Pseudo likelihood and dimension reduction for data with nonignorable nonresponse

Tang et al. (2003. Analysis of multivariate missing data with nonignorable nonresponse.Biometrika, 90(4), 747–764) and Zhao & Shao (2015. Semiparametric pseudo-likelihoods in generalized linear models with nonignorable missing data. Journal of the American Statistical Association, 110(512), 1577–1590) proposed a pseudo likelihood approach to estimate unknownparameters in a parametric density of a response Y conditioned on a vector of covariate X, whereY is subjected to nonignorable nonersponse, X is always observed, and the propensity of whetheror not Y is observed conditioned on Y and X is completely unspecified. To identify parameters, Zhao & Shao (2015. Semiparametric pseudo-likelihoods in generalized linear models withnonignorable missing data. Journal of the American Statistical Association, 110(512), 1577–1590)assumed that X can be decomposed into U and Z, where Z can be excluded from the propensitybut is related with Y even conditioned on U. The pseudo likelihood involves the estimation ofthe joint density of U and Z. When this density is estimated nonparametrically, in this paper weapply sufficient dimension reduction to reduce the dimension of U for efficient estimation. Consistency and asymptotic normality of the proposed estimators are established. Simulation resultsare presented to study the finite sample performance of the proposed estimators.