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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

Empirical likelihood for quantile regression models with response data missing at random

S. Luo
  • Corresponding author
  • School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi 710048, China
  • School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Shuxia Pang
  • School of Computer and Communication, Lanzhou University of Technology, Lanzhou, Gansu, 730000, China
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Published Online: 2017-03-27 | DOI: https://doi.org/10.1515/math-2017-0028

Abstract

This paper studies quantile linear regression models with response data missing at random. A quantile empirical-likelihood-based method is proposed firstly to study a quantile linear regression model with response data missing at random. It follows that a class of quantile empirical log-likelihood ratios including quantile empirical likelihood ratio with complete-case data, weighted quantile empirical likelihood ratio and imputed quantile empirical likelihood ratio are defined for the regression parameters. Then, a bias-corrected quantile empirical log-likelihood ratio is constructed for the mean of the response variable for a given quantile level. It is proved that these quantile empirical log-likelihood ratios are asymptotically χ2 distribution. Furthermore, a class of estimators for the regression parameters and the mean of the response variable are constructed, and the asymptotic normality of the proposed estimators is established. Our results can be used directly to construct the confidence intervals (regions) of the regression parameters and the mean of the response variable. Finally, simulation studies are conducted to assess the finite sample performance and a real-world data set is analyzed to illustrate the applications of the proposed method.

Keywords: Quantile regression; Empirical likelihood; Missing response data; Confidence interval

MSC 2010: 62G05; 62G20; 60G42

1 Introduction

Since the seminal work of Koenker [1], quantile regression (QR) has been an indispensable and versatile tool for statistical research due to its promising performance and elegant mathematical properties, and attracted immediately considerable attention, resulting in numerous papers (e.g., see [17]) devoted to various theoretical extensions of this significant topic. Moreover, QR has been widely applied to a variety of fields such as economics, finance, biology, and medicine. Compared to the mean regression model which is commonly used by the traditional least square (LS) methods, QR is able to directly estimate the effects of the covariates at different quantiles of the response variable and therefore provide more information about the distribution of the response variable. Furthermore, QR is less sensitive to outliers due to its specific estimation.

Despite the significant theoretical advances and a rapidly growing literature on QR, only scant attention has been paid to QR when the data samples contain missing values which may lead to substantial distortions on the results. In fact, missing data are a commonplace in practice due to various reasons such as loss of information caused by uncontrollable factors, unwillingness of some sampled units to supply the desired information, failure on the part of investigators to gather correct information, and so forth. Dating back to the early 1970s, spurred by the advances in computer technology that made it possible to perform laborious numerical calculations, the literature on statistical analysis of real data with missing values has flourished in applied work, see [812]. Although missing data analysis has a long history in statistics, little work on QR has taken missing data into account. Recently, Yoon [13] proposed an imputation method where the imputed values are drawn from the conditional quantile function of the response with which data are incomplete, but his method is valid only under independent and identically distributed (i.i.d.) errors. Wei [7] developed an iterative imputation procedure for the covariates with missing values in a linear QR model that is valid under non-i.i.d. error terms. Lv [5] discussed smoothed empirical likelihood analysis with missing response in partially linear quantile regression. Sherwood [6] recently proposed an inverse probability weighting QR approach for analyzing health care cost data when the covariates are MAR. Sun [14] studied QR for competing risk data when the failure type was missing. Chen [4] discussed efficient QR analysis with missing observations. Shu [15] proposed some imputation methods for quantile estimation under missing at random.

On the other hand, empirical likelihood (EL) method, introduced by Owen [16, 17], has many advantages over normal approximation methods for constructing confidence intervals. For example, the EL method produces confidence intervals or regions whose shape and orientation are determined entirely by the data, and the empirical likelihood regions are range preserving and transformation respecting. Many authors have used this method for linear, nonparametric and semiparametric regression models. About the quantile regression model, Chen [18] constructed the EL confidence intervals for population quantiles. Tang [19]developed an EL approach for estimating equations with missing data. Wang [20] considered EL for quantile regression models with longitudinal data. Whang [21] proposed a smoothed EL and discussed its higher-order properties with cross sectional data. Otsu [22] studied the first-order approximation of a smoothed conditional EL approach. The EL method has also been used for the analysis of censored survival data, for example, Zhao [23].

In this paper, a empirical-likelihood-based method is proposed to study quantile linear regression models with response data missing at random. A class of quantile empirical log-likelihood (QEL) ratios of the regression parameters are defined firstly which include QEL ratio with complete-case data, weighted QEL ratio and imputed QEL ratio. Then the statistical inference on the mean of the response for a given quantile level is further studied to obtain a bias-corrected QEL ratio of the mean of the response for a given quantile level. It is proved that the QEL ratios of both the regression parameters and the mean of the response for a given quantile level are asymptotically χ2 distribution. To compare the quantile empirical likelihood method with a normal approximation method, we also construct a class of estimators for the regression parameters and the mean of the response for a given quantile level. It is shown that this class of estimators are asymptotically normal. Furthermore, we derive consistent estimators of asymptotic variance, their confidence intervals (regions) can be constructed directly of the regressions parameters and the mean of the response for a given quantile level.

The rest of this paper is organized as follows. In Section 2, a class of QEL ratios and estimators for the regression parameters are constructed with missing response data and their asymptotic distributions are derived. In Section 3, a bias-corrected QEL ratio and the maximum empirical QEL estimator for mean of the response at a given quantile level are proposed and their asymptotic properties are studied. A simulation study is conducted in Section 4 to demonstrate the finite-sample performance of the proposed method. A real-world data set is analyzed to illustrated the applications of the proposed method in Section 5. The proof of the main results are postponed to Section 6.

2 Quantile empirical likelihood (QEL) method with missing response

Consider the quantile linear regression model Yi=XiTβτ+εi,i=1,2,,n,(1)

where Yi is the ith observation of the response Y, Xi is the ith observation of the covariates X and a d × 1 vector, τ ∈ (0,1) is the quantile level of interest, βτ is a d × 1 vector of unknown quantile regression parameters and εi is the error satisfying P(εi < 0|Xi) = τ for i = 1,2, ⋯, n.

For the model (1), we focus on the situation where some observations of Y in a sample of size n may be missing while X is observed completely. As a consequence, we have an incomplete sample {Xi,Yi,δi}i=1n with δi = 0 if Yi is missing and δi = 1, otherwise. Throughout this paper, we assume that the observations of Y are missing at random (MAR) which implies that δ and Y are conditionally independent given X. That is, P(δ = 1|X,Y) = P(δ = 1|X) = p(X). As pointed out in [24], MAR is a common assumption for statistical analysis with missing data and is reasonable in many practical situations. Hereafter, we will simply write βτ as β whenever no confusion is made.

2.1 Quantile empirical likelihood with complete-case data

In the model (1), a vector β^Q is called the complete data quantile regression estimator of β if β^Q=argminβi=0nρτ(YiXiTβτ)δi,(2)

where ρτ (u) = u(τI(u<0)) is the quantile loss function and I(·) is the indicator function. In addition, β satisfies the following estimating equation E{δiXiψ(Yi,Xi,β)}=0,i=1,2,,n,

where ψ(β)=ψ(Yi,Xi,β)=I(XiTβYi>0)τ is the quantile score function. The quantile empirical log-likelihood ratio function for β with complete-case data is defined as R^c(β)=2maxi=1nlog(npi):pi0,i=1npi=1,inpiZic(β)=0,

where Zic (β) = δi Xi Ψ (Yi, Xi, β). Furthermore, β^QEL=argmaxβR^c(β)(3)

is called the maximum quantile empirical likelihood estimator of β with complete-case data.

2.2 Weighted quantile empirical likelihood

Using the method in Section 2.1, a weighted quantile empirical log-likelihood ratio function for β is defined as R^w(β)=2maxi=1nlog(npi):pi0,i=1npi=1,inpiZiw(β)=0,

where Ziw(β)=δip(Xi)Xiψ(Yi,Xi,β)(4)

and p(x) = P(δ = 1|X = x). Here p(x) is called a selection probability function. Note that the selection probability in (4) is regarded as known. If the selection probability is unknown, it can be estimated by a kernel smoothing method. An estimator of p(x) can be defined by p^(x)=i=0nδiK((Xix)/hn)i=0nK((Xix)/hn),(5)

where K(·) is a kernel function, and hn controls the amount of smoothing used in estimations. Here, {hn}n=1 is a sequence of positive numbers tending to zero. Consequently, by replacing p(Xi) with its estimator p^(Xi), a weighted quantile estimate R^w(β)ofR^w(β) is obtained by R^w(β)=2maxi=1nlog(npi):pi0,i=1npi=1,inpiZiw(β)=0,

where Ziw(β)=δip^(Xi)Xiψ(Yi,Xi,β).

2.3 Quantile empirical likelihood with imputed values

For the quantile empirical likelihood with complete-case data and the weighted quantile empirical likelihood, the information contained in the data is not fully explored. Since incomplete-case data are discarded in constructing the empirical likelihood ratio, the coverage accuracies of the confidence regions are reduced when there are plenty of missing values. To resolve the issue, we estimate Yi by XiTβ^Q if Yi is missing. In what follows, we introduce the auxiliary random variables ZiI(β)=Xiψ(Y^i,Xi,β)=Xi(I(XiTβY^i>0)τ),i=1,2,,n,(6)

where Y^i=δiYip^(Xi)+(1δip^(Xi))XiTβ^Q. Thus, a quantile empirical log-likelihood ratio function based on imputed values is defined as R^I=2maxi=1nlog(npi):pi0,i=1npi=1,i=1npiZiI(β)=0

The ratio function is more appropriate than the quantile weighted empirical likelihood ratio function because it sufficiently uses the information contained in the data. In addition, β^Q of ZiI (β) can be substituted by β^QEL.

2.4 Asymptotic properties

In this section, some theoretical results on the asymptotic distribution of the quantile empirical likelihood ratios and their estimators proposed in Sections 2.12.3 are established. We first give the asymptotic distributions of R^c(β),R^w(β),R^I(β).

Theorem 2.1

Suppose that Conditions C1 – C6 in the Appendix all hold. If β is the true parameter, then R^(β)Lχd2, where R^(β) is taken to be R^w(β),R^w(β)orR^I(β).

Remark 1

Let χd2(1α) be the (1 – α)th quantile of χd2 with 0 < α < 1. Then it follows from Theorem 2.1 that an approximate 1 – α confidence region for β can be formulated by Rα(β~)={β~|R^(β~)χd2(1α)}.

Theorem 2.1 can also be used to test the hypothesis H0: β = β0, where H0 is rejected at level α if R^(β0)>χd2(1α).

The following theorem demonstrates that both β^Qandβ^QEL have the same asymptotic normality.

Theorem 2.2

Suppose that Conditions C1 – C6 in the Appendix hold. Then n1/2(β^QELβ^Q)=op(1)

and n(β^β)LN(0,D),

where D = A-1 BA-1, A = E{p(X)f(0 | X)XXT}, B = τ (1 – τ)E{p(X)XXT}, f (·|x) denotes the conditional density of ε on X = x, and β^ is taken to be β^QELorβ^Q.

In order to construct the confidence region of β, the asymptotic covariance matrix D can be estimated by D^=A^1B^A^1withA^=1nhi=1nδiKh(YiXiTβ^)XiXiT and B^=τ(1τ)ni=1nδiXiXiT. Obviously, D^ is a consistent estimator of D. Thus, it follows from Theorem 2.2 that D^1/2n(β^β)LN(0,Id),

which yields (β^β)TnD^1(β^β)Lχd2.(7)

Therefore, the confidence regions of β can be constructed by using (7).

3 Quantile empirical likelihood for the mean of the response

Some methods are provided firstly in this section to conduct a inference on the mean of the response θ by using empirical likelihood. Then a weighted quantile regression imputation is used to construct a weighted-corrected quantile empirical likelihood ratio of θ such that this ratio has an asymptotic χ2 distribution.

3.1 Weighted-corrected quantile empirical likelihood (WCQEL)

Above all, we introduce the auxiliary random variable Yi=δiYip(Xi)+(1δip(Xi))XiTβ

to construct the empirical likelihood ratio of θ. Since E(Yi)=θ under MAR if θ is the true parameter, a quantile empirical log-likelihood ratio function l*(θ) can be defined in the following. l(θ)=2maxi=1nlog(npi):pi0,i=1npi=1,i=1npiYi=θ.

According to the analogy work of Owen [17], it is easy to see that l* (θ) is asymptotic χ2 distributed with one degree of freedom, i.e., l(θ)Lχ12. However, β and p(·) are unknown usually, and hence l*(θ) cannot be used directly to make a statistical inference on θ. Accordingly, p(·) is replaced by its estimator defined in (5), β is computed by the following procedure.

  1. Simulate τj ~ Uniform(0,1) independently for j = 1,2,⋯, J;

  2. For each j = 1,2,⋯, J, β^(τj) is calculated by defined in (2);

  3. Finally, β can be estimated by β^SQ=1Jj=1Jβ^(τj).

As a result, an estimator of Yi denoted by Y^i, can be obtained by substituting β and p(Xi) with β^SQ and p^(Xi), that is, Y^i=δiYip^(Xi)+(1δip^(Xi))XiTβ^SQ,i=1,2,,n.(8)

Then, a weighted-corrected quantile empirical log-likelihood ratio function for θ can be defined as l^(θ)=2maxi=1nlog(npi):pi0,i=1npi=1,i=1npiY^i=θ.

The following Theorem shows that l^(θ) and l*(θ) have the same asymptotic distribution.

Theorem 3.1

Suppose that Conditions C1–C6 in the Appendix hold. If θ is the true parameter, then l^(θ)Lχ12.

Remark 2

Let χ12(1α) be the (1 – α)th quantile of the χ12 with 0 < α < 1. Then it follows from Theorem 3.1 that an approximate 1 – α confidence interval for θ can be constructed by Iα(θ~)={θ~|l^(θ~)χ12(1α)}.

Theorem 3.1 can also be used to test the hypothesis H0: θ = θ0, where H0 is rejected at level α if l^(θ0)>χ12(1α).

3.2 Normal approximation

θ^QWI is called a weighted imputation estimator of θ if θ^QWI=1ni=1nY^i,(9)

where Ŷi is defined in (8). Meanwhile, we call θ^QME=argmax{l^(θ)} a maximum quantile empirical likelihood estimator of θ. The asymptotic normality of θ^QWIandθ^QME is given in the following theorem.

Theorem 3.2

Suppose that Conditions C1 – C6 in the Appendix hold. Then n(θ^θ)LN(0,V),

where θ^ is taken to be either θ^QWIorθ^QME,V=Eσ2(X)p(X)+Var(XTβ)andσ2(x)=E(ε2|X=x).

According to Theorem 3.2, it is obtained that θ^QME=θ^QWI+op(n1/2).(10)

Furthmore, a consistent estimator of V can be formulated by V^=1ni=1n(Y^iθ^)2 with which a approximation confidence interval with confidence level 1 –α for θ can be constructed by θ^±z1α/2V^/n, where z1α/2 is the (1 – α /2)th quantile of the standard normal distribution.

4 A simulation study

In this section a simulation study is carried out to investigate the finite-sample performance of the proposed approaches. We consider the following two models.

Model 1 (homoscedastic): Yi = Xiβ + εi, i = 1, 2, ⋯, n;

Model 2 (heteroscedastic): Yi = Xiβ + ξXi εi, i = 1,2, ⋯, n,

where the observation Xi (i = 1,2,⋯, n) of the covariates X were drawn the N(0,1), εi(i=1,2,,n)iid N(0,1), and ξ and β were set to be 0.5 and 1, respectively. In simulation study, we focus on τ = 0.5,0.8. We considered the following three selection probability functions proposed by Wang and Rao (see [8]).

Case 1 : p1(x) = 0.8 + 0.2|x – 1| if |x – 1| ≤ 1, and 0.95 otherwise.

Case 2 : p2(x) = 0.9 – 0.2|x – 1| if |x – 1| ≤ 4.5, and 0.1 otherwise.

Case 3 : p3(x) = 0.6 for all x.

The average missing rates corresponding to the preceding three cases are approximately 0.09, 0.26 and 0.40, respectively. For each of the three cases, we generated 2000 Monte Carlo random samples of size n = 50, 100 and 150. The kernel function K(x) in (5) was taken to be K(x) = 0.75(1 – x2) if |x| ≤ 1; K(x) = 0, otherwise. We used the cross-validation method to select the optimal bandwidths hopt. The simulations were implemented in the following two situations.

(1) The confidence intervals of β. For the two models, we used four methods, namely, the quantile empirical likelihood with complete-case data (QCEL), the quantile weighted empirical likelihood (QWEL), the quantile empirical likelihood based on imputed values (QIEL) and the normal approximation(NA) in Theorem 2.2. For convenience, in what follows NA(β^QEL)andNA(β^Q) denote the corresponding normal approximation confidence intervals for β^QELandβ^Q. The average lengths of the confidence intervals and their corresponding empirical coverage probabilities, with a nominal level 1 – α = 0.95 and τ = 0.5, 0.8, were computed with 2000 simulation runs. The results are reported in Tables 14.

Table 1

Average lengths of the confidence intervals for β in Model 1 for different forms of the selection probability function p(x) and different values of sample size n and the quantile level τ under the nominal level 0.95.

Table 2

Emprical coverage probabilities of the intervals for β in Model 1 for different forms of the selection probability function p(x) and different values of the sample sizes n and the quantile level τ under the nominal level 0.95.

Table 3

Average lengths of the confidence intervals for β in Model 2 for different forms of the selection probability function p(x) and different values of sample size n and the quantile level τ under the nominal level 0.95.

Table 4

Emprical coverage probabilities of the intervals for β in Model 2 for different forms of the selection probability function p(x) and different values of sample sizes n and τ under nominal level 0.95.

Tables 14 show the following results. Firstly, for Case 1, QIEL yields lightly longer interval lengths but higher coverage probabilities than the other three methods. For Cases 2 and 3, QIEL performs better than the other three methods in the sense that its confidence intervals have uniformly shorter average lengths and higher coverage probabilities, which indicates that quantile regression imputation is necessary when the missing rate is large. Secondly, both QCEL and QWEL result in slightly longer interval lengths but higher coverage probabilities than NA(β^QEL)andNA(β^Q). In addition, the confidence inervals obtained by NA(β^QEL)andNA(β^Q) show nearly equal lengths and coverage accuracies in the same case. Thirdly, as expected all the interval lengths decrease and the empirical coverage probabilities increase as n increases for every given missing rate. Observably, the missing rate also affects the interval length and coverage probability. Generally, the interval length increases and the coverage probability decreases as the missing rate increases for every fixed sample size. However, the two values fail to change by a large amount for the QIEL method because the quantile regression imputation is used in QIEL. Furthermore, it is also seen that than the other methods for the heteroscedastic model QIEL still performs much better.

(2) The confidence intervals of θ. The weighted-corrected empirical likelihood(WCQEL) based on l^(θ) and NA were considered. θ^QWI defined in (9) is used to estimat θ. The empirical coverage probabilities and average lengths of the confidence intervals, with a nominal level 1 – α = 0.95 were computed with 2000 simulation runs. The results are reported in Table 5.

Table 5

The average lengths and empirical coverage probabilities of the confidence intervals for θ with different forms of the selection probability function p(x), the sample sizes n and the quantile level τ under nominal level 0.95.

It is seen from Table 5 that WQCEL produces slightly longer interval lengths, but higher coverage probabilities than NA does. All the coverage probabilities increases and the average lengths decrease as n increase. In addition, the coverage probabilities and average lengths depend on the selection probability function p(x) and the quantile level τ.

5 A real-data example

The data originally presented by [25] is investigated in this section to support the proposition that food expenditure constitutes a declining share of personal income. This data that has not any missing data consists of 235 budget surveys of 19th century European working class households. More details of the discussion on this data can be found in [26]. We consider the following linear QR model: Yi = β0(τ) + β1(τ)Xi,i = 1,2,⋯, 235, where Y is the centered annual household food expenditure and X is the centered annual household income in Belgian francs. In order to use the data set to illustrate our method, artificial missing data was created by deleting some of the response values at random. Assume that 25% of the response values in this data are missed. The missing indicator δ is generated from the probability function p(x) = 0.9 – 0.2|x – 1| if |x – 1| ≤ 4.5, and 0.1 otherwise.

We now present the estimator and the 95% confidence interval of β based on the proposed QILE method and the normal approximation method (NA) based on Theorem 2.2 with τ = 0.4 and 0.7. The results are shown in Table 6. From Table 6, we can see that the confidence interval obtained by the QIEL method has much shorter confidence interval than that obtained by the NA method, which shows that the former method is superior to the latter one.

Table 6

The estimators and confidence intervals of β based on QIEL and NA in Engel data analysis.

6 Proofs of the main results

Let r > 2 be an integer. g(x), f(·|x) and F(·|x) are used to denote the density function of X,the density and distribution functions of ε conditional on Xi = x, respectively. Let c be a positive constant which is independent of n and may take a different value in different place. The following conditions will be used in this section.

(C1) {(Yi, Xi): i = 1,2,⋯, n} are independent and identically distributed random vectors.

(C2) Both p(x) (the selection probability function) and g(x) have bounded derivatives up to order r almost surely and infx p(x) > 0:

(C3) K(·) is a kernel function of order r and is bounded and compactly supported on [– 1, 1]. Furthermore, there exist positive constants C1, C2 and ρ such that C1I[||u||ρ]K(u)C2I[||u||ρ].

(C4) P(X>Mn)=o(n1/2), where 0 < Mn →∞ as n → ∞.

(C5) The positive bandwidth parameter h satisfies nh2r →0 when n →∞.

(C6) The matrices A and B defined in Theorem 2.2 are both nonsingular. Firstly, some lemmas are introduced to derive the main results.

Lemma 6.1

(see Lemma 2 in [11]) Suppose that Conditions C1 – C6 hold. Then E{p^(Xi)p(Xi)}2=O((nhd)1Mnd)+O(h2r)+o(n1/2)

holds uniformly for i = 1,2,⋯, n.

Lemma 6.2

Suppose that Conditions C1 – C6 hold. If β is the true parameter of model (1), then 1ni=1nZi(β)LN(0,B)(11)

and E(Zi(β)/β)=A+o(1),(12)

where Zi (β) is taken to be Ziw(β),Ziw(β)orZiI(β),A=E{π(X)f(0|X)XXT} and B=τ(1τ)E{π(X)XXT} with π (x) = 1/p(x) when Zi (β) is taken to be Ziw(β) and Ziw(β); and π(x) = 1 when Zi(β) = ZiI(β).

Proof

(a) The case of Zi(β)=Ziw(β) will be proved firstly for i = 1,2,⋯, n. Some simple calculation yields 1ni=1nZiw(β)=1ni=1nδip(Xi)Xiψin(Yi,Xi,β)J.

It is easy to obtain E(J) = 0 and Cov(J) = B. Then it follows from the central limit theorem that (11) is obtained immediately. In a similar way, we can prove (12).

(b) Now, we prove the case of Zi(β)= Ziw (β) for i = 1,2,⋯, n. Because 1ni=1nZiw(β)=1ni=1nδip^(Xi)Xiψi(Yi,Xi,β)=1ni=1nδip(Xi)Xiψi(Yi,Xi,β)+1ni=1nδi(p(Xi)p^(Xi))p^(Xi)p(Xi)Xiψi(Yi,Xi,β)(13)

Similarly to the proof of Theorem 3 in [28], it follows from Conditions C2, C3 and C5 that 1ni=1nδip^(Xi)p(Xi)Xiψi(Yi,Xi,β)=Op(1)(14)

Since supx|p^(x)p(x)|=op(1), (14) indicates 1ni=1nZiwn=1ni=1nδip(Xi)Xiψi(Yi,Xi,β)+op(1)=1ni=1nZiw(β)+op(1).(15)

Then, (11) is obtained immediately. On the other hand, the proof of (12) is similar to the proof in case (a) and hence is omitted here.

(c) When Zi(β) = ZiI(β) for i = 1,2,⋯, n, direct calculation obtains XiTβY^i=δip^(Xi)(XiTβYi)+(1δip^(Xi))XiT(ββ^Q).

Then it is easily shown that 1ni=1n(1δip^(Xi))Xi=op(1) and β^Qβ=A11ni=1nZi(β)+op(n1/2)=Op(n1/2).

Therefore, we have Xi(I(xiTβY^i>0)τ)=Xi(I(xiTβYi>0)τ)=Xiψi(Yi,Xi,β)(16)

with which we can prove (11) and (12) by using the similar way in the case (a).□

Lemma 6.3

Suppose that Conditions C1 – C6 hold. If β is the true parameter of model (1), then 1ni=1nZi(β)ZiT(β)PB(17)

where Zi(β) is taken to be Ziw(β),Ziw(β)orZiI(β),andB=τ(1τ)E{π(X)XXT} with π(x) = 1/p(x) when Zi(β) is taken to be Ziw(β)andZiw(β), and π(x) = 1 when Zi (β) = ZiI(β).

Proof

(a) When Zi(β)=Ziw(β), for i = 1,2,⋯, n, some simple calculation yields 1ni=1nZiw(β)ZiwT(β)=1ni=1nAi1Ai1T,

where Ai1=δip(Xi)Xiψi(Yi,Xi,β). By the law of large numbers, we can derive the result immediately.

(b) When Zi(β)= Ziw (β) for i = 1,2,⋯,n, 1ni=1nZiw(β)ZiwT(β)=1ni=1nAi1Ai1T+1ni=1nAi1Ai2T+1ni=1nAi2Ai1T+1ni=1nAi2Ai2T=B1+B2+B3+B4,(18)

where Ai2=1ni=1nδi(p(Xi)p^(Xi))p^(Xi)p(Xi)Xiψi(Yi,Xi,β). By the law of large numbers, we can derive that B1PB. Now, B2P0 will be proved. Let B2, ks be the (k,s) component of B2, Aij,r be the rth component of Aij, j = 1,2. Then we use the Cauchy-Schwarz inequality to get |B2,ks|1ni=1nAi1,k21/21ni=1nAi2,r21/2

From Lemma 6.1 and 6.2, we can see n1i=1nAi1,k2=Op(1)andn1i=1nAi2,r2=op(1). Hence B2P0.

Using the similar argument, we can prove BiP0 for i = 3, 4.

(c) When Zi(β) = ZiI(β) for i = 1,2,⋯,n, by the (16) and the same methods as that of (a) and (b),

therefore, we can obtain the result.□

Lemma 6.4

Suppose that Conditions C1 – C6 hold. If θ is the true parameter of model (1), then 1ni=1n(Y^iθ)LN(0,V),(19) 1ni=1n(Y^iθ)2PV,(20)

and max|Y^i|=Op(n1/2).(21)

Proof

We prove (19) only. (20) and (21) can be proved similarly. It is straightforward to obtain 1ni=1n(Y^iθ)=A1+A2+A3,

where A1=1ni=1n[δiεip(Xi)+{XiTβθ}],A2=1ni=1n(1p^(Xi)1p(Xi))δiεi and A3=1ni=1n(1δip^(Xi))XiT(β^SQβ). It follows from the central limit theorem that A1LN(0,V).

To prove (19), we only need to prove that A2 = op(1) and A3 = op(1). A2 will be proved firstly. Direct calculation yields A2=1ni=1nδiεi{p(Xi)p^(Xi)}p2(Xi)+1ni=1nδiεi{p(Xi)p^(Xi)}2p2(Xi)p^(Xi)=A21+A22.

By Lemma 6.1, it is easy to show that A22 = op(1). Simple calculation yields A21=1ni=1nδiεip2(Xi)j=1nWnj(Xi){p(Xi)p(Xj)}+1ni=1nδiεip2(Xi)j=1nWnj(Xi){p(Xi)δj)}+1ni=1nδiεip(Xi){1j=1nWnj(Xi)}=A211+A212+A213,(22)

where Wnj(x)=Kh(Xjx)i=1nKh(Xix). By the Cauchy-Schwarz inequality, we get E(A2112)cni=1nEEj=1nWnj(Xi){p(Xi)p(Xj)2|Xicni=1nEEj=1nWnj(Xi)||XiXj||2|Xich20

So we prove A211 = op(1).

To handle A212, write Si=δiεip2(Xi)andSj=p(Xj)δj. We have E(A2122)=cni=1nj=1nE{Wnj(Xi)SiSj2}cni=1nEj=1nWnj2(Xi)0

Thus, it follows that A212 = op(1). It is easy to show that A213 = op(1). So A21 = op(1) is obtained immediately. In addition, Lemma 6.2 indicates A3 = op(1). This proves (19) and consequently completes the proof of Lemma 6.4.□

Proof of Theorem 2.1. By the Lagrange multiplier method, R^(β) can be represented as R^(β)=2i=1nlog(1+λT(β)Zi(β)),(23)

where λ(β) is a d × 1 vector given as the solution of the equation i=1nZi(β)1+λT(β)Zi(β))=0.(24)

According to Lemma 6.2 and the arguments in the proof of (2.14) in Owen [16], we can show that λ(β)=(Zi(β)ZiT(β))1i=1nZi(β)+op(n1/2).(25)

Applying the Taylor expansion to (23) and invoking Lemma 6.2 and (25), we obtain R^(β)=2i=1n[λT(β)Zi(β)(λT(β)Zi(β))2/2]+op(1).(26)

Then it follows from (24) that 0=i=1nZi(β)1+λT(β)Zi(β)=i=1nZi(β)i=1nZi(β)ZiT(β)λ(β)+i=1nZi(β)(λT(β)Zi(β))21+λT(β)Zi(β).

Lemma 6.2 and (25) imply i=1n(λT(β)Zi(β))2=i=1nλT(β)Zi(β)+op(1).

Therefore, it is obtained from (26) that R^(β)=(1ni=1nZiT(β))(1ni=1nZi(β)ZiT(β))1(1ni=1nZi(β))+op(1).

This together with Lemma 6.2 completes the proof of Theorem 2.1.□

Proof of Theorem 2.2. First, a Taylor expansion for Zi(β^) gives 1ni=1nZi(β^)=1ni=1nZi(β)+1ni=1nZi(β)(β^β)+op(n1/2)=Dn+A(β^β)+op(n1/2)(27)

Where Dn=1ni=1nZi(β). Then, similar to the proofs of (28)-(30) in [22], we have β^QELβ=op(n1/2) by Lemma 6.2. Finally, it follows from Lemma 6.2 and (26), (27) that the conclusion of Theorem 2.2 is obtained directly.□

Proof of Theorem 3.1. Similarly to the proof of Theorem 2.1, Theorem 3.1 can be proved by Lemma 6.3. Thus, we omit this proof.□

Proof of Theorem 3.2. It follows from (9) and (10) that n(θ^θ)=1ni=1n(Y^iθ)+op(1).

This together with (19) proves Theorem 3.2.□

Acknowledgement

This work is supported by the National Natural Science Foundations of China (Nos.11601409,11201362) and the Natural Science Foundation of Shaanxi Province of China (No. 2016JM1009).

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About the article

Received: 2016-06-28

Accepted: 2017-01-03

Published Online: 2017-03-27


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 317–330, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0028.

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© 2017 Luo and Pang. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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