Semiparametric statistical inferences for longitudinal data with nonparametric covariance modelling

ABSTRACT

In this paper, semiparametric modelling for longitudinal data with an unstructured error process is considered. We propose a partially linear additive regression model for longitudinal data in which within-subject variances and covariances of the error process are described by unknown univariate and bivariate functions, respectively. We provide an estimating approach in which polynomial splines are used to approximate the additive nonparametric components and the within-subject variance and covariance functions are estimated nonparametrically. Both the asymptotic normality of the resulting parametric component estimators and optimal convergence rate of the resulting nonparametric component estimators are established. In addition, we develop a variable selection procedure to identify significant parametric and nonparametric components simultaneously. We show that the proposed SCAD penalty-based estimators of non-zero components have an oracle property. Some simulation studies are conducted to examine the finite-sample performance of the proposed estimation and variable selection procedures. A real data set is also analysed to demonstrate the usefulness of the proposed method.

KEYWORDS:

• Additive models
• nonparametric covariance modelling
• semiparametric regression
• splines approximation

AMS SUBJECT CLASSIFICATION:

• 62H12
• 62A01

Acknowledgments

The authors would like to thank an Associate Editor and one expert referees for helpful comments which considerably improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Dr Qunfang Xu's research was supported by Ministry of Education in China Project of Humanities and Social Sciences (grant no. 15YJA910004) and grants from Natural Science Foundation of Zhejiang Province, China (grant no. LY15A010006) and Research Project from National Bureau of Statistics of China (grant no. 2013LY119). Dr Yang Bai's research is supported by Innovation Program of Shanghai Municipal Education Commission (grant no. 15ZZ035).