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

Abstract

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

Keywords

• Semi-parametric error-in-variables model
• Heteroscedastic
• Missing responses
• Strong consistent rate