Profile Inferences on Restricted Additive Partially Linear EV Models

and Applied Analysis 3 where e 1 = (1, 0) 󸀠, Ω 1 = diag( 1 h 1 K( Z 11 − z 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 where  is the response,  and  = ( 1 , . . .,   )  are covariates on   and   , respectively,  1 , . . .,   are unknown smooth functions,  = ( 1 , . . .,   )  is a -dimensional vector of unknown parameters, and  is the random error with conditional mean zero given  and .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  = 1 for model (1), and additive model, when  = 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 covariate  is measured with additive error; that is,  cannot be observed, but an unbiased measure of , denoted by , can be obtained such that where  is the measurement error with mean zero and independent of (, , ).Like Liang et al. [11], we only consider the case of  = 2 and assume that the covariance matrix of measurement error , say Σ  , is known.Otherwise, we can estimate it by repeatedly measuring  as 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: where  is a  ×  matrix of known constants, rank() = , and b is a -vector of known constants.They first proposed the restricted corrected-profile least-squares estimator β of  under the restriction  = 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  2 (1), so adjustment is needed, and the adjusted test statistic is also suggested.Though they proposed the restricted corrected-profile least-squares estimator β of  under the condition  = b, they did not investigate the asymptotic property of β .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 is obtained, for a different matrix , we can easily derive the asymptotic distribution of different constrained estimates  β with 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  on 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 Σ  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 Section 2, we first review the restricted corrected-profile least-squares estimator β of  and then study the asymptotic distribution of β .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, twostage 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.

Asymptotic Results of the
with  being a  ×  matrix of known constants and rank() =  and b being a -vector of known constants.First, we need introduce the restricted-profile leastsquares estimator of Wei and Wang [13].The notations are like those of Wang et al. [10].If  is known, the first equation of model (4) can be rewritten as Denote Hence, (6) can be written as Let   1, 1 ,   2, 2 denote the equivalent kernels for the local linear regression at  1 ,  2 : Abstract and Applied Analysis 3 where for a kernel function (⋅) and bandwidths ℎ 1 , ℎ 2 , and ) , are  × 2 design matrices.Let  1 and  2 represent the smoother matrices whose rows are the equivalent kernels at the observations Z 1 and Z 2 , respectively: Using the backfitting algorithm, we get the backfitting estimators of  1 and  2 , say F1 and F2 , which can be expressed as where    = ( − 11  /)  is the centered smoothing matrix corresponding to   ,  = 1, 2. Substituting ( 13) into (8), we have where Since   cannot be observed, the corrected-profile leastsquares estimator of  is defined as where W is defined similar to X and W = ( − )W.
When considering the restrictions  = b, Wei and Wang [13] defined the corrected Lagrange function as where  is a  × 1 Lagrange multipliers vector.Differentiating (, ) with respect to  and  and setting them to zero, we obtain Solving ( 17), we have where β is given by (15).According to Wei and Wang [13], β is called the modified restricted profile least-squares estimator of , but the asymptotic distribution of β is vacant.In the following, we will give the asymptotic distribution of β .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].
Under these assumptions, we give the following theorem that states the asymptotic distribution of β .Theorem 1. Suppose that conditions (A1)-(A5) hold and   is homoscedastic with variance  2 and independent of   .Then the modified restricted profile least-squares estimator β is asymptotically normal.Namely, where " D  →" denotes the convergence in distribution and Remark 2. From the previous theorem, it is easy to check that when  is observed exactly the asymptotic distribution of modified restricted profile least-squares estimator β is the same as the asymptotic distribution obtained by Theorem 3.1 which appeared in the works of Wei and Liu [16].
To apply Theorem 1 to inference, we need to estimate Γ | and Λ.Using plug-in method, a consistent estimator Σ of Σ can be obtained.Wei and Wang [13] proposed to estimate Γ | and Λ, respectively, by The asymptotic distribution of  β , where  is an  ×  matrix with rank() = , can be given by the following result.
Corollary 3. Suppose that conditions (A1)-(A5) hold and   is homoscedastic with variance  2 and independent of   .Then as  → ∞, one has Using Corollary 3, confidence regions of any linear combination of the parameter components given in advance can be constructed.In particular, for any 0 <  < 1, let  2  1− () denote the (1 − )-quantile of  2 ().Then constitute a confidence region of  with asymptotic coverage 1 − .

Modified Profile Lagrange Multiplier Test and Its
Asymptotic Properties.Using the estimation method described in Section 2.1, we now consider the following linear hypothesis: where  is a  × 1 positive 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 standard  2 (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 where |   , and Γ| and Λ are estimators of Γ | and Λ defined by ( 21) and ( 22), respectively.
The following theorem gives the asymptotic distribution of the modified profile Lagrange multiplier test statistic   .

Theorem 4. Suppose that conditions (A1)-(A5) hold. Then
(1) under the null hypothesis  0 of testing problem (25), (2) under the alternative hypothesis  1 of testing problem (25),   follows the asymptotic noncentral  2 (, ) distribution with k degrees of freedom, and the noncentral parameter is where 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 Σ  of measurement error is unknown.From Wei and Wang [13] we know that, when Σ  is unknown, it is difficult to estimate ( 0 ) and ( 1 ); thus their proposed test statistic is unapplicable in this case.So our proposed profile Lagrange multiplier test statistic is more attractive.
Remark 7. From Theorem 6, we known that the two-stage restricted estimator f 1 ( 1 ) is asymptotically normal and has an oracle property; that is, the estimator performs as if the other nonparametric component  2 ( 2 ) was known though it was unknown.Simulation studies further confirm our theory result.Similarly, the two-stage restricted estimator of  2 ( 2 ) and its asymptotic result can be obtained.

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.
We first study the performance of the proposed testing procedure.Consider that the null hypothesis is  0 :  = 5 with  = (1, 1) and the corresponding alternative hypothesis is  1 :  = 5 +  with  being a series of positive constants.For  = 0, the alternative hypothesis becomes the null hypothesis.We take the bandwidths ℎ 1 = ℎ 2 = 0.8⋅ −1/5 , the sample sizes  = 200, 300, and  2  = 0.25, 04.To illustrate the effectiveness of the proposed test statistics, the estimated power function curves with the significance level  = 0.05 are plotted for  2  = 0.4 and  2  = 0.25.For each case, we repeat 1000 times.The results are given in Figure 1.
From Figure 1, 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 constant  a 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 twostage restricted estimator for nonparametric components with the restricted condition that  = 5.For saving space, only the simulation results for estimators of  1 are 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 estimate  1 we assume that  2 is known.We use ℎ 1 = ℎ 2 =  −1/5 to estimate β and the optimal bandwidth ℎ chosen by the leave-one-out cross-validation method to estimate the nonparametric component  1 .The performance of estimators is evaluated by the root averaged squared errors (RASEs): We take  = 200, 300, 500 and  2  = 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.
From Table 1, it can be seen that, with the increase of the sample size , the finite sample performance of the two estimators improves and the performance of the twostage restricted estimator is close to that of the benchmark

Proofs
Firstly, some lemmas will be given.
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: where  = 1, 2. Similarly, the second formula is true for This lemma is Lemma 3.1 and Lemma 3.2 of Opsomer and Ruppert [1].
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 estimator β of  is asymptotically normal.Namely, 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 Theorem 1.Let By Lemma 10, we can obtain (50) Then under the alternative hypothesis, we have where  2 (, ) denotes the asymptotic noncentral chi-squared distribution with  degrees of freedom and the noncentral parameter is Proof of Theorem 6.Using (31), by some simple calculations, we get

Figure 1 :
Figure 1: The plot of the estimated power function for sample sizes  = 200 and  = 300, where the solid line denotes  = 300 and the dotted line denotes  = 200.(a) shows the estimated power function curves with  2  = 0.4.(b) shows the estimated power function curves with  2  = 0.25.

Restricted-Profile Least-Squares Estimator and Modified Profile Lagrange Multiplier Test
|   ,  1 ,  2 ) = 0 almost surely and the measurement errors   are i.i.d. with mean zero and independent of {  ,   = ( 1 ,  2 , . . .,   )  ,  1 ,  2 }.Besides, we also have some prior information on regression parametric vector  that can be presented by the following restricted condition:

Table 1 :
The finite sample performance of the estimators for  1 under different sample sizes  and Σ  .
estimator.So the proposed two-stage restricted estimator of the nonparametric component has an oracle property.