# Profile Inferences on Restricted Additive Partially Linear EV Models

Authors: Xiuli Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the testing problem for the parameter and restricted estimator for the nonparametric component in the additive partially
linear errors-in-variables (EV) models under additional restricted condition.
We propose a profile Lagrange multiplier test statistic based on modified profile least-squares method and two-stage restricted estimator for the nonparametric component. We derive two important results. One is that, without requiring the undersmoothing of the nonparametric components, the proposed
test statistic is proved asymptotically to be a standard chi-square distribution
under the null hypothesis and a noncentral chi-square distribution under the
alternative hypothesis. These results are the same as the results derived by Wei
and Wang (2012) for their adjusted test statistic. But our method does not
need an adjustment and is easier to implement especially for the unknown
covariance of measurement error. The other is that asymptotic distribution
of proposed two-stage restricted estimator of the nonparametric component is
asymptotically normal and has an oracle property in the sense that, though
the other component is unknown, the estimator performs well as if it was
known. Some simulation studies are carried out to illustrate relevant performances with a finite sample. The asymptotic distribution of the restricted
corrected-profile least-squares estimator, which has not been considered by
Wei and Wang (2012), is also investigated.


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## Body

## 1. Introduction

To balance the modeling bias and the “curse of dimensionality,” different kinds of semiparametric models, such as partially linear model, partially linear varying coefficient model, partially linear single-index model, and additive partially linear model, have been proposed and investigated. In this paper, we consider the semiparametric additive partially linear model, which can be written as(1)Y=X′β+∑d=1D‍fd(Zd)+ε,
where Y is the response, X and Z=(Z1,…,ZD)′ are covariates on Rp and RD, respectively, f1,…,fD are unknown smooth functions, β=(β1,…,βp)′ is ap-dimensional vector of unknown parameters, and εis the random error with conditional mean zero givenXand Z. Model (1), inheriting the interpretability of the linear model and flexibility of the additive model, has been studied by Opsomer and Ruppert [1], Li [2], and Jiang et al. [3], among others. Two important models, partially linear model, which is the case when D=1 for model (1), and additive model, whenβ=0, can be regarded as its special models.However in many practical applications, it may be difficult or impossible to measure some explanatory variables accurately, and people usually can only observe its surrogate. For the literature about errors in variables (EV) the reader can resort to Fuller [4], Cheng and Van Ness [5], Carroll et al. [6], Liang et al. [7], You and Chen [8], Li and Greene [9], Wang et al. [10], and the references therein. In this paper, we consider model (1) with the case that only covariateXis measured with additive error; that is, X cannot be observed, but an unbiased measure ofX, denoted byW, can be obtained such that
(2)W=X+U,
where U is the measurement error with mean zero and independent of (X,Z,Y). Like Liang et al. [11], we only consider the case of D=2 and assume that the covariance matrix of measurement error U, say Σuu, is known. Otherwise, we can estimate it by repeatedly measuringWas mentioned by Liang et al. [7, 11].In many practical applications such as production and consumption studies, in addition to the sample information, people always have some prior information on regression parametric vector which can be formed into some constraint conditions and used to improve the parametric estimators. For more detailed discussion see Jorgenson [12]. It is well known that for general linear models and nonlinear regression models there are restricted estimation and corresponding results, but for semi-parametric regression models, the results of restricted estimation are very few. Recently, Wei and Wang [13] considered the testing problem under the following linear hypothesis on the parametric component:
(3)H0:Aβ=bversusH1:Aβ≠b,
where A is ak×pmatrix of known constants, rank(A)=k, and bis ak-vector of known constants. They first proposed the restricted corrected-profile least-squares estimator β^r of βunder the restriction Aβ=b. Based on the difference between the corrected residual sums of squares under the null and alternative hypotheses, a test statistic is suggested, but the limiting distribution is a weighted sum of independent standard χ2(1), so adjustment is needed, and the adjusted test statistic is also suggested. Though they proposed the restricted corrected-profile least-squares estimator β^r of βunder the condition Aβ=b, they did not investigate the asymptotic property of β^r. In particular they did not give the estimator of nonparametric component. So in this paper, we will investigate the above problems. Once the asymptotic distribution of β^ris obtained, for a different matrixA, we can easily derive the asymptotic distribution of different constrained estimates Aβ^rwith different purposes. In addition to the just-mentioned problems, our other aim is to suggest the profile Lagrange multiplier test statistic for the unknown parameter βon testing problem and show that its limiting distribution is a standard chi-squared distribution under the null hypothesis and a noncentral chi-square distribution under the alternative hypothesis. These results are the same as the results derived by Wei and Wang [13] for their adjusted test statistic, but our method does not need an adjustment and is easier to implement especially when the covariance Σuu is unknown. For the Lagrange multiplier test method, see Wei and Wu [14] and Zhang et al. [15].The rest of this paper is organized as follows. In Section2, we first review the restricted corrected-profile least-squares estimator β^r of β and then study the asymptotic distribution of β^r. After that, we construct the modified profile Lagrange multiplier test statistic and derive its asymptotic distribution under the null and alternative hypotheses. In Section 3, two-stage restricted estimators for the nonparametric components are proposed and their asymptotic distribution are presented. Some simulation studies are carried out to assess the performance of the derived results in Section 4. Proofs of the main results are given in Section 5.


## 2. Asymptotic Results of the Restricted-Profile Least-Squares Estimator and Modified Profile Lagrange Multiplier Test

### 2.1. The Restricted-Profile Least-Squares Estimator and Its Asymptotic Property

For the case ofD=2, like Liang et al. [11] and Wang et al. [10], we assume E{f1(Z1)}=E{f2(Z2)}=0 to ensure identifiability of the nonparametric functions, andXandYare centered for simplicity. Suppose the observable data {Yi,Zi1,Zi2,Wi=(Wi1,Wi2,…,Wip)′,i=1,…,n} are independent and identically distributed (i.i.d.) and are generated from the following model:
(4)Yi=Xi′β+f1(Zi1)+f2(Zi2)+εi,Wi=Xi+Ui,
where the model errors {εi,i=1,…,n} are i.i.d. with E(εi∣Xi,Zi1,Zi2)=0 almost surely and the measurement errorsUiare i.i.d. with mean zero and independent of {Yi,Xi=(Xi1,Xi2,…,Xip)′,Zi1,Zi2}. Besides, we also have some prior information on regression parametric vectorβthat can be presented by the following restricted condition:
(5)Aβ=b
withAbeing a k×p matrix of known constants and rank(A)=k and b being a k-vector of known constants.First, we need introduce the restricted-profile least-squares estimator of Wei and Wang [13]. The notations are like those of Wang et al. [10]. If β is known, the first equation of model (4) can be rewritten as
(6)Yi-Xi′β=f1(Zi1)+f2(Zi2)+εi,i=1,…,n.Denote(7)Y=(Y1,…,Yn)′,ε=(ε1,…,εn)′,X=(X1,…,Xn)′,Z1=(Z11,…,Zn1)′,Z2=(Z12,…,Zn2)′,F1=(f1(Z11),…,f1(Zn1))′,F2=(f2(Z12),…,f2(Zn2))′.
Hence, (6) can be written as
(8)Y-Xβ=F1+F2+ε.
Let s1,z1′,s2,z2′ denote the equivalent kernels for the local linear regression atz1,z2:
(9)s1,z1′=e1′(𝒵1′Ω1𝒵1)-1𝒵1′Ω1,s2,z2′=e1′(𝒵2′Ω2𝒵2)-1𝒵2′Ω2,
wheree1=(1,0)′,
(10)Ω1=diag(1h1K(Z11-z1h1),…,1h1K(Zn1-z1h1)),Ω2=diag(1h2K(Z12-z2h2),…,1h2K(Zn2-z2h2)),
for a kernel functionK(·)and bandwidthsh1,h2, and
(11)𝒵1=(1Z11-z1⋮⋮1Zn1-z1),𝒵2=(1Z12-z2⋮⋮1Zn2-z2)
aren×2design matrices. LetS1andS2represent the smoother matrices whose rows are the equivalent kernels at the observationsZ1andZ2, respectively:
(12)S1=(s1,Z11′⋮s1,Zn1′),S2=(s1,Z12′⋮s2,Zn2′).
Using the backfitting algorithm, we get the backfitting estimators ofF1andF2, sayF^1andF^2, which can be expressed as
(13)F^1={I-(I-S1cS2c)-1(I-S1c)}(Y-Xβ),F^2={I-(I-S2cS1c)-1(I-S2c)}(Y-Xβ),
whereSic=(I-11′/n)Siis the centered smoothing matrix corresponding toSi,i=1,2. Substituting (13) into (8), we have
(14)(I-S)Y=(I-S)Xβ+ε,
whereS={I-(I-S1cS2c)-1(I-S1c)}+{I-(I-S2cS1c)-1(I-S2c)}.SinceXicannot be observed, the corrected-profile least-squares estimator ofβis defined as
(15)β^=argminβ[(Y-Wβ)′(I-S)′(I-S)(Y-Wβ)22222222222-nβ′Σuuβ]=[W~′W~-nΣuu]-1W~′(I-S)Y,
whereWis defined similar toXandW~=(I-S)W.When considering the restrictionsAβ=b, Wei and Wang [13] defined the corrected Lagrange function as
(16)F(β,λ)=[(Y-Wβ)′(I-S)′(I-S)(Y-Wβ)-nβ′Σuuβ]+2λ′(Aβ-b),
whereλis a k×1 Lagrange multipliers vector. DifferentiatingF(β,λ)with respect toβandλand setting them to zero, we obtain
(17)∂F(β,λ)∂β=-2W~′(I-S)Y+2W~′W~β-2nΣuuβ+2A′λ=0,∂F(β,λ)∂λ=2(Aβ-b)=0.
Solving (17), we have
(18)β^r=β^-(W~′W~-nΣuu)-1A′[A(W~′W~-nΣuu)-1A′]-1×(Aβ^-b),λ^=[A(W~′W~-nΣuu)-1A′]-1(Aβ^-b),
whereβ^is given by (15).According to Wei and Wang [13],β^ris called the modified restricted profile least-squares estimator of β, but the asymptotic distribution ofβ^ris vacant. In the following, we will give the asymptotic distribution ofβ^r. The following assumptions will be required to derive the main results. These assumptions are common and can be found in the works of Liang et al. [11], Wang et al. [10], and Wei and Wang [13].(A1)
The density functions ofZ1andZ2are bounded away from 0 and have bounded continuous second partial derivatives.(A2)
LetX˘i=Xi-E(Xi∣Zi1)-E(Xi∣Zi2). The matrix ΓX|Z=^E(X˘1X˘1′) is positive-definite, E(εi∣Xi,Zi1,Zi2)=0, and E(|εi|3∣Xi,Zi1,Zi2)<∞.(A3)
The bandwidthsh1 and h2 are of order n-1/5.(A4)
The functionK(·) is a symmetric density function with compact support and satisfies ∫K(u)du=1,∫‍uK(u)du=0,∫‍u2K(u)du=1, and ∫‍u4K(u)du<∞.(A5)
EUi=0andE(∥Ui∥3)<∞.Under these assumptions, we give the following theorem that states the asymptotic distribution ofβ^r.Theorem 1.
Suppose that conditions (A1)–(A5) hold andεiis homoscedastic with varianceσ2and independent ofUi. Then the modified restricted profile least-squares estimatorβ^ris asymptotically normal. Namely,
(19)n(β^r-β)⟶𝒟N(0,Σ),n⟶∞,
where “→𝒟” denotes the convergence in distribution and
(20)Σ=ΓX∣Z-1ΛΓX∣Z-1-ΓX∣Z-1ΛΓX∣Z-1ΞΓX∣Z-1-ΓX∣Z-1ΞΓX∣Z-1ΛΓX∣Z-1+ΓX∣Z-1ΞΓX∣Z-1ΛΓX∣Z-1ΞΓX∣Z-1,Λ=E(ε1-U1′β)2ΓX∣Z+E{(U1U1′-Σuu)β}⊗2+Σuuσ2,Ξ=A′(AΓX∣Z-1A′)-1A,R⊗2=RR′.Remark 2.
From the previous theorem, it is easy to check that whenXis observed exactly the asymptotic distribution of modified restricted profile least-squares estimatorβ^ris the same as the asymptotic distribution obtained by Theorem 3.1 which appeared in the works of Wei and Liu [16].
To apply Theorem1 to inference, we need to estimateΓX|ZandΛ. Using plug-in method, a consistent estimatorΣ^ofΣcan be obtained. Wei and Wang [13] proposed to estimateΓX|ZandΛ, respectively, by
(21)Γ^X∣Z=1n∑i=1n‍W~iW~i′-Σuu,(22)Λ^=1n∑i=1n‍{W~i(Y~i-W~i′β^)+Σuuβ^}⊗2.The asymptotic distribution ofBβ^r, whereBis ans×pmatrix withrank(B)=s, can be given by the following result.Corollary 3.
Suppose that conditions (A1)–(A5) hold andεiis homoscedastic with varianceσ2and independent ofUi. Then asn→∞, one has
(23)n(Bβ^r-Bβ)⟶𝒟N(0,BΣB′).Using Corollary3, confidence regions of any linear combination of the parameter components given in advance can be constructed. In particular, for any0<α<1, letχ1-α2(p)denote the(1-α)-quantile ofχ2(p). Then
(24){Bβ∈Rs:(Bβ-Bβ^r)′(1nBΣ^B′)-1(Bβ-Bβ^r)≤χ1-α2(p)(1nBΣ^B′)-1}
constitute a confidence region ofBβwith asymptotic coverage1-α.


### 2.2. Modified Profile Lagrange Multiplier Test and Its Asymptotic Properties

Using the estimation method described in Section2.1, we now consider the following linear hypothesis:
(25)H0:Aβ=b,H1:Aβ=b+δ,
whereδis ak×1positive constant vector. Wei and Wang [13] constructed a test statistic based on the difference between the corrected residual sums of squares under the null and alternative hypotheses and showed that the asymptotic null distribution of proposed statistic is a weighted sum of independent standardχ2(1), so adjustment is needed. Notice that the statistics proposed in Wei and Wu [14] and Zhang et al. [15] can achieve the standard chi-squared limit, so in this paper, we use the similar idea to construct the profile Lagrange multiplier test statistic.Using the estimator of Lagrange multiplier defined in (18), the modified profile Lagrange multiplier test statistic can be constructed by
(26)Tn=1nλ^′(C0L0C0′)-1λ^,
whereC0=(AΓ^X|Z-1A′)-1,L0=AΓ^X|Z-1Λ^Γ^X|Z-1A′, and Γ^X|Z and Λ^ are estimators of ΓX|Z and Λ defined by (21) and (22), respectively.The following theorem gives the asymptotic distribution of the modified profile Lagrange multiplier test statisticTn.Theorem 4.
Suppose that conditions (A1)–(A5) hold. Then(1)
under the null hypothesisH0 of testing problem (25),
(27)Tn⟶𝒟χ2(k),asn⟶∞,(2)
under the alternative hypothesisH1of testing problem (25),Tnfollows the asymptotic noncentral χ2(k,ζ) distribution with k degrees of freedom, and the noncentral parameter is
(28)ζ=limn⟶∞n(Aβ-b)′(CLC′)-1(Aβ-b),where C=(AΓX|Z-1A′)-1,L=AΓX|Z-1ΛΓX|Z-1A′.Remark 5.
From the above theorem, we known that our results are the same to the results of adjusted test statistic derived by Wei and Wang [13], but our proposed test statistic is more easier to perform, especially for the case when the covariance matrix Σuu of measurement error is unknown. From Wei and Wang [13] we know that, when Σuu is unknown, it is difficult to estimate RSS(H0) and RSS(H1); thus their proposed test statistic is unapplicable in this case. So our proposed profile Lagrange multiplier test statistic is more attractive.


## 2.1. The Restricted-Profile Least-Squares Estimator and Its Asymptotic Property

For the case ofD=2, like Liang et al. [11] and Wang et al. [10], we assume E{f1(Z1)}=E{f2(Z2)}=0 to ensure identifiability of the nonparametric functions, andXandYare centered for simplicity. Suppose the observable data {Yi,Zi1,Zi2,Wi=(Wi1,Wi2,…,Wip)′,i=1,…,n} are independent and identically distributed (i.i.d.) and are generated from the following model:
(4)Yi=Xi′β+f1(Zi1)+f2(Zi2)+εi,Wi=Xi+Ui,
where the model errors {εi,i=1,…,n} are i.i.d. with E(εi∣Xi,Zi1,Zi2)=0 almost surely and the measurement errorsUiare i.i.d. with mean zero and independent of {Yi,Xi=(Xi1,Xi2,…,Xip)′,Zi1,Zi2}. Besides, we also have some prior information on regression parametric vectorβthat can be presented by the following restricted condition:
(5)Aβ=b
withAbeing a k×p matrix of known constants and rank(A)=k and b being a k-vector of known constants.First, we need introduce the restricted-profile least-squares estimator of Wei and Wang [13]. The notations are like those of Wang et al. [10]. If β is known, the first equation of model (4) can be rewritten as
(6)Yi-Xi′β=f1(Zi1)+f2(Zi2)+εi,i=1,…,n.Denote(7)Y=(Y1,…,Yn)′,ε=(ε1,…,εn)′,X=(X1,…,Xn)′,Z1=(Z11,…,Zn1)′,Z2=(Z12,…,Zn2)′,F1=(f1(Z11),…,f1(Zn1))′,F2=(f2(Z12),…,f2(Zn2))′.
Hence, (6) can be written as
(8)Y-Xβ=F1+F2+ε.
Let s1,z1′,s2,z2′ denote the equivalent kernels for the local linear regression atz1,z2:
(9)s1,z1′=e1′(𝒵1′Ω1𝒵1)-1𝒵1′Ω1,s2,z2′=e1′(𝒵2′Ω2𝒵2)-1𝒵2′Ω2,
wheree1=(1,0)′,
(10)Ω1=diag(1h1K(Z11-z1h1),…,1h1K(Zn1-z1h1)),Ω2=diag(1h2K(Z12-z2h2),…,1h2K(Zn2-z2h2)),
for a kernel functionK(·)and bandwidthsh1,h2, and
(11)𝒵1=(1Z11-z1⋮⋮1Zn1-z1),𝒵2=(1Z12-z2⋮⋮1Zn2-z2)
aren×2design matrices. LetS1andS2represent the smoother matrices whose rows are the equivalent kernels at the observationsZ1andZ2, respectively:
(12)S1=(s1,Z11′⋮s1,Zn1′),S2=(s1,Z12′⋮s2,Zn2′).
Using the backfitting algorithm, we get the backfitting estimators ofF1andF2, sayF^1andF^2, which can be expressed as
(13)F^1={I-(I-S1cS2c)-1(I-S1c)}(Y-Xβ),F^2={I-(I-S2cS1c)-1(I-S2c)}(Y-Xβ),
whereSic=(I-11′/n)Siis the centered smoothing matrix corresponding toSi,i=1,2. Substituting (13) into (8), we have
(14)(I-S)Y=(I-S)Xβ+ε,
whereS={I-(I-S1cS2c)-1(I-S1c)}+{I-(I-S2cS1c)-1(I-S2c)}.SinceXicannot be observed, the corrected-profile least-squares estimator ofβis defined as
(15)β^=argminβ[(Y-Wβ)′(I-S)′(I-S)(Y-Wβ)22222222222-nβ′Σuuβ]=[W~′W~-nΣuu]-1W~′(I-S)Y,
whereWis defined similar toXandW~=(I-S)W.When considering the restrictionsAβ=b, Wei and Wang [13] defined the corrected Lagrange function as
(16)F(β,λ)=[(Y-Wβ)′(I-S)′(I-S)(Y-Wβ)-nβ′Σuuβ]+2λ′(Aβ-b),
whereλis a k×1 Lagrange multipliers vector. DifferentiatingF(β,λ)with respect toβandλand setting them to zero, we obtain
(17)∂F(β,λ)∂β=-2W~′(I-S)Y+2W~′W~β-2nΣuuβ+2A′λ=0,∂F(β,λ)∂λ=2(Aβ-b)=0.
Solving (17), we have
(18)β^r=β^-(W~′W~-nΣuu)-1A′[A(W~′W~-nΣuu)-1A′]-1×(Aβ^-b),λ^=[A(W~′W~-nΣuu)-1A′]-1(Aβ^-b),
whereβ^is given by (15).According to Wei and Wang [13],β^ris called the modified restricted profile least-squares estimator of β, but the asymptotic distribution ofβ^ris vacant. In the following, we will give the asymptotic distribution ofβ^r. The following assumptions will be required to derive the main results. These assumptions are common and can be found in the works of Liang et al. [11], Wang et al. [10], and Wei and Wang [13].(A1)
The density functions ofZ1andZ2are bounded away from 0 and have bounded continuous second partial derivatives.(A2)
LetX˘i=Xi-E(Xi∣Zi1)-E(Xi∣Zi2). The matrix ΓX|Z=^E(X˘1X˘1′) is positive-definite, E(εi∣Xi,Zi1,Zi2)=0, and E(|εi|3∣Xi,Zi1,Zi2)<∞.(A3)
The bandwidthsh1 and h2 are of order n-1/5.(A4)
The functionK(·) is a symmetric density function with compact support and satisfies ∫K(u)du=1,∫‍uK(u)du=0,∫‍u2K(u)du=1, and ∫‍u4K(u)du<∞.(A5)
EUi=0andE(∥Ui∥3)<∞.Under these assumptions, we give the following theorem that states the asymptotic distribution ofβ^r.Theorem 1.
Suppose that conditions (A1)–(A5) hold andεiis homoscedastic with varianceσ2and independent ofUi. Then the modified restricted profile least-squares estimatorβ^ris asymptotically normal. Namely,
(19)n(β^r-β)⟶𝒟N(0,Σ),n⟶∞,
where “→𝒟” denotes the convergence in distribution and
(20)Σ=ΓX∣Z-1ΛΓX∣Z-1-ΓX∣Z-1ΛΓX∣Z-1ΞΓX∣Z-1-ΓX∣Z-1ΞΓX∣Z-1ΛΓX∣Z-1+ΓX∣Z-1ΞΓX∣Z-1ΛΓX∣Z-1ΞΓX∣Z-1,Λ=E(ε1-U1′β)2ΓX∣Z+E{(U1U1′-Σuu)β}⊗2+Σuuσ2,Ξ=A′(AΓX∣Z-1A′)-1A,R⊗2=RR′.Remark 2.
From the previous theorem, it is easy to check that whenXis observed exactly the asymptotic distribution of modified restricted profile least-squares estimatorβ^ris the same as the asymptotic distribution obtained by Theorem 3.1 which appeared in the works of Wei and Liu [16].
To apply Theorem1 to inference, we need to estimateΓX|ZandΛ. Using plug-in method, a consistent estimatorΣ^ofΣcan be obtained. Wei and Wang [13] proposed to estimateΓX|ZandΛ, respectively, by
(21)Γ^X∣Z=1n∑i=1n‍W~iW~i′-Σuu,(22)Λ^=1n∑i=1n‍{W~i(Y~i-W~i′β^)+Σuuβ^}⊗2.The asymptotic distribution ofBβ^r, whereBis ans×pmatrix withrank(B)=s, can be given by the following result.Corollary 3.
Suppose that conditions (A1)–(A5) hold andεiis homoscedastic with varianceσ2and independent ofUi. Then asn→∞, one has
(23)n(Bβ^r-Bβ)⟶𝒟N(0,BΣB′).Using Corollary3, confidence regions of any linear combination of the parameter components given in advance can be constructed. In particular, for any0<α<1, letχ1-α2(p)denote the(1-α)-quantile ofχ2(p). Then
(24){Bβ∈Rs:(Bβ-Bβ^r)′(1nBΣ^B′)-1(Bβ-Bβ^r)≤χ1-α2(p)(1nBΣ^B′)-1}
constitute a confidence region ofBβwith asymptotic coverage1-α.


## 2.2. Modified Profile Lagrange Multiplier Test and Its Asymptotic Properties

Using the estimation method described in Section2.1, we now consider the following linear hypothesis:
(25)H0:Aβ=b,H1:Aβ=b+δ,
whereδis ak×1positive constant vector. Wei and Wang [13] constructed a test statistic based on the difference between the corrected residual sums of squares under the null and alternative hypotheses and showed that the asymptotic null distribution of proposed statistic is a weighted sum of independent standardχ2(1), so adjustment is needed. Notice that the statistics proposed in Wei and Wu [14] and Zhang et al. [15] can achieve the standard chi-squared limit, so in this paper, we use the similar idea to construct the profile Lagrange multiplier test statistic.Using the estimator of Lagrange multiplier defined in (18), the modified profile Lagrange multiplier test statistic can be constructed by
(26)Tn=1nλ^′(C0L0C0′)-1λ^,
whereC0=(AΓ^X|Z-1A′)-1,L0=AΓ^X|Z-1Λ^Γ^X|Z-1A′, and Γ^X|Z and Λ^ are estimators of ΓX|Z and Λ defined by (21) and (22), respectively.The following theorem gives the asymptotic distribution of the modified profile Lagrange multiplier test statisticTn.Theorem 4.
Suppose that conditions (A1)–(A5) hold. Then(1)
under the null hypothesisH0 of testing problem (25),
(27)Tn⟶𝒟χ2(k),asn⟶∞,(2)
under the alternative hypothesisH1of testing problem (25),Tnfollows the asymptotic noncentral χ2(k,ζ) distribution with k degrees of freedom, and the noncentral parameter is
(28)ζ=limn⟶∞n(Aβ-b)′(CLC′)-1(Aβ-b),where C=(AΓX|Z-1A′)-1,L=AΓX|Z-1ΛΓX|Z-1A′.Remark 5.
From the above theorem, we known that our results are the same to the results of adjusted test statistic derived by Wei and Wang [13], but our proposed test statistic is more easier to perform, especially for the case when the covariance matrix Σuu of measurement error is unknown. From Wei and Wang [13] we know that, when Σuu is unknown, it is difficult to estimate RSS(H0) and RSS(H1); thus their proposed test statistic is unapplicable in this case. So our proposed profile Lagrange multiplier test statistic is more attractive.


## 3. Two-Stage Restricted Estimator for the Nonparametric Component

According toβ^r, the corrected backfitting estimators, say F~1=(f~1(Z11),…,f~1(Zn1))′ and F~2=(f~2(Z12),…,f~2(Zn2))′, of F1 and F2 can be defined by
(29)F~1={I-(I-S1cS2c)-1(I-S1c)}(Y-Wβ^r),F~2={I-(I-S2cS1c)-1(I-S2c)}(Y-Wβ^r).From (29), we know thatF~1andF~2are only the estimators of f1(·) and f2(·) at the observations (Z11,…,Zn1)′ and (Z12,…,Zn2)′, respectively. We next give the two-stage restricted estimator of f1(·).Noting the fact thatf1(z1)=E[Y-W′β-f2(Z2)∣Z1=z1] and using the local linear expansion, we define a two-stage restricted estimator of (f1(z1),f1′(z1))′ by
(30)(f^1R(z1),f^1′R(z1))′=argmina0,a1∑i=1n‍[Yi-Wi′β^r-f~2(Zi2)-a02222222222222-a1(Zi1-z1)Yi-Wi′β^r-f~2(Zi2)-a0]2Kh(Zi1-z1),
whereKh(x)=(1/h)K(x/h)andhis a bandwidth. Then the two-stage restricted estimator f^1R(z1) of f1(z1) can be written as
(31)f^1R(z1)=(1,0)(𝒵1′Ω𝒵1)-1𝒵1′Ω(I-S2cS1c)-1(I-S2c)×(Y-Wβ^r),
where Ω=diag((1/h)K((Z11-z1)/h),…,(1/h)K((Zn1-z1)/h)).Letp1(z1)denote the density function ofZ1, v0=∫‍K2(u)du,μ2=∫‍u2K(u)du. The following theorem gives the asymptotic normality of the two-stage restricted estimator.Theorem 6.
Suppose that conditions (A1)–(A5) hold andnh5=O(1). If σ2(z1)=E(ε2∣Z1=z1) has continuous derivative, then as n→∞, one has
(32)nh(f^1R(z1)-f1(z1)-h22μ2f1′′(z1))⟶𝒟N(0,Σ(z1)),
where Σ(z1)=((σ2(z1)+β′Σuuβ)/p1(z1))v0.Remark 7.
From Theorem6, we known that the two-stage restricted estimator f^1R(z1) is asymptotically normal and has an oracle property; that is, the estimator performs as if the other nonparametric component f2(z2) was known though it was unknown. Simulation studies further confirm our theory result. Similarly, the two-stage restricted estimator of f2(z2) and its asymptotic result can be obtained.


## 4. Simulation

In this section, some simulations are carried out to evaluate the finite sample performance of the testing procedure and the proposed two-stage restricted estimator for nonparametric components.In our simulation studies, consider the following additive partially linear EV model:(33)Yi=Xi1β1+Xi2β2+f1(Zi1)+f2(Zi2)+εi,Wij=Xij+Uij,j=1,2,
where Xi1,Xi2,Zi1,andZi2 are mutually independent variables and β1=3,β2=2,Xi1~N(0,1), and Xi2~N(2,1). f1(Zi1)=Zi1exp(-Zi12),f2(Zi2)=2sin(Zi2), and Zi1,Zi2 are generated from a uniform distribution U[-1,1]. The model error εi~N(0,0.22) and measure error Uij~N(0,σu2). The kernel function is taken as the Epanechnikov kernel K(t)=(3/4)(1-t2)I(|t|≤1).We first study the performance of the proposed testing procedure. Consider that the null hypothesis isH0:Aβ=5 with A=(1,1) and the corresponding alternative hypothesis is H1:Aβ=5+δ with δ being a series of positive constants. For δ=0, the alternative hypothesis becomes the null hypothesis. We take the bandwidths h1=h2=0.8·n-1/5, the sample sizes n=200,300, and σu2=0.25,04. To illustrate the effectiveness of the proposed test statistics, the estimated power function curves with the significance level α=0.05 are plotted for σu2=0.4 and σu2=0.25. For each case, we repeat 1000 times. The results are given in Figure 1.The plot of the estimated power function for sample sizesn=200 and n=300, where the solid line denotes n=300 and the dotted line denotes n=200. (a) shows the estimated power function curves with σu2=0.4. (b) shows the estimated power function curves with σu2=0.25.
(a)
(b)From Figure1, we can see the following results.(1)
Our proposed test statistic is sensitive to the corresponding alternative hypothesis. This can be seen from the fact that if we increase the constantδa little, the estimated power function increases rapidly.(2)
The measurement errors affect the power function. For the same sample sizes, increasing the variance of the measurement error results in the decrease of the estimated power function. For the same measurement error, when the observation sample sizes increase, the estimated power function also increases. So our proposed procedure is feasible and easy to perform.Second, we give the finite sample performance of the two-stage restricted estimator for nonparametric components with the restricted condition thatAβ=5. For saving space, only the simulation results for estimators of f1are presented. We compare two estimators: one is the proposed two-stage restricted estimator for nonparametric component by (30) and the other is the benchmark estimator, which has the same form as the two-stage restricted estimator except that when we estimatef1we assume thatf2is known. We use h1=h2=n-1/5 to estimate β^r and the optimal bandwidth h chosen by the leave-one-out cross-validation method to estimate the nonparametric component f1. The performance of estimators is evaluated by the root averaged squared errors (RASEs):
(34)RASE(f^1)=n-1∑i=1n‍[f^1(Zi1)-f1(Zi1)]2.
We take n=200,300,500 and σu2=0.25,0.4,0.6, respectively. For a given sample size, the sample means (SMs) and standard deviations (STDs) of RASEs are calculated. In each simulation, the repeated number is 1000. The simulation results are listed in Table 1.Table 1
The finite sample performance of the estimators forf1 under different sample sizes n and Σuu.
Σ
u
u
n
=
200
n
=
300
n
=
500
Σ
u
u
=
0.25
I
2
Two-stage
SM
0.2249
0.1887
0.1530
STD
0.1152
0.0915
0.0552
Benchmark
SM
0.2239
0.1762
0.1494
STD
0.0958
0.0764
0.0533
Σ
u
u
=
0.4
I
2
Two-stage
SM
0.2787
0.2298
0.1923
STD
0.1245
0.1077
0.0930
Benchmark
SM
0.2784
0.2284
0.1905
STD
0.1235
0.1043
0.0916
Σ
u
u
=
0.6
I
2
Two-stage
SM
0.3253
0.2609
0.2162
STD
0.1728
0.1375
0.1038
Benchmark
SM
0.3200
0.2602
0.2153
STD
0.1707
0.1344
0.1008From Table1, it can be seen that, with the increase of the sample sizen, the finite sample performance of the two estimators improves and the performance of the two-stage restricted estimator is close to that of the benchmark estimator. So the proposed two-stage restricted estimator of the nonparametric component has an oracle property.


## 5. Proofs

Firstly, some lemmas will be given.Lemma 8.
Let(X1,Y1),…,(Xn,Yn) be i.i.d random vectors, where Yi is scalar random variable. Further assume that E|Y|s<∞ and supx∫‍|y|sp(x,y)dy<∞, where p(x,y) denotes the joint density of (X,Y). Let K(·) be a bounded positive function with a bounded support, satisfying a Lipschitz condition. Given that n2δ-1h→∞ for some δ<1-s-1, then
(35)supx|1n∑i=1n‍{Kh(Xi-x)Yi-E[Kh(Xi-x)Yi]}|=Op(log(1/h)nh)1/2.This is Lemma 7.1 of Fan and Huang [17].Lemma 9.
Suppose that assumptions (A1)–(A5) hold. Then the following asymptotic approximations hold uniformly over all the elements of the matrices:(36)Sdc=Sd-11′n+op(11′n),(I-S1cS2c)-1=I+op(1n),
where d=1,2. Similarly, the second formula is true for (I-S2cS1c)-1.This lemma is Lemma 3.1 and Lemma 3.2 of Opsomer and Ruppert [1].Lemma 10.
Suppose that assumptions (A1)–(A5) hold. Then asn→∞, it holds that
(37)1nW′(I-S)′(I-S)W-Σuu=1nW~′W~-Σuu⟶PΓX∣Z,
where W~=(I-S)W.The proof of this lemma can be found in the works of Wang et al. [10].Lemma 11.
Suppose that assumptions (A1)–(A5) hold. Then the corrected-profile least-squares estimatorβ^ofβis asymptotically normal. Namely,
(38)n(β^-β)⟶𝒟N(0,ΓX∣Z-1ΛΓX∣Z-1),n⟶∞.This lemma is just Theorem 1 of Liang et al. [11] and is same as Theorem 3.2 of Wang et al. [10].Proof of Theorem1.
Let(39)Jn=I-(W~′W~-nΣuu)-1A′[A(W~′W~-nΣuu)-1A′]-1A=I-(1nW~′W~-Σuu)-1A′[A(1nW~′W~-Σuu)-1A′]-1A.
By Lemma10, we can obtain
(40)Jn⟶PJ=I-ΓX∣Z-1A′(AΓX∣Z-1A′)-1A.
By the first equation of (18), we have
(41)β^r-β={I-(W~′W~-nΣuu)-1A′22222×[A(W~′W~-nΣuu)-1A′]-1A}×(β^-β)=J(β^-β)+(Jn-J)(β^-β).
Using the results thatJn-J=oP(1) and β^-β=OP(n-1/2), we get
(42)(Jn-J)(β^-β)=oP(n-1/2).
By the Slutsky theorem and Lemma11, we can derive the result after some calculations.Proof of Theorem4.
(1) Using Lemma10, we know that
(43)C0=(A(1nW~′W~-Σuu)-1A′)-1⟶P(AΓX∣Z-1A′)-1=C.
Similarly, we can prove that(44)L0=AΓ^X∣Z-1Λ^Γ^X∣Z-1A′⟶PAΓX∣Z-1ΛΓX∣Z-1A′=L.
Under the null hypothesis of testing problem (25) and by applying Lemma 11, we can prove that
(45)n(Aβ^-b)⟶𝒟N(0,L).
By the second equation of (18), we get
(46)1nλ^=[A(1nW~′W~-Σuu)-1A′]-1n(Aβ^-b),
Using (43), (31), and the Slutsky theorem, we derive that
(47)1nλ^⟶𝒟N(0,CLC′).
By (43)–(45), it is easy to derive that, under the null hypothesis of testing problem (25),
(48)Tn=1nλ^′(C0L0C0′)-1λ^⟶𝒟χ2(k).
(2) Under the alternative hypothesis and again by applying Lemma11, we have
(49)n(Aβ^-b)=nA(β^-β)+n(Aβ-b)⟶𝒟N(n(Aβ-b),L).
Using the similar argument as in (47), we can prove that
(50)1nλ^=[A(1nW~′W~-Σuu)-1A′]-1n(Aβ^-b)⟶𝒟N(n(Aβ-b),CLC′).
Then under the alternative hypothesis,we have(51)Tn=1nλ^′(C0L0C0′)-1λ^⟶𝒟χ2(k,ζ),
where χ2(k,ζ) denotes the asymptotic noncentral chi-squared distribution with k degrees of freedom and the noncentral parameter is
(52)ζ=limn→∞n(Aβ-b)′(CLC′)-1(Aβ-b).Proof of Theorem6.
Using (31), by some simple calculations, we get
(53)f^1R(z1)=(1,0)(𝒵1′Ω𝒵1)-1𝒵1′Ω(I-S2cS1c)-1×(I-S2c)(Y-Wβ^r)=(1,0)(𝒵1′Ω𝒵1)-1𝒵1′Ω(I-S2cS1c)-1(I-S2c)×[(F1+F2)+(ε-Uβ)+W(β-β^r)]=T1+T2+T3,
where
(54)T1=(1,0)(𝒵1′Ω𝒵1)-1𝒵1′Ω(I-S2cS1c)-1×(I-S2c)(F1+F2),T2=(1,0)(𝒵1′Ω𝒵1)-1𝒵1′Ω(I-S2cS1c)-1×(I-S2c)(ε-Uβ),T3=(1,0)(𝒵1′Ω𝒵1)-1𝒵1′Ω(I-S2cS1c)-1×(I-S2c)W(β-β^r).
Referring to the proof of Theorem 4.1 in Opsomer and Ruppert [1] and using Lemma 9, we obtain that
(55)(I-S2cS1c)-1(I-S2c)(F1+F2)=F1+Op(h12+h22),(I-S2cS1c)-1(I-S2c)(ε-Uβ)=(ε-Uβ)(I+op(1)),(I-S2cS1c)-1(I-S2c)W=(X+U-E(X∣Z2))(I+op(1)).
For Zi1 in a small neighborhood of z1 and |Zi1-z1|<h, for each f1(Zi1), we use the following Taylor expansion:
(56)f1(Zi1)=f1(z1)+f1′(z1)(Zi1-z1)+f1′′(z1)2(Zi1-z1)2+o(h2).
Then we derive that
(57)F1=(f1(Z11)f1(Z21)⋮f1(Zn1))=𝒵1(f1(z1)f1′(z1))+(f1′′(z1)2(Z11-z1)2+o(h2)⋮f1′′(z1)2(Zn1-z1)2+o(h2)).
By Lemma 8 and the usual nonparametric regression results, we can easily get
(58)T1=f1(z1)+f1′′(z1)2μ2h2+op(h2).
Using the result that(59)(𝒵1′Ω𝒵1)-1=1np1(z1)H-1(100μ2)-1H-1(I+op(1))
with H=diag(1,h), after some simple calculations, we can get
(60)nhT2=1np1(z1)nh∑i=1n‍Kh(Zi1-z1)(εi-Ui′β)+op(1).
Note that nh(1/n)∑i=1n‍Kh(Zi1-z1)(εi-Ui′β) is asymptotically normal with mean 0 and variance
(61)v0p1(z1)E[(ε1-U1′β)2∣Z1=z1]=v0p1(z1)(σ2(z1)+β′Σuuβ).
By the Slutsky theorem, we can derive that
(62)nhT2⟶𝒟N(0,σ2(z1)+β′Σuuβp1(z1)v0).
Using similar calculation to that ofT2 and the result derived from Theorem 1 that β-β^r=Op(n-1/2), then we have
(63)nhT3=nh(p1(z1))-11n∑i=1n‍Kh(Zi1-z1)×(Xi-E(Xi∣Z2))Op(n-1/2)+op(1)=op(1).
By using the Slutsky theorem again, we derive the desired result.



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*Source: 594391-2013-09-25.xml*