Annals of Statistics

Joint asymptotics for semi-nonparametric regression models with partially linear structure

Guang Cheng and Zuofeng Shang

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We consider a joint asymptotic framework for studying semi-nonparametric regression models where (finite-dimensional) Euclidean parameters and (infinite-dimensional) functional parameters are both of interest. The class of models in consideration share a partially linear structure and are estimated in two general contexts: (i) quasi-likelihood and (ii) true likelihood. We first show that the Euclidean estimator and (pointwise) functional estimator, which are re-scaled at different rates, jointly converge to a zero-mean Gaussian vector. This weak convergence result reveals a surprising joint asymptotics phenomenon: these two estimators are asymptotically independent. A major goal of this paper is to gain first-hand insights into the above phenomenon. Moreover, a likelihood ratio testing is proposed for a set of joint local hypotheses, where a new version of the Wilks phenomenon [ Ann. Math. Stat. 9 (1938) 60–62; Ann. Statist. 1 (2001) 153–193] is unveiled. A novel technical tool, called a joint Bahadur representation, is developed for studying these joint asymptotics results.

Article information

Ann. Statist., Volume 43, Number 3 (2015), 1351-1390.

Received: May 2014
Revised: January 2015
First available in Project Euclid: 15 May 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties 62F12: Asymptotic properties of estimators
Secondary: 62F03: Hypothesis testing

Joint asymptotics joint Bahadur representation local likelihood ratio test semi-nonparametric models smoothing spline


Cheng, Guang; Shang, Zuofeng. Joint asymptotics for semi-nonparametric regression models with partially linear structure. Ann. Statist. 43 (2015), no. 3, 1351--1390. doi:10.1214/15-AOS1313.

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