Asymptotic Properties for Estimators in a Semiparametric EV Model with NA Errors and Missing Responses

Abstract

This article deals with the semiparametric errors-in-variables (EV) model $y_i={\xi }_i\beta +g\left(t_i\right)+{ϵ}_i$ , $x_i={\xi }_i+{\mu }_i$ , where $y_i$ are the random missing response variables, $\left({\xi }_i,t_i\right)$ are the design points, ${\xi }_i$ are the potential variables observed with measurement errors ${\mu }_i$ , and random errors ${ϵ}_i$ are negatively associated (NA) variables. In addition, we need to estimate the unknown slope parameter $\beta$ and nonparametric component $g\left(·\right)$ . In order to solve the missing responses, we introduce three different approaches to get the estimators of $\beta$ and $g\left(·\right)$ . Meanwhile, we study the asymptotic properties for all the estimators. The final results show that the strong consistent rates for all the estimators can achieve $o\left(n^{-1/4}\right)$ . The asymptotic normality for all the estimators is considered. We compare the finite sample performance of the three estimators by simulation as well.