Profile Inferences on Restricted Additive Partially Linear EV Models

Xiuli Wang

doi:10.1155/2013/594391

Abstract

We consider the testing problem for the parameter and restricted estimator for the nonparametric component in the additive partially linear errors-in-variables (EV) models under additional restricted condition. We propose a profile Lagrange multiplier test statistic based on modified profile least-squares method and two-stage restricted estimator for the nonparametric component. We derive two important results. One is that, without requiring the undersmoothing of the nonparametric components, the proposed test statistic is proved asymptotically to be a standard chi-square distribution under the null hypothesis and a noncentral chi-square distribution under the alternative hypothesis. These results are the same as the results derived by Wei and Wang (2012) for their adjusted test statistic. But our method does not need an adjustment and is easier to implement especially for the unknown covariance of measurement error. The other is that asymptotic distribution of proposed two-stage restricted estimator of the nonparametric component is asymptotically normal and has an oracle property in the sense that, though the other component is unknown, the estimator performs well as if it was known. Some simulation studies are carried out to illustrate relevant performances with a finite sample. The asymptotic distribution of the restricted corrected-profile least-squares estimator, which has not been considered by Wei and Wang (2012), is also investigated.

Highlights

To balance the modeling bias and the “curse of dimensionality,” different kinds of semiparametric models, such as partially linear model, partially linear varying coefficient model, partially linear single-index model, and additive partially linear model, have been proposed and investigated
For the case of D = 2, like Liang et al [11] and Wang et al [10], we assume E{f1(Z1)} = E{f2(Z2)} = 0 to ensure identifiability of the nonparametric functions, and X and Y are centered for simplicity
Independent Besides, we have some prior information on regression parametric vector β that can be presented by the following restricted condition: Aβ = b with A being a k × p matrix of known constants and rank(A) = k and b being a k-vector of known constants

Summary

Introduction

To balance the modeling bias and the “curse of dimensionality,” different kinds of semiparametric models, such as partially linear model, partially linear varying coefficient model, partially linear single-index model, and additive partially linear model, have been proposed and investigated. Wei and Wang [13] constructed a test statistic based on the difference between the corrected residual sums of squares under the null and alternative hypotheses and showed that the asymptotic null distribution of proposed statistic is a weighted sum of independent standard χ2(1), so adjustment is needed. We known that our results are the same to the results of adjusted test statistic derived by Wei and Wang [13], but our proposed test statistic is more easier to perform, especially for the case when the covariance matrix Σuu of measurement error is unknown. Our proposed profile Lagrange multiplier test statistic is more attractive

Two-Stage Restricted Estimator for the Nonparametric Component

Simulation

Proofs