Semiparametric inference for estimating equations with nonignorably missing covariates

ABSTRACT

We consider statistical inference of unknown parameters in estimating equations (EEs) when some covariates have nonignorably missing values, which is quite common in practice but has rarely been discussed in the literature. When an instrument, a fully observed covariate vector that helps identifying parameters under nonignorable missingness, is available, the conditional distribution of the missing covariates given other covariates can be estimated by the pseudolikelihood method of Zhao and Shao [(2015), ‘Semiparametric pseudo likelihoods in generalised linear models with nonignorable missing data’, Journal of the American Statistical Association, 110, 1577–1590)] and be used to construct unbiased EEs. These modified EEs then constitute a basis for valid inference by empirical likelihood. Our method is applicable to a wide range of EEs used in practice. It is semiparametric since no parametric model for the propensity of missing covariate data is assumed. Asymptotic properties of the proposed estimator and the empirical likelihood ratio test statistic are derived. Some simulation results and a real data analysis are presented for illustration.

Keywords:

• Empirical likelihood
• likelihood ratio statistics
• moment condition model
• nonresponse instrument
• not missing at random
• pseudolikelihood

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Note that the concept of instrument discussed in these missing data literature is similar but not the same as the concept of instrumental variable widely used in economics and other social sciences.

Additional information

Funding

The authors acknowledge the research support from the National Natural Science Foundation of China (Grant 71473040, 71661137005, 11571081 and 11601156).