Empirical likelihood for nonlinear regression models with nonignorable missing responses

Abstract

This article develops three empirical likelihood (EL) approaches to estimate parameters in nonlinear regression models in the presence of nonignorable missing responses. These are based on the inverse probability weighted (IPW) method, the augmented IPW (AIPW) method and the imputation technique. A logistic regression model is adopted to specify the propensity score. Maximum likelihood estimation is used to estimate parameters in the propensity score by combining the idea of importance sampling and imputing estimating equations. Under some regularity conditions, we obtain the asymptotic properties of the maximum EL estimators of these unknown parameters. Simulation studies are conducted to investigate the finite sample performance of our proposed estimation procedures. Empirical results provide evidence that the AIPW procedure exhibits better performance than the other two procedures. Data from a survey conducted in 2002 are used to illustrate the proposed estimation procedure. The Canadian Journal of Statistics 48: 386–416; 2020 © 2020 Statistical Society of Canada

Bayesian inference for nonprobability samples with nonignorable missingness

“…However, the Equation () cannot be used due to the missingness of the study variable. According to Yang and Tang [22], and Sang and Kim [23], the following estimating equation can be adopted to obtain the estimator $$$ \hat{\theta} $$$. $$$$ {\displaystyle \begin{array}{ll}{S}_0\left(\theta \right)& =\frac{1}{n_1}\sum \limits_{i=1}^{n_1}\left\{{\delta}_is\left({\delta}_i,{x}_i,{y}_i;\theta \right)\right.\\ {}& \kern1em \left.+\left(1-{\delta}_i\right){E}_0\left[s\right({\delta}_i,{x}_i,{y}_i;\theta \Big)\mid {x}_i\Big]\right\}=0,\end{array}} $$$$ where $$$ {E}_0\left[\cdot \mid {x}_i\right]=E\left(\cdot \mid {x}_i,{\delta}_i=0\right) $$$, and …”

“…Consider a relationship between the response variable $$$ {y}_i $$$ and the covariate vector $$$ {x}_i $$$ as follows: $$$$ {y}_i=f\left({x}_i,\beta \right)+{\varepsilon}_i, $$$$ where $$$ \beta \in \mathcal{B} $$$ is a p‐dimensional vector of unknown parameters, and $$$ {\varepsilon}_i $$$ is the random error, satisfying $$$ E\left({\varepsilon}_i\mid {x}_i\right)=0 $$$ and $$$ \mathrm{Var}\left({\varepsilon}_i\mid {x}_i\right)={\sigma}^2 $$$. According to Yang and Tang [22], $$$ \beta $$$ can be estimated by solving the following equation …”

“…However, the Equation ( 5) cannot be used due to the missingness of the study variable. According to Yang and Tang [22], and Sang and Kim [23], the following estimating equation can be adopted to obtain the estimator θ.…”

“…where 𝛽 ∈  is a p-dimensional vector of unknown parameters, and 𝜀 i is the random error, satisfying E(𝜀 i |x i ) = 0 and Var(𝜀 i |x i ) = 𝜎 2 . According to Yang and Tang [22], 𝛽 can be estimated by solving the following equation…”