Abstract
This article deals with the semiparametric errors-in-variables (EV) model y i = ξ i β + g t i + ϵ i , x i = ξ i + μ i , where y i are the random missing response variables, ξ i , t i are the design points, ξ i are the potential variables observed with measurement errors μ i , and random errors ϵ i are negatively associated (NA) variables. In addition, we need to estimate the unknown slope parameter β and nonparametric component g · . In order to solve the missing responses, we introduce three different approaches to get the estimators of β and g · . 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 n − 1 / 4 . The asymptotic normality for all the estimators is considered. We compare the finite sample performance of the three estimators by simulation as well.
Highlights
Introduction e following semiparametricEV model is considered: yi ξiβ + g ti + εi, (1)xi ξi + μi, where yi are the response variables, are design points, ξi are the potential variables observed with measurement errors μi, and Eμi 0 are random errors with Eεi 0. e unknown parameter β(∈ R) needs to be estimated. g(·) is an unknown function defined on a close interval [0, 1], and h(·) is a known function defined on [0, 1] satisfying ξi h ti + vi, (2)where vi are design points
Xi ξi + μi, where yi are the response variables, are design points, ξi are the potential variables observed with measurement errors μi, and Eμi 0 are random errors with Eεi 0. e unknown parameter β(∈ R) needs to be estimated. g(·) is an unknown function defined on a close interval [0, 1], and h(·) is a known function defined on [0, 1] satisfying ξi h ti + vi, (2)
For model (1) with responses missing randomly, the most direct way to construct the estimators of β and g(·) is deleting all the missing data
Summary
We list some assumptions, which will be used in the proof of main results as follows. (i) (A0) Let εi, 1 ≤ i ≤ n be a NA sequence, μi, 1 ≤ i ≤ n be independent random variables satisfying (i) Eεi 0, Eε2i 1, Eμi 0, andEμ2i Ξ2μ > 0; (ii) supiE|εi|p < ∞, supiE|μi|p < ∞ for some p > 4; (iii) εi, 1 ≤ i ≤ n, μi, 1 ≤ i ≤ n are independent of each other. J n) be weight functions defined on [0, 1] and satisfy (i) max1≤j≤n ni 1 δjWcnj(ti) O(1); (ii) supt nj 1 δjWcnj(t)I(|t − tj| > a · n− 1/4). Where Fji denotes the σ-algebra generated by Xt, i ≤ t ≤ j, and L2(Fji ) consists of Fji -measurable random variables with the finite second moment. E sequence Xi is said to be α-mixing or strong mixing if α(n) ⟶ 0, as n ⟶ ∞
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