Nonparametric regression estimation using weak separability

Abstract

In this paper I propose three new estimators of nonparametric regression functions subject to weak separability (WS). The use of WS reduces the curse of dimensionality. WS nests other separability concepts such as (generalized) additive separability ((G)AS). The advantage of WS over (G)AS is that WS allows for interactions between regressors whereas (G)AS does not permit any interactions. The estimators use marginal integration and are shown to have a limiting normal distribution and a convergence rate which is the same as that of an unconstrained nonparametric estimator of a regression function of lower dimension. An attractive and unusual feature of two of my estimators is that regressors can have arbitrary convex support and that the integration regions can depend on the values of the remaining variables. The estimators can be iterated and I show that under strong assumptions further asymptotic efficiency improvements are possible. The computation of the estimators is simple. The performance of one of the estimators is studied in a simulation study.