Semi-parametric regression: Efficiency gains from modeling the nonparametric part

Abstract

It is widely admitted that structured nonparametric modeling that circumvents the curse of dimensionality is important in nonparametric estimation. In this paper we show that the same holds for semi-parametric estimation. We argue that estimation of the parametric component of a semi-parametric model can be improved essentially when more structure is put into the nonparametric part of the model. We illustrate this for the partially linear model, and investigate efficiency gains when the nonparametric part of the model has an additive structure. We present the semi-parametric Fisher information bound for estimating the parametric part of the partially linear additive model and provide semi-parametric efficient estimators for which we use a smooth backfitting technique to deal with the additive nonparametric part. We also present the finite sample performances of the proposed estimators and analyze Boston housing data as an illustration.

Efficient semiparametric estimation in generalized partially linear additive models

“…A rigorous proof of Theorem 1 can be done as in Yu et al (2009) by using Halperin's Theorem for iterative alternating projections and Proposition A.3.5 in Bickel, Klaassen, Ritov, and Wellner (1993). The information bound (8) can be decomposed as…”

“…We will take an approach similar to Yu et al (2009) and present a heuristic argument here. The rigorous proof can be done as in Yu et al (2009).…”

“…The rigorous proof can be done as in Yu et al (2009). To avoid unnecessary complexity, we assume m 0 = 0 and start with the case where p = 1.…”

“…Härdle, Huet, Mammen, and Sperlich (2004) provided an estimator which does not utilize the additive structure and thus results in inefficiency. Yu et al (2009) also discussed a special case when G is the identity function. In this paper, we discuss efficiency under this model and this is the first discussion about efficiency under this model, to the best of our knowledge.…”

“…Speckman (1988). Recently, Yu, Mammen, and Park (2009) studied semiparametric approaches in regression analysis and found that putting some structure on the nonparametric part of the regression function is helpful not only for circumventing the dimensionality problem but also for gaining efficiency in estimating the parametric part.…”