Generalized empirical likelihood for nonsmooth estimating equations with missing data

In missing data analysis, it is challenging to estimate the propensity score (PS) function. Traditional parametric, nonparametric or semiparametric approaches to estimate the PS function may be subject to model misspecification or lead to inefficient estimation. To address the aforementioned issues, we here assume that the PS function is unknown, and it is estimated by a series estimation method. To our knowledge, there is little work developed in utilizing the series estimation approach to estimate PS function even though it is widely used to estimate a regression function in the literature. The augmented inverse probability weighted (AIPW) estimating equations (EEs) are constructed via the estimated PS functions. Based on the constructed AIPW EEs, we propose generalized empirical likelihood (GEL) estimators of parameters, which are semiparametric efficient. Under some regularity conditions, we establish asymptotic properties of GEL estimators of parameters and GEL ratio statistics for parametric restrictions enjoying the Wilks’ phenomenon. In particular, we construct a pseudo-residual-based stochastic process to assess the plausibility of the posited nonsmooth EEs, their asymptotic properties are shown under null and local alternative hypotheses. A resampling procedure is proposed to evaluate the p-values of the proposed goodness-of-fit test statistics. Simulation studies and real data are used to illustrate the proposed methodologies.