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Abstract:
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Consider the partially linear model (PLM) with random design: Y=X^T\beta^*+g(W)+u, where g(.) is an unknown nonlinear real function, X is p-dimensional, W is one-dimensional, and \beta^* is s-sparse. Our aim is to efficiently estimate \beta^* based on n i.i.d. observations of (Y,X,W) with n< p. For this, the best theoretical results to date rest on the following three assumptions: (i) g(.) belongs to some smooth enough function class G (e.g., globally Lipschitz); (ii) s^2\log p/n\to 0; and (iii) u is subgaussian. These assumptions are related to long-standing problems in nonparametric statistics, and are arguably difficult to relax. In this paper, Honore and Powell's pairwise difference approach (plus a lasso-type penalty) is shown to attain rate-optimality with all three assumptions relaxed. Particularly, rate-optimality proves attainable in a certain regime when g(.) is even a sharply discontinuous piecewise Holder function, a result new to both statistics and econometrics communities. The proof rests on a general method for determining the estimation accuracy of a "contaminated" M-estimator, new U-statistics tools, and function perturbation theory.
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