Empirical Likelihood For Nonlinear Regression Models With Nonignorably Nonresponse

In this paper, we employ an empirical likelihood approach to study nonlinear regression model under nonignorable missing response data. A class of empirical likelihood estimators and functions for parameter of interest are established by two steps. In the first step, we assume a parametric logistic model for the response prob-ability model and employ a maximum likelihood method proposed to estimate the propensity score. Once a consistent estimator of propensity score is obtained, three modified asymptotically unbiased EEs based on the exponential tilting model are constructed respectively, which is the key idea of our empirical likelihood approach introduced next. Here, specifically, our three EEs are defined based on inverse prob-ability weighting approach (IPW), nonparametric imputation (NI), and augmented inverse probability weighting approach (AIPW) respectively. Consequently, the de-sired empirical likelihood estimators and functions are computed in the second step. Based on three modified asymptotically unbiased EEs, we construct a class of em-pirical likelihood-based confidence intervals and regions for regression coefficients. The limiting distributions of the proposed empirical likelihood ratio statistics are investigated. Also, a class of empirical likelihood-based estimators of the parame-ters of interest are constructed, and the asymptotic distributions of the proposed estimators are obtained. In our research framework, some population characteristics of covariates may be known in practice. Hence, we further propose a more efficient two-step empirical likelihood-based estimator of parameter by incorporating auxil-iary information on the covariates. After using auxiliary information, the new class of empirical likelihood estimators for parameter of interest are similarly established by two steps, yet the only difference between the two methods is that the latter incorporates the auxiliary information into the new EEs can derive a better and more efficient estimator. By using the parametrically estimated propensity scores, our proposed empirical likelihood procedure can alleviate the dimensionality issue in certain sense. Furthermore, estimation method of parametrical propensity scores is more stable than that of validation sample. We systematically investigate the asymptotic properties of the proposed empirical likelihood estimators under this new setting. Two numerical simulations and a real data analysis are conducted to demonstrate the theoretical properties and practical performance of our approach.