE and the LLSE. For both the PLSE and the LLSE, we establish root-n convergence and semiparametric efficiency of the parametric component under mild assumptions. We illustrate the developed R package simest to compute the proposed estimators. Our proposed algorithm works even when n is modest and d is large (e.g., n = 500, and d = 100). This is joint work with Arun Kumar Kuchibhotla and Bodhisattva Sen.
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