A consistent model specification test for a partial linear model with covariates missing at random

Abstract

In this paper, we discuss the model checking problem for a partial linear model when some covariates are missing at random. A weighted model-adjustment method is applied to estimate the regression coefficients and the nonparametric function for the null hypothetical partial linear model. A testing procedure based on a residual-marked empirical process is developed to check the adequacy of the partial linear model. It is shown that the proposed test is consistent and can detect the local alternatives converging to the null hypothetical model at the rate n ^−1/2. Since the asymptotic null distribution of the testing statistics is case-dependent, an adjusted wild bootstrap method is used to decide the critical value, which is proved to be consistent. A simulation study and a real data analysis are conducted to show that the proposed procedure works well.

Keywords:

• covariates missing at random
• empirical process
• model check
• partial linear model

Acknowledgements

The authors thank the Editor-in-Chief, Professor Suojin Wang, an associate editor and two reviewers for their constructive and insightful comments and suggestions that greatly improved the paper. They also thank Professor Hua Liang for valuable suggestions. Tianfa Xie's research was supported by the National Natural Science Foundation of China (10971007, 11101015), the specialised research fund for the doctoral program of higher education (No. 20091103120012) and the fond from the government of Beijing (No. 2011D005015000007). Zhihua Sun's research was supported by the National Natural Science Foundation of China (10901162, 10926073), China Postdoctoral Science Foundation and Foundation of the Key Laboratory of Random Complex Structures and Data Science (RCSDS), CAS. Liuquan Sun's research was partly supported by the National Natural Science Foundation of China Grants (No. 11171330, 10731010, 10971015 and 11021161) and Key Laboratory of RCSDS, CAS (No. 2008DP173182).