Random weighting M -estimation for linear errors-in-variables models

Journal of the Korean Statistical Society 41 (2012) 505–514

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Random weighting M-estimation for linear errors-in-variables models Rong Jiang, Xiaohan Yang ∗ , Weimin Qian Department of Mathematics, Tongji University, Shanghai, 200092, PR China

article

info

Article history: Received 24 October 2011 Accepted 5 March 2012 Available online 23 March 2012 AMS 2000 subject classifications: primary 62G20 secondary 62N02 Keywords: Errors-in-variables M-estimation Randomly weighting method Linear hypothesis M-test Random weighting M-test

abstract In this paper, we extend the random weighting method to linear errors-in-variables models and propose random weighting M-estimators (RWME) for parameters. Its large sample properties are studied and the consistency and asymptotic normality are proved under mild conditions. In addition, the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters. Extensive simulations are reported, showing that the proposed method works well in practical settings. The proposed methods are also applied to a data set from an AIDS clinical trial group study. Crown Copyright © 2012 Published by Elsevier B.V. on behalf of The Korean Statistical Society. All rights reserved.

1. Introduction Consider a linear errors-in-variables (EV) model as follows:



Y = xT β0 + ε, X = x + u,

(1.1)

where x is a p-dimensional vector of unobserved latent covariates which is measured in an error-prone way, X is the observed surrogate of x, β0 is a p-dimensional unknown parameter vector, Y is the response vector, (ε, uT )T is a p + 1-dimensional spherical error vector, which means that (ε, uT )T =d RUp+1 (R is a nonnegative random variable, Up+1 is a uniform random vector on Ωp = {a : a ∈ Rp+1 , ∥a∥ = 1}, R and Up+1 are independent), and (ε, uT )T and x are independent. Model (1.1) belongs to a kind of model called the EV model or the measurement error model which was proposed by Deaton (1985) to correct for the effects of sampling errors and is somewhat more practical than the ordinary regression model. Fuller (1987) gave a systematic survey on this research topic and present many applications of measurement error data. M-estimation refers to a general method of estimation, where the estimates are obtained by optimizing some objective functions. The most widely used M-estimators include maximum likelihood, ordinary least-squares, and least absolute deviation estimators. The asymptotic theories and inference procedures for M-estimation have been extensively studied in Cheng and Huang (2010), Cui (1997), Delecroix, Hristache, and Patilea (2006), Lee and Pun (2006) and Ma and Kosorok (2005). It is well known that the asymptotic distribution of the estimators by M-estimation is generally related to nuisance parameters which cannot be conveniently estimated. The randomly weighting method can provide a way of assessing the distribution of the estimators without estimating the nuisance parameter.

∗

Corresponding author. E-mail addresses: [email protected] (R. Jiang), [email protected] (X.H. Yang), [email protected] (W.M. Qian).

1226-3192/$ – see front matter Crown Copyright © 2012 Published by Elsevier B.V. on behalf of The Korean Statistical Society. All rights reserved. doi:10.1016/j.jkss.2012.03.001

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R. Jiang et al. / Journal of the Korean Statistical Society 41 (2012) 505–514

The random weighting method was first proposed by Zheng (1987). In fact, the random weighting method was referred to as the Bayesian bootstrap method (Rubin, 1981), exchangeably weighted bootstrap (Praestgaard & Wellner, 1993) and weighted bootstrap (Barbe & Bertail, 1995). An advantage of the random weighting method is that no observation is repeatedly used within each replica of the random weighting, though each observation may be weighted unequally. This method has been used in many applications as an alternative to the bootstrap method. For example, Rao and Zhao (1992) used this method to derive the approximate distribution of the M-estimator in the linear regression model. Cui, Li, Yang, and Wu (2008) proposed a random weighting method for the proportional hazards model. Wang, Wu, and Zhao (2009) extended the method to the censored regression model. Jiang, Qian, and Zhou (2011) discussed randomly weighting least square estimators for the unknown parameters in the semi-linear EV model. The random weighting M-estimation is different from the generalized bootstrap method proposed by Chatterjee and Bose (2005) the estimators of which are obtained by solving estimating equations. In this paper, our objective is to apply the random weighting M-estimation to errors-in-variables models, and establish the asymptotic normality of the RWME for the parameter. These results can be used to construct confidence intervals for β0 . Furthermore, we propose a M-test for errors-in-variables models. The M-test has been used by Zhao and Chen (1991) to test linear hypotheses in the linear model. But the critical values of the test statistic are related to estimators of nuisance parameters. Chen, Ying, Zhang, and Zhao (2008) proposed an easy and convenient randomly weighting resampling method to determine the critical values for testing linear hypotheses in least absolute deviation regression. Motivated by this idea, we also use the random weighting method to determine the critical values for testing hypothesis in errors-in-variables models. This paper is organized as follows. The RWME method for linear errors-in-variables models is proposed in Section 2, the asymptotic properties of the proposed estimators are also given in this section. Some simulations and a real data application are conducted in Section 3 to illustrate our methodology. Final remarks are given in Section 4. All the conditions and technical proofs are collected in the Appendix. 2. Methodology and main results 2.1. Random weighting M-estimation method for EV model Based on the observations {Yi , Xi }ni=1 , Cui (1997) proposed the M-estimation βˆ to estimate β by

βˆ = arg min β

n 

 ρ

Yi − XiT β





1 + ∥β∥2

i =1

,

(2.1)

where ρ(·) is a non-monotonic convex function defined on R = (−∞, +∞). It was shown in Cui (1997) that under the certain regularity conditions, we have

√

L

n(βˆ − β0 ) − → N (0, c0−2 (1 + ∥β0 ∥2 )Σx−1 S Σx−1 ),

(2.2)

L

where − → stands for convergence in distribution, β0 is the true value of β , c0 = E ψ ′ (ε), Σx = E (xxT ) and S = E ψ 2 (ε)Σx + β0 β T

E [ψ 2 (ε)u211 ](Ip − 1+∥β 0∥2 ), where u11 is the first component of u and 0

ψ(t ) =

 ψ1 (t ), −ψ2 (−t ),

t ≥0 t <0

where ψ1 (t ) and ψ2 (t ) are nonnegative functions defined on [0, +∞). Define the ρ(·) function to be ρ(t ) =

t 0

ψ(x)dx.

But the asymptotic covariance matrix of βˆ involves the density of the errors and nuisance parameters; therefore it is difficult to estimate reliably. To overcome this problem, and motivated by Rao and Zhao (1992), we define

β = arg min ∗

β

n 

 wi ρ

i =1

Yi − XiT β



1 + ∥β∥2

 ,

(2.3)

as the RWME of β0 , where ω1 , . . . , ωn are independently and identically distributed (i.i.d.) non-negative random variables with E (ω1 ) = Var(ω1 ) = 1. In this paper, notations L∗ and P ∗ denote the corresponding probability calculations conditionally on {Yi , Xi }ni=1 . Our main results are as follows. Theorem 1. Assume that under the model (1.1), the conditions A1–A5 hold. Then

√

n(β ∗ − β0 ) = c0−1 εi −uTi β0

where Ai = ψ( √

1+∥β0

∥2



n 1  1 + ∥β0 ∥2 Σx−1 √ wi Ai + op (1), n i =1

)(xi + ui +

(εi −uTi β0 )β0 ). 1+∥β0 ∥2

(2.4)

R. Jiang et al. / Journal of the Korean Statistical Society 41 (2012) 505–514

507

Particularly, when w ≡ 1, we have

√

n(βˆ − β0 ) = c0−1



n 1  1 + ∥β0 ∥2 Σx−1 √ Ai + op (1). n i =1

(2.5)

Theorem 2. Under the conditions of Theorem 1. Then

√

ˆ = c0−1 n(β ∗ − β)



n 1  L∗ 1 + ∥β0 ∥2 Σx−1 √ (wi − 1)Ai + op (1) − → N (0, c0−2 (1 + ∥β0 ∥2 )Σx−1 S Σx−1 ). n i =1

Further by (2.2) and (2.6), the multi-dimensional Kolmogorov–Smirnov distance between

√

√

ˆ and n(β ∗ − β)

(2.6)

√

n(βˆ − β0 ) is

√

ˆ ≤ u) − P ( n(βˆ − β0 ) ≤ u)| → 0, sup |P ∗ ( n(β ∗ − β)

in probability

(2.7)

u

as n → ∞, where u runs over all p-vectors, and the inequality between vectors means coordinatewise inequality. Remark 1. From Theorems 1 and 2, it is clear that β ∗ is a consistent estimator of β0 and the conditionally limiting distribution of β ∗ for observations given is the same as that of βˆ . Consequently, we can take the conditional distribution of β ∗ as an approximation to that of βˆ without estimating the asymptotic covariance matrix when making confidence interval for parameters. In practical applications, this can be done by the Monte Carlo method. Specifically, one can generate random weights repeatedly for (2.3) and then obtain RWME of the regression parameters. Then the empirical distribution of the produced estimates is used as an approximation to the distribution of the M-estimator of β0 . For example, in deriving the (1 − α)100% confidence interval for β0 , one can implement the random weighting N times to obtain the estimates β ∗(1) , β ∗(2) , . . . , β ∗(N ) and hence use the lower and upper α/2 quantiles of these quantities as the approximation of lower and upper limits of the confidence interval. 2.2. M-test In this paper, we are also interested in hypothesis testing H0 : H T (β − b0 ) = 0 ←→ H1 : H T (β − b0 ) ̸= 0,

(2.8)

where H is a known p × q matrix of rank q, and b0 is a known p-vector (0 < q ≤ p). Zhao and Chen (1991) established the limiting distribution of M-test statistic under the local alternatives. Zhao (2004) extended the method to the censored model. It is natural to consider test statistic Mn =

n 



Yi − XiT β˜

ρ 

i=1

˜ 2 1 + ∥β∥

where β˜ = arg minH T (β−b0 )=0

 −

n 



ρ 

i=1

n

i =1

Yi − XiT βˆ

Y −XiT β

ρ( √i

1+∥β∥2

ˆ 2 1 + ∥β∥

 ,

).

Now we present the asymptotic properties of the test statistic Mn in the following. Theorem 3. Suppose that A1–A4 hold. Then, under the null hypothesis (2.8),

2



n   1  1  T −1/2  Mn = H n Σx Ai  + op (1), √  2c0  n i=1 1/2

where Hn = Σx

1

H (H T Σx H )− 2 and √1n

n

i =1

(2.9) −1/2

HnT Σx

L

−1/2

Ai − → N (0, HnT Σx

−1/2

S Σx

Hn ).

2.3. Random weighting M-test It is rather difficult and inaccurate to estimate the nuisance parameter c0 and S. Moreover, the limiting distribution of Mn has a rather complex expression. Recently, Chen et al. (2008) proposed an easy and convenient randomly weighting resampling method to determine the critical values for testing linear hypotheses in least absolute deviation regression. Zhao, Wu, and Yang (2008) extended the method to the linear model in M-method. Motivated by this idea, we use randomly weighting method to directly determine the critical values instead of estimating the distribution of Mn for testing hypothesis

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R. Jiang et al. / Journal of the Korean Statistical Society 41 (2012) 505–514

in the EV model. Define Mn∗ =

 n 



Yi − XiT β˜ ∗

wi ρ  





n 

Yi − XiT β ∗

 

− wi ρ  1 + ∥β ∗ ∥2  ∗ 2 ˜ i = 1 1 + ∥β ∥      n n    Yi − XiT β˜ Yi − XiT βˆ −  , − wi ρ   wi ρ    i =1 ˜ 2 ˆ 2  i=1 1 + ∥β∥ 1 + ∥β∥  i =1

where β˜ ∗ = arg minH T (β−b0 )=0

n

Y −XiT β

i=1

wi ρ( √i

1+∥β∥2

).

We intend to use the resampling distribution of Mn∗ to approximate the distribution of Mn . This is justified if it can be shown that the conditional distribution of the given Mn∗ data converges to the same limiting distribution as that of Mn . Because the resampling distribution can be approximated arbitrarily close by repeatedly generating a large number of i.i.d. sequences {wi }ni=1 , we can use the conditional empirical distribution of Mn∗ to get critical regions for Mn . Clearly this approach avoids any density estimation. Now we present the asymptotic properties of the random weighting test statistic Mn∗ in the following. Theorem 4. Suppose that A1–A5 hold. Then, under the null hypothesis (2.8),

2



n   1  1   (wi − 1)HnT Σx−1/2 Ai  + op (1) Mn = √  2c0  n i=1

∗

where √1n

n

i =1

−1/2

(wi − 1)HnT Σx

L∗

−1/2

Ai − → N (0, HnT Σx

(2.10) −1/2

S Σx

Hn ). Further by (2.9) and (2.10), we have

L∗ (Mn∗ ) → L(Z ) ← L(Mn ), as n → ∞, where Z is sum of squares of q normal random variables. Remark 2. Theorems 3 and 4 show that the limiting distribution of Mn∗ under the null hypothesis (2.8) is the same as the null limiting distribution of Mn . Therefore, we can directly use conditional distribution of Mn∗ as an approximation to the null distribution of Mn and determine critical values of the test statistic Mn without estimating the nuisance parameters. It is desired to determine a sequence cn (α) such that limn→∞ P (Mn > cn (α)) = α under H0 , for the given level α ∈ (0, 1). As shown in the sequel, the (1 − α) quantile cn∗ (α) of the conditional distribution of Mn∗ for the given {Yi , Xi }ni=1 can be taken as an approximation to cn (α), and this can be carried out by the following procedure. Take N large enough and generate N ∗ independent replicas of random weights to obtain N randomly weighting estimates Mnj , j = 1, . . . , N, then the p-value of ∗ testing hypothesis is approximately equal to ♯{j : Mnj > Mn , j = 1, . . . , N }/N. A test at nominal significance level α is to ∗ ∗ ∗ reject H0 if Mn is larger than the sample (1 − α) quantile of Mn1 , Mn2 , . . . , MnN and accept H0 otherwise. It is easy to show that, for the given nominal significant level α ∈ (0, 1), the test Mn with critical value cn∗ (α) has the same asymptotic level and asymptotic power as the test with critical value cn (α) obtained by estimating nuisance parameters. 3. Simulation In this section, we conduct simulation studies to assess the finite sample performance of the proposed procedures and illustrate the proposed methodology on AIDS clinical trials. Example 3.1. The data are generated from model (1.1), where the random error variable is taken to be the standard normal distribution N (0, 1) and t3 distribution. The explanatory variable x is generated from uniform distribution on (3, 5) and β = 1. The randomly weighting variables ω are taken to be the exponential distribution and the Poisson distribution with means 1 (Exp(1) and P(1) respectively). All of the simulations are run for 500 replicas and the number of randomly weighting variables is N = 500. We first study the performance of parameter estimators by using our proposed method (RWME). Throughout our simulation study, the convex function is taken to be ρ(·) = | · |. The mean values of parameter estimators and their standard errors are respectively reported Table 1. Table 1 shows that the performance of β ∗ is very close to the true value in all terms. Moreover, β ∗ is much more accurate when sample sizes increase. We next investigate the length of confidence intervals and empirical coverage rates by the random weighting method at the nominal levels 90% and 95%. Simulation results are reported in Tables 2 and 3 respectively. From Table 2, it can be seen that the empirical coverage rates are reasonably close to the true values in all cases, which indicates that the randomly weighting method is valid. As expect, the coverage levels based on the different cases are much closer to the nominal levels when sample sizes increase. Table 3 shows that the length of confidence intervals are small. Not unexpectedly, the length

R. Jiang et al. / Journal of the Korean Statistical Society 41 (2012) 505–514

509

Table 1 Simulation results for β ∗ .

ε

w

50

100

200

N (0, 1)

Exp(1) P(1)

1.0059 (0.0580) 1.0061 (0.0578)

1.0033 (0.0412) 1.0032 (0.0411)

1.0027 (0.0298) 1.0027 (0.0297)

t3

Exp(1) P(1)

1.0437 (0.0417) 1.0436 (0.0427)

1.0413 (0.0316) 1.0411 (0.0317)

1.0409 (0.0204) 1.0409 (0.0224)

n

Table 2 Simulation results for coverage probability of confidence intervals.

ε

w

0.90

0.95

50

100

200

50

100

200

N (0 , 1 )

Exp(1) P(1)

0.8879 0.8882

0.8890 0.8890

0.8951 0.8938

0.9368 0.9368

0.9399 0.9406

0.9414 0.9410

t3

Exp(1) P(1)

0.8874 0.8862

0.8917 0.8897

0.8916 0.8913

0.9356 0.9347

0.9401 0.9393

0.9394 0.9405

Table 3 Simulation results for length of confidence intervals.

ε

w

0.90

0.95

50

100

200

50

100

200

N (0 , 1 )

Exp(1) P(1)

0.2153 0.2161

0.1511 0.1512

0.1012 0.1016

0.2582 0.2597

0.1800 0.1799

0.1221 0.1216

t3

Exp(1) P(1)

0.1579 0.1588

0.1109 0.1098

0.0767 0.0771

0.1887 0.1905

0.1301 0.1312

0.0921 0.0926

Table 4 Empirical significant levels and powers.

β2

n

N (0, 1)

t2

0.01

0.05

0.01

0.05

50

0 0.1 0.2 0.5

0.0720 0.2300 0.3700 0.9900

0.0440 0.1300 0.3100 0.9600

0.0540 0.2200 0.4800 0.8700

0.0260 0.1200 0.3800 0.8300

100

0 0.1 0.2 0.5

0.0840 0.2400 0.5700 0.9900

0.0540 0.1500 0.4800 0.9900

0.0660 0.4100 0.7300 0.9600

0.0340 0.2900 0.6100 0.9500

200

0 0.1 0.2 0.5

0.0800 0.4200 0.8800 1.0000

0.0520 0.3300 0.8100 1.0000

0.0880 0.6900 1.0000 1.0000

0.0380 0.6100 0.9700 0.9700

of confidence intervals decreases with sample sizes. Finally, Tables 1–3 show that the performances of Poisson weights are exactly similar to that of exponential weights. Example 3.2. We perform simulations to study properties of the proposed test with practical sample sizes. The simulation results presented here are under model



Y = x1 β1 + x2 β2 + ε, X = x + u,

where x is generated from standard normal distribution and the error ε and u are generated from N (0, 1) and t2 distribution. The null hypothesis is H0 : β2 = 0. The β1 is taken to be 1. Here, the randomly weighting variables are only taken to be exponential distribution with means 1. Fig. 1 is quantile–quantile plots of Mn and Mn∗ when sample sizes are n = 200. From Fig. 1, we can see that the conditional distributions of Mn∗ are closer to distributions of Mn . Table 4 lists the power functions at significance level α = 0.05 and 0.01. From Table 4, we can see that the empirical significance levels are close to the nominal significance levels when the null is true, which indicates that the randomly weighting test is a valid test.

510

R. Jiang et al. / Journal of the Korean Statistical Society 41 (2012) 505–514

(a) ε ∼ N (0, 1).

(b) ε ∼ t2 .

Fig. 1. Q–Q plot of Mn∗ vs. Mn .

Example 3.3. In this section, we present an analysis of an AIDS clinical trial group (ACTG 315) study. One of the purposes of this study is to investigate the relationship between virologic and immunologic responses in AIDS clinical trials. In general, it is believed that the virologic response RNA (measured by viral load) and immunologic responses (measured by CD4+ cell counts) are negatively correlated during treatment. Our preliminary investigations suggested that viral load depends linearly on CD4+ cell count. We therefore model the relationship between viral load and CD4+ cell counts by model (1.1). Let Yi be the viral load and let xi be the CD4+ cell count for subject i. To reduce the marked skewness of CD4+ cell counts, and make treatment times equal space, we take log-transformations of both variables. The xi are measured with error (Liang, Wu, & Carroll, 2003). The model we used is Y = β0 + xβ1 + ε,

X = x + u,

where X is the observed CD4 cell counts. The parameter estimators by using our proposed methods are (β0 , β1 ) = (2.7252, −0.0895) and the M estimator is (2.7090, −0.0842). The 95% confidence interval of β0 is (2.6214, 2.7965) and that of β1 is (−0.1092, −0.0592). It can be seen that the length of confidence intervals is small. Furthermore, we test linear hypothesis H0 : β1 = 0. The resulting p-value is 0, suggesting that β1 is significant. 4. Discussion The aim of this paper is to provide convenient inference and linear hypothesis testing for the linear EV model based on M-estimate. The proposed inference procedure via resampling avoids the difficulty of density estimation and is convenient to implement with the availability of the standard linear programming and computing power. All simulation studies confirm that the performance of the random weighting method works well. In addition, we believe that the method can be extended to cases with presence of censorship, which are common in survival analysis (see Ma & Yin, 2011). Appendix To prove main results in this paper, the following technical conditions are imposed. A1. ∀t , ψ(t ) = {y : ψ(y) = t } is a union of V disjoint intervals (V is a positive integer and a single point is defined as a closed interval). A2. E ψ(ε) = 0, sign(t )E ψ(ε + t ) ≥ 0, and there exists ∆ > 0 such that |E ψ(ε + t )| > 0, 0 ̸= t ∈ [−∆, ∆]. A3. The discontinuous points of ψ(t ) are at most countable, E ψ(ε) = 0, 0 < E ψ 2 (ε) < ∞ and there exist positive constants c0 , c1 such that |E ψ(ε + t )| ≤ c1 |t |, and E ψ(ε + t ) = c0 t + o(t ), (t → 0). A4. P {R = 0} = 0, Σx = E (xxT ) > 0 and E (R2 + ∥x∥2 )[1 + sup|t |≤R+∥x∥ ψ 2 (ε + t )] < +∞. A5. The random weights ω1 , . . . , ωn are i.i.d. with P (ω1 ≥ 0), E (ω1 ) = Var(ω1 ) = 1, and the sequence {ωi } and {Yi , Xi , xi } are independent. Remark 3. Conditions A1–A4 are standard conditions in the M-estimation for linear errors-in-variables models, see Cui (1997). In addition, conditions A5 is commonly assumed in the random weighting method, see Wang et al. (2009).

R. Jiang et al. / Journal of the Korean Statistical Society 41 (2012) 505–514

511

To prove the theorem, we first introduce the following lemma. Lemma 1. Under the conditions of Theorem 1, we have n 1

Q (β) =

n i=1

  wi ρ





1 + ∥β∥2



1 + ∥β0 ∥2

(β − β0 ) Σx (β − β0 ) − T

2(1 + ∥β0 ∥2 )



Yi − XiT β0

−ρ



c0

=

Yi − XiT β

n 1

n i =1

Ai (β − β0 )

wi 

1 + ∥β0 ∥2

+ o(∥β − β0 ∥ ) + op 2



 ∥β − β0 ∥ . √ n

Proof.

 ρ



Yi − XiT β



 −ρ

1 + ∥β∥2



Yi − XiT β0



1 + ∥β0 ∥2

 

   εi − uTi β xTi (β − β0 ) εi − uTi β0 = ρ  − −ρ  1 + ∥β∥2 1 + ∥β∥2 1 + ∥β0 ∥2    εi − uTi β0 εi − uTi β xTi (β − β0 ) εi − uTi β0  − − −ψ  1 + ∥β0 ∥2 1 + ∥β∥2 1 + ∥β∥2 1 + ∥β0 ∥2    εi − uTi β0 xTi (β − β0 ) εi − uTi β0 εi − uTi β  +ψ  − − 1 + ∥β0 ∥2 1 + ∥β∥2 1 + ∥β∥2 1 + ∥β0 ∥2 , f1i (β) + f2i (β).

Consequently, Q (β) =

n 1

n i=1

wi f1i (β) +

n 1

n i =1

wi f2i (β).

Apply the proof of Theorem 3 in Cui (1997), we can obtain EQ (β) = n 1

n i=1 n 1

n i=1

c0 2(1 + ∥β0 ∥2 )

(β − β0 )T Σx (β − β0 ) + o(∥β − β0 ∥2 ),

[wi f1i (β) − E wi f1i (β)] = op [wi f2i (β) − E wi f2i (β)] = −



 ∥β − β0 ∥ , √ n

n 1

n i=1

Ai (β − β0 )



wi 

1 + ∥β0 ∥2

+ op

 ∥β − β0 ∥ . √ n

Then Q (β) = EQ (β) +

=

n 1

n 1

n i=1

n i=1

c0 2(1 + ∥β0 ∥2 )

[wi f1i (β) − E wi f1i (β)] +

(β − β0 )T Σx (β − β0 ) −

n 1

n i =1

[wi f2i (β) − E wi f2i (β)] Ai (β − β0 )

wi 

1 + ∥β0 ∥2

+ o(∥β − β0 ∥2 ) + op



 ∥β − β0 ∥ .  √ n

Next we proceed to prove the theorems. Proof of Theorem 1. Write

√

n(β¯ − β0 ) = c0−1



n 1  wi Ai . 1 + ∥β0 ∥2 Σx−1 √ n i =1

It is easily seen that under the conditions of Theorem 1, by the Central Limit Theorem, we have

√

L

n(β¯ − β0 ) − → N (0, c0−2 (1 + ∥β0 ∥2 )Σx−1 S Σx−1 ).

Thus, to prove the theorem, it suffices to show that

√ √ ¯ = op (1), ∥ nβ ∗ − nβ∥

(A.1)

512

R. Jiang et al. / Journal of the Korean Statistical Society 41 (2012) 505–514

as n → ∞. By Lemma 1, we can obtain c0

¯ = Q (β)

2(1 + ∥β0 ∥ ) 2

c0

(β¯ − β0 )T Σx (β¯ − β0 ) −

Ai

wi 

n i=1 c0

1 + ∥β0 ∥2

(β¯ − β0 ) + op

  1 n

(β¯ − β0 )T Σx (β¯ − β0 ) + op 2(1 + ∥β0 ∥2 ) 1 + ∥β0 ∥2   c0 1 ¯ − β0 )T Σx (β¯ − β0 ) + op =− ( β , 2(1 + ∥β0 ∥2 ) n =

(β¯ − β0 )T Σx (β¯ − β0 ) −

n 1

  1 n

and Q (β ∗ ) =

c0 2(1 + ∥β0 ∥ ) 2

(β ∗ − β0 )T Σx (β ∗ − β0 ) −

c0

(β¯ − β0 )T Σx (β ∗ − β0 ) + op 1 + ∥β0 ∥2

  1 n

.

Noting that

¯ = Q (β ∗ ) − Q (β)

c0 2(1 + ∥β0 ∥ ) 2

¯ Σx (β ∗ − β) ¯ + op (β ∗ − β)

 

Therefore, (A.2) implies that (A.1) holds true. The rest results of completed. 

1 n

≤ 0.

(A.2)

√

n(βˆ − β0 ) are Theorem 3 in Cui (1997). The proof is

Proof of Theorem 2. By the result of Theorem 1, we have

√

ˆ = c0−1 n(β ∗ − β)



n 1  1 + ∥β0 ∥2 Σx−1 √ (wi − 1)Ai + op (1). n i =1

(A.3)

From Lemma 2.9.5 in Van der Vaart and Wellner (1996), it follows that conditionally on {Yi , Xi }ni=1 , n 1 

√

n i =1

L∗

(wi − 1)Ai − → N (0, S )

(A.4)

for almost every sequence {Yi , Xi }ni=1 . Thus, by (A.3) and (A.4), it is easy to show that (2.6) holds true.

√

L∗

ˆ − n(β ∗ − β) → N (0, c0−2 (1 + ∥β0 ∥2 )Σx−1 S Σx−1 ).

By using the similar argument as in Rao and Zhao (1992), (2.7) can be shown to hold true. This completes the proof. 1/2

 1

Proof of Theorem 3. Define K is a known p×(p−q) matrix of rank p−q and satisfy H T K = 0. Write Kn = Σx K (K T Σx K )− 2 , so KnT Kn = Ip−q , HnT Hn = Iq and HnT Kn = 0. Let H T (b − b0 ) = 0 and H T (β0 − b0 ) = 0. It is easy to see that there exists a

unique γ ∈ Rp−q such that b − β0 = K γ , then set β˜ − β0 = K γ˜ . We take wi ≡ 1, then similar to the proof of Theorem 1, we can obtain

˜ = Q (β) =

c0 2(1 + ∥β0 ∥2 ) c0 2(1 + ∥β0 ∥2 )

(β˜ − β0 )T Σx (β˜ − β0 ) − 

n 

1

n 1 + ∥β0 ∥2 i=1

γ˜ T K T Σx K γ˜ − 

n 

1

n 1 + ∥β0 ∥2 i=1

ATi (β˜ − β0 ) + op (1)

ATi K γ˜ + op (1).

Then, we can obtain n  1 T γ˜ = c0−1 1 + ∥β0 ∥2 (K T Σx K )−1 K Ai + op (1),

n i =1

and n  1 T β˜ − β0 = K γ˜ = c0−2 1 + ∥β0 ∥2 K (K T Σx K )−1 K Ai + op (1).

n i=1

Similarly,

ˆ = Q (β)

c0 2(1 + ∥β0 ∥ ) 2

(βˆ − β0 )T Σx (βˆ − β0 ) − 

n 

1

n 1 + ∥β0 ∥

2

i=1

ATi (βˆ − β0 ) + op (1).

R. Jiang et al. / Journal of the Korean Statistical Society 41 (2012) 505–514

513

Hence, under the null hypotheses, we have Mn =

n 



Yi − XiT β˜

ρ 

˜ 2 1 + ∥β∥

i=1

=

c0 2(1 + ∥β0 ∥2 )

−

 −

n 



Yi − XiT βˆ

ρ 

ˆ 2 1 + ∥β∥

i =1

  = Q (β) ˜ − Q (β) ˆ

[(β˜ − β0 )T Σx (β˜ − β0 ) − (βˆ − β0 )T Σx (βˆ − β0 )]

1

·

1 + ∥β0 ∥2

n 1

n i =1

Ai ([β˜ − β0 ] − [βˆ − β0 ]) + op (1)

2



n   1  1  T −1/2  = Hn Σx Ai  + op (1). √  2c0  n i=1

The theorem is proved.



Proof of Theorem 4. Similar to the proof of Theorem 3, under the null hypotheses, we can obtain

ˆ = − Q (β ∗ ) − Q (β) −

c0 2(1 + ∥β0 ∥ ) c0 2

(β ∗ − β0 )T Σx (β ∗ − β0 ) +

2(1 + ∥β0 ∥2 ) c0

c0 1 + ∥β0 ∥2

(β ∗ − β0 )T Σx (βˆ − β0 )

(βˆ − β0 )T Σx (βˆ − β0 ) + op (1)

ˆ T Σx (β ∗ − β) ˆ + op (1) (β ∗ − β)  2 n   1  1  −1/2  = − (wi − 1)Σx Ai  + op (1), √  2c0  n i=1  2 n   1  1   ∗ T − 1 / 2 ˜ =− Q (β˜ ) − Q (β) (wi − 1)Kn Σx Ai  + op (1). √  2c0  n i=1 = −

2(1 + ∥β0 ∥2 )

Then, we can obtain

˜ − Q (β)} ˆ = {Q (β˜ ∗ ) − Q (β)} ˜ − {Q (β ∗ ) − Q (β)} ˆ Mn∗ = {Q (β˜ ∗ ) − Q (β ∗ )} − {Q (β) 2



n   1 1  1   (wi − 1)KnT Σx−1/2 Ai  + =− √  2c0  n i=1 2c0

 2 n  1    −1/2  (wi − 1)Σx Ai  + op (1) √  n i=1 

2



n   1  1   = (wi − 1)HnT Σx−1/2 Ai  + op (1). √  2c0  n i=1

The proof of Theorem 4 is completed.



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