Strong consistency rates for the estimators in a heteroscedastic EV model with missing responses

Abstract

AbstractThis article is concerned with the semi-parametric error-in-variables (EV) model with missing responses: $y_{i}= \xi _{i}\beta +g(t_{i})+\epsilon _{i}$yi=ξiβ+g(ti)+ϵi, $x_{i}= \xi _{i}+\mu _{i}$xi=ξi+μi, where $\epsilon _{i}=\sigma _{i}e_{i}$ϵi=σiei is heteroscedastic, $f(u_{i})=\sigma ^{2}_{i}$f(ui)=σi2, $y_{i}$yi are the response variables missing at random, the design points $(\xi _{i},t_{i},u_{i})$(ξi,ti,ui) are known and non-random, β is an unknown parameter, $g(\cdot )$g(⋅) and $f(\cdot )$f(⋅) are functions defined on closed interval $[0,1]$[0,1], and the $\xi _{i}$ξi are the potential variables observed with measurement errors $\mu _{i}$μi, $e_{i}$ei are random errors. Under appropriate conditions, we study the strong consistent rates for the estimators of β, $g(\cdot )$g(⋅) and $f(\cdot )$f(⋅). Finite sample behavior of the estimators is investigated via simulations.