Rate optimal estimation with the integration method in the presence of many covariates

By N. Hengartner, Stefan Sperlich. 2005

64 citations recorded by the source corpus.

For multivariate regressors, integrating the Nadaraya-Watson regression smoother produces estimators of the lower-dimensional marginal components that are asymptotically normally distributed, at the optimal rate of convergence. Some heuristics, based on consistency of the pilot estimator, suggested that the estimator would not converge at the optimal rate of convergence in the presence of more than four covariates. This paper shows first that marginal integration with its internally normalized counterpart leads to rate-optimal estimators of the marginal components. We introduce the necessary modifications and give central limit theorems. Then, it is shown that the method apply also to more general models, in particular we discuss feasible estimation of partial linear models. The proofs reveal that the pilot estimator shall over-smooth the variables to be integrated, and, that the resulting estimator is itself a lower-dimensional regression smoother. Hence, finite sample properties of the estimator are comparable to those of low-dimensional nonparametric regression. Further advantages when starting with the internally normalized pilot estimator are its computational attractiveness and better performance (compared to its classical counterpart) when the covatiates are correlated and nonuniformly distributed. Simulation studies underline the excellent performance in comparison with so far known methods.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8cb3bddcda57722f826e2459b84e58cb6505d923

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

Citations in the corpus

Spline-backfitted kernel smoothing of nonlinear additive autoregression model (2006, unknown): “For instance, there are four types of kernel-based estimators: the classic backfitting estimators (CBE) of [9] and [19]; marginal integration estimators (MIE) of [6, 17, 16, 22, 30] and a kernel-based method of estimating rate to optimality of [10]; the smoothing backfitting estimators (SBE) of [18]”
Estimating Semiparametric Panel Data Models by Marginal Integration (2012, unknown): “For model identiﬁcation, we follow Hengartner and Sperlich (2005) and assume R C f ( u ) q ( u ) du = 0.”
Smooth backfitting in practice (2005, background): “…et al. (1999) or Dette et al. (2004)) that classical backfitting as well as classical marginal integration (but not so for the modified version of Hengartner and Sperlich (2001)) are strongly affected by increasing correlation in the design, i.e. they lose efficiency and finally break down.”
Estimation in semiparametric spatial regression (2006, methodology): “It is a special case of the conditions used by Napahetian (1987) and Lin and Lu (1996). Condition (3.”
SPLINE-BACKFITTED KERNEL SMOOTHING OF ADDITIVE COEFFICIENT MODEL (2010, background): “…Härdle, and Nielsen (1999), Sperlich, Tjøstheim, and Yang (2002), Yang, Sperlich, and Härdle (2003), Rodrı́guez-Póo, Sperlich, and Vieu (2003), and Hengartner and Sperlich (2005), satisfy criterion (iii) and partly (ii) as they are asymptotically normal at any given point, but do not…”
UNIFORM BAHADUR REPRESENTATION FOR LOCAL POLYNOMIAL ESTIMATES OF M-REGRESSION AND ITS APPLICATION TO THE ADDITIVE MODEL (2007, unknown): “As noted by Linton and Härdle (1996) and Hengartner and Sperlich (2005), cautious choice of the bandwidth is crucial for φn1(.) to be asymptotically normal.”