Efficient Estimation of the Nonparametric Mean and Covariance Functions for Longitudinal and Sparse Functional Data

ABSTRACT

We consider the estimation of mean and covariance functions for longitudinal and sparse functional data by using the full quasi-likelihood coupling a modification of the local kernel smoothing method. The proposed estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient in terms of their linear functionals. Their superiority to the competitors is further illustrated numerically through simulation studies. The method is applied to analyze AIDS study and atmospheric study. Supplementary materials for this article are available online.

KEYWORDS:

• Curse of dimensionality
• Full quasi-likelihood function
• Gaussian process
• Longitudinal data
• Nonparametric structure
• Semiparametric efficiency

Acknowledgments

The authors thank the co-editors, an associate editor, and three referees for their insightful suggestions and comments that have substantially improved an earlier version of this article.

Additional information

Funding

Lin’s research was partially supported by Fund of National Natural Science (Nos. 11571282 and 11528102) and Fundamental Research Funds for the Central Universities (No. JBK120509 and 14TD0046) of China. Liang’s research was partially supported by NSF grants DMS-1418042 and DMS-1620898, and by Award Number 11529101, made by National Natural Science Foundation of China.