Strong Consistency of Estimators under Missing Responses

In this article, we focus on the semi-parametric error-in-variables model with missing responses: ( ) i i i i y g t ξ β = + +  , i i i x ξ μ = + , where i y are the response variables missing at random, ( ) , i i t ξ are design points, i ξ are the potential variables observed with measurement errors i μ , the unknown slope parameter β and nonparametric component ( ) g ⋅ need to be estimated. Here we choose two different approaches to estimate β and ( ) g ⋅ . Under appropriate conditions, we study the strong consistency for the proposed estimators.

. β ∈  is an unknown parameter that needs to be estimated.( ) g ⋅ is a unknown function defined on close interval [ ] 0,1 , ( ) where i v are also design points.
Model (1.1) and its special forms have gained much attention in recent years.
When 0 i µ ≡ , i ξ are observed exactly, the model (1.1) reduces to the general semi-parametric model, which was first introduced by Engle et al. [1].However, in many applications, there are often covariates measurement errors.So the EV models are somewhat more practical than the ordinary regression model.In addition, when i y are complete observed and ( ) 0 g ⋅ ≡ , the model (1.1) reduces to the usual linear EV model, which has been studied by Liu and Chen [2], Miao et al. [3], Miao and Liu [4], Fan et al. [5] and so on.For complete data, the model (1) itself has also been studied by many authors: See Cui and Li [6], Liang et al. [7], Zhou et al. [8] and so on.In recent years, the semi-parametric EV models have been widely concerned.
On the other hand, we often encounter incomplete data in the practical application of the models.In particular, some response variables may be missing, by design or by happenstance.For example, the responses i y may be very expen- sive to measure and only part of i y are available.Actually, missing of responses is very common in opinion polls, social-economic investigations, market research surveys and so on.Therefore, we focus our attention on the case that missing data occur only in the response variables.When i ξ can fully be ob- served, the model (1.1) reduces to the usual semi-parametric model which has been studied by many scholars in the literature: See Wang et al. [9], Wang and Sun [10], Bianco et al. [11].
To deal with missing data, one method is to impute a plausible value for each missing datum and then analyze the results as if they are complete.In regression problems, common imputation approaches include linear regression imputation by Healy and Westmacott [12], nonparametric kernel regression imputation by Cheng [13], semi-parametric regression imputation by Wang et al. [9], Wang and Sun [10], among others.We here extend the methods to the estimation of β and ( ) g ⋅ under the semi-parametric EV model (1.1).We obtain two approaches to estimate β and ( ) g ⋅ with missing responses and study the strong consistency for the estimators.
In this paper, suppose we obtain a random sample of incomplete data Throughout this paper, we assume that i y is missing at random.The assumption implies that i δ and i y are independent.That is, ( ) ( ) This assumption is a common assumption for statistical analysis with missing data and is reasonable in many practical situations.
The paper is organized as follows.In Section 2, we list some assumptions.The main results are given in Section 3. Some preliminary lemmas are stated in Section 4. Proofs of the main results are provided in Sections 5.

Assumptions
In this section, we list some assumptions which will be used in the main results.

Main Results
For model (1.1), we want to seek the estimators of β and ( ) g ⋅ .The most natural idea is to delete all the missing data.Therefore, one can get model ( ) can be observed, we can apply the least squares estimation method to estimate the parameter β .If the parameter β is known, using the complete data ( ) W t are weight functions satisfying (A3).On the other hand, under the condition of the semi-parametric EV model, Liang et al. [7] improved the least squares estimator (LSE) on the basis of the usual partially linear model, and employ the estimator of parameter β to minimize the following formula: ∑ Therefore, we can achieve the modified LSE of β as follow: ( ) where ( ) .
Apparently, the estimators ˆc β and ( ) ˆc n g t are formed without taking all sample information into consideration.Hence, in order to make up for the missing data, we imply an imputation method from Wang and Sun [10], and let one can get another estimators for β and ( ) . where

Proof of Main Results
Firstly, we introduce some notations, which will be used in the proofs below.
) ( ) ( )) ) ) potential variables observed with measurement errors i µ , the unknown slope parameter β and nonparametric component ( ) g ⋅ need to be estimated.Here we choose two different approaches to estimate β and ( ) g ⋅ .1.IntroductionConsider the following semi-parametric error-in-variables (EV) model some finite positive constants, whose values are unimportant and may change.Now, we introduce several lemmas, which will be used in the proof of the main results.