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June 2015 Joint asymptotics for semi-nonparametric regression models with partially linear structure
Guang Cheng, Zuofeng Shang
Ann. Statist. 43(3): 1351-1390 (June 2015). DOI: 10.1214/15-AOS1313


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.


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Guang Cheng. Zuofeng Shang. "Joint asymptotics for semi-nonparametric regression models with partially linear structure." Ann. Statist. 43 (3) 1351 - 1390, June 2015.


Received: 1 May 2014; Revised: 1 January 2015; Published: June 2015
First available in Project Euclid: 15 May 2015

zbMATH: 1320.62087
MathSciNet: MR3346706
Digital Object Identifier: 10.1214/15-AOS1313

Primary: 62F12 , 62G20
Secondary: 62F03

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

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 3 • June 2015
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