Confidence Intervals for Nonparametric Regression Functions with Missing Data

Abstract

Suppose that we have a nonparametric regression model Y = m(X) + ε with X ∈ R^p, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x₀) for fixed x₀ ∈ R^p are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x₀). We also construct an empirical likelihood (EL) statistic for m(x₀) with limiting distribution of χ²₁, which is used to construct an EL confidence interval for m(x₀).

Keywords:

• Confidence interval
• Empirical likelihood
• Inverse probability weighted imputation
• Nonparametric regression model
• Missing at random

Mathematics subject classification::

• Primary 62G05
• Secondary 62E20

Acknowledgments

The authors are thankful to the referees for constructive suggestions.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (11271088, 11361011, 11201038), the Natural Science Foundation of Guangxi (2013 GXNSFAA 019004, 2013GXNSFAA 019007, 2013GXNSFBA 019001) and the Program for Excellent Talents in Guangxi Higher Education Institutions.