Estimation for a hybrid model of functional and linear measurement errors regression with missing response

Abstract

In this paper, we consider least squares estimation and empirical likelihood inference for partial functional linear models when the covariates in the non-functional linear component are measured with additive error and the responses are missing at random. Asymptotic properties of the proposed estimators for the parametric and nonparametric components are established. A class of empirical log-likelihood ratio functions of the parametric component and response mean are developed, and the corresponding maximum empirical likelihood estimators are constructed. We also prove that the empirical log-likelihood ratio functions are asymptotic standard chi-squared distribution. The results can be used to construct confidence intervals for the parametric component and response mean. The asymptotic distributions of the corresponding maximum empirical likelihood estimators are also established. Meanwhile, a simulation study is conducted to demonstrate the finite sample performance of the proposed procedure. A real data analysis is also used to illustrate our methods.

Keywords:

• Asymptotical properties
• errors-in-variables
• empirical likelihood
• missing at random
• partial functional linear models

2000 AMS Subject Classifications:

• Primary 62M99
• Secondary 62H30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [Grant Nos. 1210010203, 11831008, 11971171 and 72071068, the National Statistical Science Research Project [Grant No. 2021LZ41], the China Postdoctoral Science Foundation [Grant No. 2019M651422], the National Social Science Foundation Key Program [Grant No. 17ZDA091], the 111 Project of China [Grant No. B14019], the Natural Science Foundation of Shanghai [Grant Nos. 17ZR1409000 and 20ZR1423000] and the Project of Humanities and Social Science Foundation of Ministry of Education [Grant No. 20YJC910003].