Stochastic Restricted Estimation in Partially Linear Measurement Error Models

As a generalization of nonparametric regression model, partially linear model has been studied extensively in the last decades. This paper considers estimation of the semiparametric model under the situation that the covariates are measured with additive error in the linear part and some additional stochastic linear restrictions exist on the parametric component. Based on the corrected profile least-squares approach and mixed regression method, we propose a stochastic restricted estimator named the corrected profile mixed estimator for the parametric component, and discuss its statistical properties. We also construct a weighted stochastic restricted estimation for the parametric component. Finally, the proposed procedure is illustrated by simulation studies.


Introduction
In the last two decades, semiparametric measurement error (errors-in-variables) models have received considerable attention in statistics and econometrics.In this paper, the following structural partially linear errors-in-variables model is considered where y i are response, x i = (x i1 , x i2 , • • • , x ip ) T is a vector of random explanatory variables, t i is a scalar covariate, β = (β 1 , β 2 , • • • , β p ) T is a vector of p-dimensional parameters which are unknown, f (•) is an unknown smooth function, and the residuals ε i 's are independent and identical distribution with zero mean and finite variance σ 2 .The measurement errors ξ i are independent and identically distributedwith mean zero and covariance matrix Σ ξ .Σ ξ is assumed to be known.In the meantime, ε i 's are independent of (x T i , t i )'s and each ξ i is independent of (y i , x i , t i ).Obviously, model (1.1) is a generalization of the linear measurement error model and partially linear model, and has been discussed by many authors in the last two decades.To estimate β, Liang et al.(1999) proposed a corrected two-step method, Wang(1999) constructed an estimator using validation data.When the ratio of Σ ξ to σ 2 is known, Cui and Li (1998) proposed a profile total least-square estimator.
In practice, some outside sources can provide some prior information about the regression coefficients.Such information can be expressed as some exact or stochastic restrictions on the unknown regression parameters.Use of such restrictions on parameters may improve upon the efficiency of the estimator, and details can be found in Toutenburg(1982) and Rao et al. (2008).
In recent years, measurement error models with exact or stochastic restrictions have been studied by many authors.for the linear measurement error models, Shalabh et al. (2007) and Shalabh et al. (2009) studied the exact restricted estimation.Shalabh et al. (2010) and Li and Yang (2013) studied the estimation of the model when stochastic linear restrictions on regression coefficients are available.For the partially linear measurement error model with exact linear restrictions, Wei et al. (2013) proposed a restricted corrected profile least-squares estimator for the parametric component.In this paper, we consider the estimation of model (1.1) with the stochastic linear restrictions.The stochastic restrictions can be defined as where b is a k × 1 known vector and A is a k × p known matrix of full row rank.Further, the random error η is assumed to be distributed with mean vector 0 and covariance matrix σ 2 Ω, where Ω is a known positive definite matrix.Based on the corrected profile least-squares approach of Liang et al.(1999), and the mixed regression estimation method of Theil and Goldberger (1961) and Theil (1963), a stochastic restricted estimator named corrected profile mixed estimator for the parametric component β is defined.As a result, we extend, on one hand, the results of Shalabh et al. (2010) and Li and Yang (2013) from linear measurement error models to semiparametric measurement error models, and on the other hand, the results of Wei et al. (2013) from exact restrictions to the case of stochastic restrictions.
This paper is organized as follows.In Section 2, we construct the stochastic restricted estimators for the parametric component, and discuss their statistical properties.To examine the performance of the proposed approaches, we conduct some simulations in Section 3 .The proofs of main results are given in Section 4.

Corrected Profile Mixed Estimation
Let y * i = y i − x T i β, then we can get the following standard nonparametric regression model We apply the local linear approach to estimate the function f (•).For a given point t 0 , we have for t in a neighborhood of t 0 is available.Let K(•) be a kernel function and h be a bandwidth, the local linear regression approach gives the solution of local parameters where where Replacing f (t i ) in (2.1) with f (t i ), we can obtain the synthetic linear regression model after some algebraic operation where εi The profile least squares estimator for β can be obtained with linear model (2.4).While x i cannot be exactly observed in our case, the resulting estimator is inconsistent if we ignore the measurement error and replaces x i by v i in (2.4).By the correction for attenuation technique, the corrected profile least squares estimator of β can be defined as According to Theil and Goldberger (1961), based on the stochastic prior information, the corrected profile least-squares mixed estimator can be given by minimizing the following cost function with respect to β.
By (2.7), we can obtain the following equation Solving the equation (2.8), the corrected profile mixed estimator of β is obtained as βcpm Denote E = VT V − nΣ ξ , by the Theorem A.18 of Rao et al (2008)(Page.494), we have Then, by (2.9) and (2.10), βm can be rewritten as where β is the corrected profile least-squares estimator of β in (2.6).
The asymptotic normality of βcpm can be proven by the following theorem .
Theorem 2.1 Under the conditions in the Section 5, the corrected profile mixed estimator of β is asymptotically normal, namely, where 1 , and A ⊗2 means AA T .The estimator βcpm is defined under the assumption that both the sample information and the prior information which is expressed as stochastic restrictions are equally important and weight equally in the statistical procedure.In practice, this assumption may violate.Schaffrin and Toutenburg (1990) proposed the weighted mixed regression estimation to fix the problem.Li and Yang (2013) have studied the weighted stochastic restricted estimation for linear errors-in-variables models.In the following, we consider the weighted stochastic restricted estimation for model (1.1).Similar to Li and Yang (2013), the weighted corrected profile least-squares mixed estimator can be obtained by minimizing the following objective function (2.12) with respect to β, and ω is a non-stochastic and non-negative scalar weight ranging from 0 to 1.By differentiating function F w (β) with respect to β, we have the following equation Solving the equation (2.13), the weighted corrected profile mixed estimator of β is obtained as The following theorem gives the asymptotic normality of βwcpm .
Theorem 2.2 Under the conditions in the Section 5, the corrected profile mixed estimator of β is asymptotically normal, namely, Remark 2.1 Theorems 2.1 and 2.2 indicate that both the corrected profile mixed estimator βcpm and the weighted corrected profile mixed estimator βwcpm and the corrected profile least-squares estimator β have the same asymptotic distributions.
The results is consistent with the results of Shalabh et al. (2010) and Li and Yang (2013) on the linear ultrastructural measurement error model with stochastic linear restrictions.As sample size increases, the results suggest that the effect of stochastic restrictions on the properties of estimators vanishes.
The simulation applys the mean squared error (MSE) criterion to compare the performance of the estimators.For each setting, the simulation replicates 1000.In each replication, the corrected profile least-squares estimator β, and the corrected profile mixed estimator βcpm can be obtained.The estimated mean squared error (EMSE) for both β and βcpm is given, and more details are presented in Table 3.1.The EMSEs for the different estimators are calculated by: where β * k j denotes the estimate of the kth parameter in jth replication and β k , k = 1, 2, 3, 4 are the true parameter values above.
The simulation results can be summarized as follows.The EMSE of all the estimators decrease, as the sample size increases.In all these cases, the corrected profile mixed estimators show better performance than the corrected profile least-squares estimators.

Proof of the Main Results
To derive the main results, the following assumptions are required.They are also assumed in Wei et al. (2013).These mild assumptions can be easily satisfied in most cases.
Assumption 1 The random variable t has a bounded support Ψ. Its density function g(•) is Lipschitz continuous and bounded away from 0 on its support.
Assumption 4 The function K(•) is a symmetric density function with compact support and the bandwidth h satisfies nh 8 → 0 and nh 2 /(log n) 2 → ∞.