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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
J Am Stat Assoc. Author manuscript; available in PMC 2010 December 1.
Published in final edited form as:
J Am Stat Assoc. 2009 December 1; 104(488): 1631–1645.
doi:  10.1198/jasa.2009.tm09055
PMCID: PMC2908045
NIHMSID: NIHMS142014

Local Rank Inference for Varying Coefficient Models1

Abstract

By allowing the regression coefficients to change with certain covariates, the class of varying coefficient models offers a flexible approach to modeling nonlinearity and interactions between covariates. This paper proposes a novel estimation procedure for the varying coefficient models based on local ranks. The new procedure provides a highly efficient and robust alternative to the local linear least squares method, and can be conveniently implemented using existing R software package. Theoretical analysis and numerical simulations both reveal that the gain of the local rank estimator over the local linear least squares estimator, measured by the asymptotic mean squared error or the asymptotic mean integrated squared error, can be substantial. In the normal error case, the asymptotic relative efficiency for estimating both the coefficient functions and the derivative of the coefficient functions is above 96%; even in the worst case scenarios, the asymptotic relative efficiency has a lower bound 88.96% for estimating the coefficient functions, and a lower bound 89.91% for estimating their derivatives. The new estimator may achieve the nonparametric convergence rate even when the local linear least squares method fails due to infinite random error variance. We establish the large sample theory of the proposed procedure by utilizing results from generalized U-statistics, whose kernel function may depend on the sample size. We also extend a resampling approach, which perturbs the objective function repeatedly, to the generalized U-statistics setting; and demonstrate that it can accurately estimate the asymptotic covariance matrix.

Keywords: Asymptotic relative efficiency, Local linear regression, Local rank, Varying coefficient model

1 Introduction

As introduced in Cleveland, Crosse and Shyu (1992) and Hastie and Tibshirani (1993), the varying coefficient model provides a natural and useful extension of the classical linear regression model by allowing the regression coefficients to depend on certain covariates. Due to its flexibility to explore the dynamic features which may exist in the data and its easy interpretation, the varying coefficient model has been widely applied in many scientific areas. It has also experienced rapid developments in both theory and methodology, see Fan and Zhang (2008) for a comprehensive survey. Fan and Zhang (1999) proposed a two-step estimation procedure for the varying coefficient model when the coefficient functions have possibly different degrees of smoothness. Kauermann and Tutz (1999) investigated the use of varying coefficient models for diagnosing the lack-of-fit of regression, regarding the varying coefficient model as an alternative to a parametric null model. Cai, Fan and Li (2000) developed a more efficient estimation procedure for varying coefficient models in the framework of generalized linear models. As special cases of varying coefficient models, time-varying coefficient models are particularly appealing in longitudinal studies, survival analysis and time series data since they allow one to explore the time-varying effect of covariates over the response. Pioneering works on novel applications of time-varying coefficient models to longitudinal data include Brumback and Rice (1998), Hoover, et al. (1998), Wu, et al. (1998) and Fan and Zhang (2000), among others. For more details, readers are referred to Fan and Li (2006) and the references therein. Time-varying coefficient models are also popular in modeling and predicting nonlinear time series data and survival data, see Fan and Zhang (2008) for related literature.

Estimation procedures in the aforementioned papers are built on either local least squares type or local likelihood type methods. Although these estimators remain asymptotically normal for a large class of random error distributions, their efficiency can deteriorate dramatically when the true error distribution deviates from normality. Furthermore, these estimators are very sensitive to outliers. Even a few outlying data points may introduce undesirable artificial features in the estimated functions. These considerations motivate us to develop a novel local rank estimation procedure that is highly efficient, robust and computationally simple. In particular, the proposed local rank regression estimator may achieve the nonparametric convergence rate even when the local linear least squares method fails to consistently estimate the regression coefficient functions due to infinite random error variance, which occurs for instance when the random error has a Cauchy distribution.

The new approach can substantially improve upon the commonly used local linear least squares procedure for a wide class of error distributions. Theoretical analysis reveals that the asymptotic relative efficiency (ARE), measured by the asymptotic mean squared error (or the asymptotic mean integrated squared error), of the local rank regression estimator in comparison with the local linear least squares estimator has an expression that is closely related to that of the Wilcoxon-Mann-Whitney rank test in comparison with the two-sample t-test. However, different from the two-sample test scenario where the efficiency is completely determined by the asymptotic variance, in the current setting of estimating an infinite-dimensional parameter both bias and variance contribute to the asymptotic efficiency. The value of ARE is often significantly greater than one. For example, the ARE is 167% for estimating the regression coefficient functions when the random error has a t3 distribution, is 240% for exponential random error distribution, and is 493% for lognormal random error distribution.

A striking feature of the local rank procedure is that its pronounced efficiency gain comes with only a little loss when the random error actually has a normal distribution, for which the ARE of the local rank regression estimator relative to the local linear least squares estimator is above 96% for estimating both the coefficient functions and their derivatives. For estimating the regression coefficient functions, the ARE has a sharp lower bound 88.96%, which implies that the efficiency loss is at most 11.04% in the worst case scenario. For estimating the first derivative of the regression coefficient functions, the ARE possesses a lower bound 89.91%. Kim (2008) developed a quantile regression procedure for varying coefficient models when the random errors are assumed to have a certain quantile equal to zero. She used the regression splines method and derived the convergence rate, but the lack of asymptotic normality result does not allow the comparison of the relative efficiency. On the other hand, one may extend the local quantile regression approach (Yu and Jones, 1998) to the varying coefficient models. However, this is expected to yield an estimator which still suffers from loss of efficiency and may have near zero ARE relative to the local linear least squares estimator in the worst case scenario.

The new estimator proposed in this paper minimizes a convex objective function based on local ranks. The implementation of the minimization can be conveniently carried out using existing functions in the R statistical software package via a simple algorithm (§4.1). The objective function has the form of a generalized U-statistic whose kernel varies with the sample size. Under some mild conditions, we establish the asymptotic representation of the proposed estimator and further prove its asymptotic normality. We derive the formula of the asymptotic relative efficiency of the local rank estimator relative to the local linear least squares estimator, which confirms the efficiency advantage of the local rank approach. We also extend a resampling approach, which perturbs the objective function repeatedly, to the generalized U-statistics setting; and demonstrate that it can accurately estimate the asymptotic covariance matrix.

This paper is organized as follows. Section 2 presents the local rank procedure for estimating the varying coefficient models. Section 3 discusses its large sample properties and proposes a resampling method for estimating the asymptotic covariance matrix. In Section 4, we address issues related to practical implementation and present Monte Carlo simulation results. We further illustrate the proposed procedure via analyzing an environment data set. Regularity conditions and technical proofs are presented in the Appendix.

2 Local rank estimation procedure

Let Y be a response variable, and U and X be the covariates. The varying coefficient model is defined by

Y=a0(U)+XTa(U)+ε,
(1)

where a0(·) and a(·) are both unknown smooth functions. The random error ε has probability density function g(·) which has finite Fisher information, i.e., {g(x)}−1 g′(x)2dx < ∞. In this paper, it is assumed that U is a scalar and X is a p-dimensional vector. The proposed procedures can be extended to the case of multivariate U with more complicated notations by following the same idea in this paper.

Suppose that {Ui, Xi, Yi}, i = 1, …, n, is a random sample from model (1). Write Xi = (Xi1, …, Xip)T and a(·) = (a1(·), …, ap(·))T. For u in a neighborhood of any given u0, we locally approximate the coefficient function by a Taylor expansion

am(u)am(u0)+am(u0)(uu0),m=0,1,,p.
(2)

Denote α1 = a0(u0), α2=a0(u0), βm = am(u0) and βp+m=am(u0), for m = 1, …, p. Based on the above approximation, we obtain the residual for estimating Yi at Ui = u0

ei=Yiα1α2(Uiu0)m=1p[βm+βp+m(Uiu0)]Xim.
(3)

We define the local rank objective function to be

Qn(β,α2)=1n(n1)1i,jneiejKh(Uiu0)Kh(Uju0),
(4)

where β = (β1, …, βp, βp+1, …, β2p)T, and for a given kernel function K(·) and a bandwidth h, Kh(t) = h−1K(t/h). Note that Qn(β, α2) does not depend on α1 because α1 is canceled out in eiej. The objective function Qn(β, α2) is a local version of Gini’s mean difference, which is a classical measure of concentration or dispersion (David, 1998). Without the kernel functions, [n(n − 1)]−1 Σ1 ≤ i, j n|ejej| is the global rank objective function that leads to the classical rank estimator in linear models based on Wilcoxon scores. Rank-based statistical procedures have played a fundamental role in nonparametric analysis of linear models due to its high efficiency and robustness. We refer to the review paper of McKean (2004) for many useful references.

For any given u0, minimizing Qn(β, α2) yields the local Wilcoxon rank estimator for (β0T,α2)T, where β0=β(u0)=(a1(u0),,ap(u0),a1(u0),,ap(u0))T. Denote the minimizer of Qn(β, α2) by ([beta]T, [alpha]2)T. Then for m = 1, …, p,

a^m(u0)=β^m,a^m(u0)=β^p+manda^0(u0)=α^2,

In the sequel, we also use the vector notation [alpha](u0) = (â1(u0), …, âp(u0))T and a^(u0)=(a^1(u0),,a^p(u0))T when convenient.

The location parameter a0(u0) needs to be estimated separately. This is analogous to the scenario of global rank estimation of intercept in the linear regression model. In order to make the intercept identifiable, it is essential to have additional location constraint on the random errors. We adopt the commonly used constraint that εi has median zero. Given ([beta]T, [alpha]2)T, we estimate a0(u0) by [alpha]1, the value of α1 that minimizes

n1i=1n|Yiα1α^2(Uiu0)m=1p[β^m+β^p+m(Uiu0)]Xim|Kh(Uiu0),
(5)

which is a local version of a weighted L1-norm objective function.

3 Theoretical Properties

3.1 Large sample distributions

In this subsection, we investigate the asymptotic properties of [beta] and [alpha]2. The main challenge comes from the non-smoothness of the objective function Qn(β, α2). To overcome this difficulty, we first derive an asymptotic representation of [beta] and [alpha]2 via a quadratic approximation of Qn(β, α2), which holds uniformly in a local neighborhood of the true parameter values. Aided with this asymptotic representation, we further establish the asymptotic normality of the local rank estimator.

Let us begin with some new notation. Let γn = (nh)−1/2, and define

β=γn1(β1a1(u0),,βpap(u0),h(βp+1a1(u0)),,h(β2pap(u0)))T,α=(α1,α2)T=γn1(α1a0(u0),h(α2a0(u0)))TΔi(u0)=m=1p[am(Ui)am(u0)am(u0)(Uiu0)]Xim+[a0(Ui)a0(u0)a0(u0)(Uiu0)].

Let (β^nT,α^2n)T be the value of (βT,α2)T that minimizes the following reparametrized objective function

Qn(β,α2)=1n(n1)1i,jn|(εiγnα2(Uiu0)/hγnβTZi+Δi(u0))(εjγnα2(Uju0)/hγnβTZj+Δj(u0))|Kh(Uiu0)Kh(Uju0),
(6)

Where Zi=(XiT,((Uiu0)/h)XiT)T. Let H = diag(1, h)[multiply sign in circle]Ip, where [multiply sign in circle] denotes the operation of Kronecker product and Ip denotes a p × p identity matrix. Then it can be easily seen that

β^n=nhH(β^β0)andα^2n=nh3[α^2a0(u0)].

We next show that the non-smooth function Qn(β,α2) can be locally approximated by a quadratic function of (βT,α2)T. Let μi = ∫ tiK(t)dt, i = 1, 2, and νi = ∫ tiK2(t)dt, i = 0, 1, 2. In this paper, we assume that the kernel function K(·) is symmetric. This is not restrictive considering that most of the commonly used kernel functions, such as the Epanechnikov kernel K(t) = 0.75(1 − t2)I(|t| < 1), are symmetric. We use Sn(β,α2)=(Sn1T(β,α2),Sn2(β,α2))T to denote the gradient function of Qn(β,α2), i.e., Sn1(β,α2)=βQn(β,α2) and Sn2(β,α2)=α2Qn(β,α2). More specifically,

Sn1(β,α2)=2γn[n(n1)]1ij[I(εiγnα2(Uiu0)/hγnβTZi+Δi(u0)εjγnα2(Uju0)/hγnβTZj+Δj(u0))1/2](ZiZj)Kh(Uiu0)Kh(Uju0),

and

Sn2(β,α2)=2γn[n(n1)]1ij[I(εiγnα2(Uiu0)/hγnβTZi+Δi(u0)εjγnα2(Uju0)/hγnβTZj+Δj(u0))1/2]((UiUj)/h)Kh(Uiu0)Kh(Uju0).

Furthermore, we consider the following quadratic function of (βT,α2)T:

Bn(β,α2)=γn1(βT,α2)(Sn1(0,0)Sn2(0,0))+12γn(βT,α2)A(βα2)+γn1Qn(0,0),
(7)

where

A=4τf2(u0)((u0)000μ2(u0)000μ2),
(8)

(u0)=E[XiXiTUi=u0], 0 denotes a matrix (or vector) of zeroes whose dimension is determined by the context, τ = ∫ g2(t)dt is the Wilcoxon constant, and g(·) is the density function of the random error ε.

Lemma 3.1

Suppose that Conditions (C1) — (C4) in the Appendix hold. Then ∀ ε > 0, ∀ c > 0,

P[sup(βT,α2)c|γn1Qn(β,α2)Bn(β,α2)|ε]0,

where || · || denotes the Euclidean norm.

Lemma 3.1 implies that the non-smooth objective function Qn(β,α2) can be uniformly approximated by a quadratic function Bn(β,α2) in a neighborhood around 0. In the appendix, it is also shown that the minimizer of Bn(β,α2) is asymptotically within a o(1) neighborhood of (β^nT,α^n2)T. This further allows us to derive the asymptotic distribution.

The local linear Wilcoxon estimator of a(u0) = (a1(u0), …, ap(u0))T is â(u0). The theorem below provides an asymptotic representation of â(u0) and the asymptotic normal distribution. Let Sn1(0,0)=(Sn11T(0,0),Sn12T(0,0))T, where Sn11(0, 0) and Sn12(0, 0) are both p × 1 vectors.

Theorem 3.2

Suppose that Conditions (C1) — (C4) in the Appendix hold. Then we have the following asymptotic representation

nh[a^(u0)a(u0)]=γn2[4τf2(u0)(u0)]1Sn11(0,0)+oP(1),
(9)

where f(u) is the density function of U. Furthermore,

nh[a^(u0)a(u0)μ22a(u0)h2+o(h2)]N(0,ν012τ2f(u0)1(u0))
(10)

in distribution, where a(u0)=(a1(u0),,ap(u0))T.

Remark

For the estimators of the derivatives of the coefficient functions, we have the following asymptotic representations:

nh3[α^2a0(u0)]=γn2[4τf2(u0)μ2]1Sn2(0,0)+oP(1),
(11)

nh3[a^(u0)a(u0)]=γn2[4τf2(u0)μ2(u0)]1Sn12(0,0)+oP(1).
(12)

Following similar proof as that for Theorem 3.2 in the appendix, it can be shown that nh3[α^n2a0(u0)] and nh3[a^(u0)a(u0)] are both asymptotically normal. The proof of the asymptotic normality of [alpha]2 and â′(u0) is given in the technical report version of this paper ((Wang, Kai and Li, 2009).

3.2 Asymptotic relative efficiency

We now compare the estimation efficiency of the local rank estimator (denoted by b âR(u0)) with that of the local linear least squares estimator (denoted by âLS(u0)) for estimating a(u0) in the varying coefficient model. To measure efficiency, we consider both the asymptotic mean squared error (MSE) at a given u0 and the asymptotic mean integrated squared error (MISE). When evaluating both criteria, we plug in the theoretical optimal bandwidth.

Zhang and Lee (2000) gives the asymptotic MSE of âLS(u0) for estimating a(u0):

MSELS(h;u0)=Ea^LS(u0)a(u0)2=μ22a(u0)24h4+ν0σ2f(u0)tr{1(u0)}1nh,

where σ2 = var(ε) is assumed to be finite and positive. Thus, the theoretical optimal bandwidth, which minimizes the asymptotic MSE of âLS(u0), is

hLSopt(u0)=[ν0σ2tr{1(u0)}μ22a(u0)2f(u0)]1/5n1/5.
(13)

From (10), the asymptotic MSE of the local rank estimator âR(u0) is

MSER(h;u0)=Ea^R(u0)a(u0)2=μ22a(u0)24h4+ν012τ2f(u0)tr{1(u0)}1nh.

The theoretical optimal bandwidth for the local rank estimator thus is

hRopt(u0)=[ν0tr{1(u0)}12τ2μ22a(u0)2f(u0)]1/5n1/5.
(14)

This allows us to calculate the local asymptotic relative efficiency.

Theorem 3.3

The asymptotic relative efficiency of the local rank estimator to the local linear least squares estimator for a(u0) is

ARE(u0)=MSELS{hLSopt(u0),u0}MSER{hRopt(u0),u0}=(12σ2τ2)4/5.

This asymptotic relative efficiency has a lower bound 0.8896, which is attained at the random error density f(t)=3205(5x2)I(x5).

Remark 1

Alternatively, we may consider the asymptotic relative efficiency obtained by comparing the MISE, which is defined as MISE(h) = ∫ E||â(u) − a(u)||2w(u) du with a weight function w(·). This provides a global measurement. Interestingly, it leads to the same relative efficiency. This follows by observing that the theoretical optimal global bandwidths for the local linear least squares estimator and the local rank estimator are

hLSopt=[ν0σ2w(u)tr{1(u)}/f(u)duμ22a(u)2w(u)du]1/5n1/5,
(15)

and

hRopt=[ν0w(u)tr{1(u)}/f(u)du12τ2μ22a(u)2w(u)du]1/5n1/5,
(16)

respectively. Thus, with the theoretical optimal bandwidths,

ARE=MISELS(hLSopt)MISER(hRopt)=(12σ2τ2)4/5.

Define [var phi] = (12σ2τ2)4/5. Then ARE(u0) = ARE = [var phi].

Note that the above ARE is closely related to the asymptotic relative efficiency of the Wilcoxon-Mann-Whitney rank test in comparison with the two-sample t-test. Table 1 depicts the value of [var phi] for some commonly used error distributions. It can be seen that the desirable high efficiency of traditional rank methods for estimating a finite-dimensional parameter completely carries over to the local rank method for estimating an infinite dimensional parameter.

Table 1
Asymptotic Relative Efficiency

By a similar calculation, we can show that the asymptotic relative efficiencies of the local rank estimator to the local linear estimator for a′(u0) and a′(·) both equal ψ = (12σ2τ2)8/11, which has a lower bound 0.8991. This value is also reported in Table 1 for some common error distributions.

Remark 2

We may also apply the local median approach (Yu and Jones, 1998) to estimate the coefficient functions and their first derivatives. Similarly, we can prove that such estimators are asymptotically normal. The ARE of the local median estimator versus the local linear least squares estimator is closely related to that of the sign test versus the t-test. It is known that the ARE of the sign test versus the t-test for the normal distribution is 0.63. Thus we expect the efficiency loss of the local median procedure to be substantial for normal random error.

3.3 Asymptotic normality of [alpha]1

Following (5), α^1=nh{α^1a0(u0)} is the value of α1 that minimizes

Qn0(α1,α^2,β^)=n1i=1n|εiγnα1(α^2a0(u0))(Uiu0)m=1p[(β^mam(u0))+(β^p+mam(u0))(Uiu0)]Xim+Δi(u0)|Kh(Uiu0).

Similarly as Lemma 3.1, we can establish the following local quadratic approximation which holds uniformly in a neighborhood around 0:

γn1Qn0(α1,α^2,β^)=γn1α1Sn0+γng(0)f(u0)α12+γn1Qn0(0,a0(u0),β0)+op(1),
(17)

where

Sn0=2γnn1i=1n[I(εiΔi(u0))1/2]Kh(Uiu0).
(18)

This further leads to an asymptotic representation of [alpha]1:

nh(α^1a0(u0))=γn2[2g(0)f(u0)]1Sn0+op(1).
(19)

The theorem below gives the asymptotic distribution of [alpha]1.

Theorem 3.4

Under the conditions of Theorem 3.2, we have

nh[α^1a0(u0)μ2a0(u0)2h2+o(h2)]N(0,[12g2(0)f(u0)]1ν0).

3.4 Estimation of the standard errors

To make statistical inference based on the local rank methodology, one needs to estimate the standard error of the resulting estimator. As indicated by Theorem 3.2, the asymptotic covariance matrix of the local rank estimator is rather complex and involves unknown functions. Here we propose a standard error estimator using a simple resampling method proposed by Jin, Ying and Wei (2001).

Let V1, …, Vn be independent and identically distributed nonnegative random variables with mean 1/2 and variance 1. We consider a stochastic perturbation of (4):

Q¯n(β,α2)=1n(n1)1i,jn(Vi+Vj)eiejKh(Uiu0)Kh(Uju0),
(20)

where ei is defined in (3). In Qn(β, α2), the data {Yi, Ui, Xi} are considered to be fixed, and the randomness comes from the Vi’s. Let ([beta with macron above]T, [alpha]2)T be the value of (βT, α2)T that minimizes Qn(β, α2). It is easy to obtain ([beta with macron above]T, [alpha]2)T by applying a simple algorithm described in Section 3.1.

Jin, Ying and Wei (2001) established the validity of the resampling method when the objective function has a U-statistics structure. Although their theory covers many important applications, they require that the U-statistic has a fixed kernel. We extend their result to our setting, where the U-statistic involves variable kernel due to nonparametric smoothing. Let ā(u0) be the local rank estimator of a(u0) based on the perturbed objective function (20), i.e., it is the subvector that consists of the first p components of [beta with macron above]. Its asymptotic normality is given in the theorem below.

Theorem 3.5

Under the conditions of Lemma 3.1, conditional on almost surely every sequence of data {Yi; Ui; Xi},

nh[a¯(u0)a^(u0)]N(0,ν012τ2f(u0)1(u0))

in distribution.

This theorem suggests that to estimate the asymptotic covariance matrix of b â(u0), one can repeatedly perturb (4) by generating a large number of independent random samples {Vi}i=1n. For each perturbed objective function, one solves for ā(u0). The sample covariance matrix of ā(u0) based on a large number of independent perturbations provides a good approximation. The accuracy of the resulting standard error estimate will be tested in the next section.

The perturbed estimator has conditional bias equal to zero. It has been found that standard bootstrap method, which resamples from the empirical distribution of the data, also estimates bias as zero when estimating nonparametric curves (Hall and Kang, 2001). It is possible to use more delicate bootstrap technique to estimate the bias of a nonparametric curve estimator. Although some of the ideas may be adapted to the method of disturbing the objective function, this is beyond the scope of our paper and is not pursued further here.

4 Numerical Studies

4.1 A pseudo-observation algorithm

The local rank estimator can be obtained by applying an efficient and reliable algorithm. Note that the local rank estimator of (β0T,a0(u0))T can be solved by fitting a weighted L1 regression on n(n1)2 pseudo observations ( xixj, Yi-Yj) with weights wij = K((Uiu0)/h)K((Uju0)/h), where xi=(Uiu0,XiT,(Uiu0)XiT)T, 1 ≤ i < jn. Given ([beta]T, [alpha]2)T, the estimator of a0(u0) can be obtained by another weighted L1 regression on (1, Yiα^2(Uiu0)m=1p[β^m+β^p+m(Uiu0)]Xim) With weights wi = K((Uiu0)/h), 1 ≤ in. Many statistical software packages can implement weighted L1 regression. In our numerical studies, we use the function “rq” in the R package quantreg.

4.2 Bandwidth selection

Bandwidth selection is an important issue for all statistical models that involve nonparametric smoothing. Although we have derived the theoretical optimal bandwidth for the local rank estimator in (14) and (16), it is difficult to use the “plug-in“ method to estimate it due to many unknown quantities.

We propose below an alternative bandwidth selection method that is practically feasible. This approach is based on the relationship between hRopt and hLSopt. From Section 2.3, we see that

hRopt(u0)=(112τ2σ2)1/5hLSopt(u0)andhRopt=(112τ2σ2)1/5hLSopt.
(21)

Thus, we can first use existing bandwidth selectors (e.g. Zhang and Lee 2000) to estimate hLSopt(u0) or hLSopt. The error variance σ2 can be estimated based on the residuals; in particular when robustness is of concern it can be estimated using the MAD of the residuals. Hettmansperger and Mckean (1998, p.181) discussed in details how to estimate τ, which can be obtained by the function “wilcoxontau” in the R software developed by Terpstra and McKean (2005). In the end, we plug in these estimators into (21) to get the bandwidth for local rank estimator.

Alternatively, instead of the above two-step procedure, we may directly use the computationally intensive cross-validation to estimate the bandwidth for the local rank procedure. Note that under outlier contamination, standard cross-validation can lead to extremely biased bandwidth estimates because it can be adversely influenced by extreme prediction errors. Robust cross-validation method, such as that developed by Leung (2005), is therefore preferred.

4.3 Examples

We conduct Monte Carlo simulations to access the finite sample performance, and illustrate the proposed methodology on a real environmental data set. In the analysis, we use the Epanechnikov kernel K(u) =.75(1 − u2)I(|u| < 1).

Example 1

We generate random data from

Y=a0(U)+a1(U)X1+a2(U)X2+ε,

where a0(u) = exp(2u − 1), a1(u) = 8u(1 − u) and a2(u) = 2 sin2(2πu). The covariate U follows a uniform distribution on [0, 1], and is independent from (X1, X2), where the covariates X1 and X2 are standard normal random variables with correlation coefficient 2−1/2. The coefficient functions and the mechanism to generate U and (X1, X2) are the same as those in Cai, Fan and Li (2000). We consider six different error distributions: N(0, 1), Laplace, standard Cauchy, t-distribution with 3 degrees of freedom, mixture of normals 0.9N(0, 1) + 0.1N(0, 102) and lognormal distribution. Except for Cauchy error, all the other generated random errors are standardized to have median 0 and variance 1. We consider sample sizes n =400 and 800, and conduct 400 simulations for each case.

We compare the performance of the local rank estimate with the local least squares estimate using the square root of average squared errors (RASE), defined by

RASE={1ngridm=1pk=1ngrid{a^m(uk)am(uk)}2}1/2,

where {uk: k = 1, ···, ngrid} is a set of grid points uniformly placed on [0, 1] with ngrid = 200. The sample mean and standard deviation of the RASEs over 400 simulations are presented in Figure 1 and Figure 2, for sample size n = 400 and 800, respectively. The two figures clearly demonstrate that the local rank estimator performs almost as well as the local least squares estimator for normal random error; and has smaller RASE for other heavier-tailed error distributions. The efficiency gain can be substantial, see for example the mixture normal case where the observed relative efficiency for the local rank estimator versus the local least squares estimator is above 2 for most choices of bandwidth. For Cauchy random error, the local rank estimator yields a n-consistent estimator but the local least squares estimator is inconsistent, which is reflected by the extremely large value of RASE for the local least squares estimator.

Figure 1
Bar graphs of the RASE with standard error for sample size n = 400 over 400 simulations. The light gray bar denotes the local least squares method and the dark gray bar denotes the local rank method. The horizontal axis is in the unit of hopt, which is ...
Figure 2
Bar graphs of the RASE with standard error for sample size n = 800 over 400 simulations. The light gray bar denotes the local least squares method and the dark gray bar denotes the local rank method. The horizontal axis is in the unit of hopt, which is ...

Figure 3 depicts the estimated coefficient functions for the normal random error and the mixture normal random error for a typical sample, which is selected in such a way that its RASE value is the median of the 400 RASE values. For this typical sample, we observe that the local rank estimator is almost identical to the local least squares estimator for normal random error; but falls much closer to the truth than the local least squares estimator does for mixture normal random error. Figure 4 plots the estimated coefficient functions for all 400 simulations when the random error has a mixture of normal distribution. It is clear that the local rank estimator has smaller variance. In these two figures, we set the bandwidth to be the theoretic optimal one hopt, calculated using (15) and (16), for both the local rank estimator and the local least squares estimator.

Figure 3
Plot of estimated coefficient functions for a typical data set
Figure 4
(a) and (c) are plots of 400 local least squares estimators of a1(·) and a2(·) over 400 simulation, respectively. (b) and (d) are plots of 400 local rank estimators of a1(·) and a2(·), respectively.

At the end, we evaluate the resampling method (Section 3.4) for estimating the standard errors. We randomly perturb the objective function 1000 times; each time the random variables Vi in (20) are generated from the Gamma(0.25, 2) distribution. Table 2 summarizes the simulation results at three points u0 = 0.25, 0.50 and 0.75. In the table, ‘SD’denotes the standard deviation of 400 estimated âm(u0) and can be regarded as the true standard error; ‘SE(std(SE))’ denotes the mean (standard deviation) of 400 estimated standard errors from the resampling method. Bandwidths are set to be the optimal ones. We observe that the resampling method estimates the standard error very accurately.

Table 2
Standard deviations of the local rank estimators with n = 400

Example 2

As an illustration, we now apply the local rank procedure to the environmental data set in Fan and Zhang (1999). Of interest is to study the relationship between levels of pollutants and the number of total hospital admissions for circulatory and respiratory problems on every Friday from January 1, 1994 to December 31, 1995. The response variable is the logarithm of the number of total hospital admissions, and the covariates include the level of sulfur dioxide (X1), the level of nitrogen dioxide (X2) and the level of dust (X3). A scatter plot of the response variable over time is given in Figure 5(a). We analyze this data set using the following varying coefficient model

Figure 5
(a) Scatterplot of the log of number of total hospital admissions over time, and the solid curve is an estimator of the expected log of number of hospital admissions over time at the average pollutant levels, i.e., â0(u) + â1(u)X ...

Y=a0(u)+a1(u)X1+a2(u)X2+a3(u)X3+ε,

where u denotes time and is scaled to the interval [0,1].

We select the bandwidth via the relation (21). More specifically, we first use 20-fold cross-validation to select a bandwidth ĥLS for the local least squares estimator. We then use the function ‘wilcoxontau’ in the R package for rank regression by Terpstra and McKean to estimate (12τ2)−1/2 and use the MAD of the residuals to robustly estimate σ. All these lead to the selected bandwidth for the local rank estimator: ĥR = 0.26.

The estimated coefficient functions are depicted in Figures 5(b), (c) and (d), where the two dashed curves around the solid line are the estimated function plus/minus twice the standard errors estimated by the resampling method. These two dashed lines can be regarded as a pointwise confidence interval with bias ignored. The figures suggest clearly that the coefficient functions vary with time. The fitted curve is shown in Figure 5(a).

Now we demonstrate the robustness of the local rank procedure. To this end, we artificially perturb the dataset by moving the response value of the 68th observation from 5.89 to 6.89, and the response value of the 34th observation from 5.07 to 3.07. We refit the data with both the local least squares procedure and the local rank procedure, see Figure 6. We observe that the local least squares estimator changes dramatically due to the presence of these two artificial outliers. In contrast, the local rank estimator is nearly unaffected.

Figure 6
(a) Scatterplot of the perturbed data (without the two outliers shown as they are outside of the range), and the local LS and the local rank estimators of the expected log of number of hospital admissions. (b), (c) and (d) are the local LS and the rank ...

Supplementary Material

WKL

Appendix: Proofs

Regularity conditions

  • (C1). Assume that {Ui, Xi, Yi} are independent and identically distributed, and that the random error ε and the covariate {U, X} are independent. Furthermore, assume that ε has probability density function g(·) which has finite Fisher information, i.e., {g(x)}−1g′(x)2dx < ∞; and that U has probability density function f(·).
  • (C2). The function am(·), m = 0, 1,…, p, has continuous second-order derivative in a neighborhood of u0.
  • (C3). Assume that E(Xi|Ui = u0) = 0 and that (u)=E(XiXiTUi=u) is continuous at u = u0. The matrix Σ(u0) is positive definite.
    (C4). The kernel function K(·) is symmetric about the origin, and has a bounded support. Assume that h → 0 and nh2 → ∞, as n → ∞

These conditions are used to facilitate the proofs, but may not be the weakest ones. The assumptions on the random errors in (C1) are the same as those for multiple linear rank regression (Hettmansperger and McKean, 1998). (C2) imposes smoothness requirement on the coefficient functions. In (C3), the assumption E(Xi|Ui = u0) = 0 (also adopted by Kim, 2007) makes the presentation simpler but can be relaxed. It can be shown that the asymptotic normality still holds without this assumption. The conditions on the kernel function and bandwidth in (C4) are common for nonparametric kernel smoothing.

In our proofs, we will use some results on generalized U-statistic, where the kernel function is allowed to depend on the sample size n. The generalized U-statistic has the form Un = [n(n − 1)]−1ΣΣij Hn(Di, Dj), where {Di}i=1n is a random sample and Hn is symmetric in its arguments, i.e., Hn(Di, Dj) = Hn(Dj, Di). In this paper, Di=(XiT,Ui,εi)T. Define rn(Di) = E[Hn(Di, Dj)|Di], [r with macron]n = E[rn(Di)] and U^n=r¯n+2n1i=1n[rn(Di)r¯n]. We will repeatedly use the following lemma taken from Powell, Stock and Stoker (1989).

Lemma A.1

If E[||Hn(Di, Dj)||2] = o(n), then n(UnU^n)=op(1) and Un = [r with macron]n + op(1).

We need the following two lemmas to prove Lemma 3.1. Denote

An11=2h2E{(ZiZj)(ZiZj)TK(Uiu0h)K(Uju0h)},An12=2h2E{(ZiZj)[(UiUj)/h]K(Uiu0h)K(Uju0h)},An21=An12T,An22=2h2E{[(UiUj)2/h2]K(Uiu0h)K(Uju0h)},

and define

An=τ(An11An12An21An22).

Lemma A.2

Suppose that Conditions (C1) — (C4) hold, then AnA, where A is defined in (8).

Proof

We can write An11=(An111An112An113An114). Let

An111=2h2E[(XiXj)(XiXj)TK(Uiu0h)K(Uju0h)].

Calculating the expectation by conditional on Ui and Uj first, An11 becomes

2h2E[(XiXj)(XiXj)TUi=u,Uj=v]K(uu0h)K(vu0h)f(u)f(v)dudv.

Using Condition (C3), straightforward calculation gives An1114f2(u0)(u0). Let

An112=2h2E{(XiXj)[Xi(Uiu0)/hXj(Uju0)/h]TK(Uiu0h)K(Uju0h)}.

Using Condition (C3) and notice that K(·) is symmetric, it can be shown that An1120. By symmetry, An1130. Similarly, we have

An114=2h2E{[Xi(Uiu0)/hXj(Uju0)/h][Xi(Uiu0)/hXj(Uju0)/h]TK(Uiu0h)K(Uju0h)}4f2(u0)(u0)μ2.

Thus An114f2(u0)(u0)(Ip00μ2Ip). Similarly, we can show that An12=An21T0, and

An22=2(t1t2)2K(t1)K(t2)f(u0+t1h)f(u0+t2h)dt1dt24f2(u0)μ2.

Lemma A.3

Under Conditions (C1) — (C4), we have

γn1[Sn(β,α2)Sn(0,0)]=γnA(βα2)+op(1).

Proof

Let Un=γn1[Sn(β,α2)Sn(0,0)]=[n(n1)]1ijWn(Di,Dj), where

Wn(Di,Dj)=2[I(εiγnα2(Uiu0)/hγnβTZi+Δi(u0)εjγnα2(Uju0)/hγnβTZj+Δj(u0))1/2](ZiZj(UiUj)/h)Kh(Uiu0)Kh(Uju0).

Let Hn(Di, Dj) = [Wn(Di, Dj)+Wn(Dj, Di)]/2, then Un = [n(n − 1)]−1 ΣΣij Hn(Di, Dj) has the form of a generalized U-statistic. Note that E[Hn(Di,Dj)2]12E[Wn(Di,Dj)2]+12E[Wn(Di,Dj)2]=E[Wn(Di,Dj)2]. Furthermore,

E[Wn(Di,Dj)2]4h4E{[(ZiZj)T(ZiZj)+[(UiUj)/h]2]K2(Uiu0h)K2(Uju0h)}=O(h2)=o(n)sincenh2byassumption.

Thus Un = E[Hn(Di, Dj)] + op(1) by Lemma A.1. Furthermore,

E[Hn(Di,Dj)]=2h2E{[G(ε+Δj(u0)Δi(u0)γnα2(UjUi)/hγnβT(ZjZi))G(ε)]g(ε)dε(ZiZj(UiUj)/h)K(Uiu0h)K(Uju0h)}=2h2γnE{g[ε+Δj(u0)Δi(u0)]g(ε)dε(ZiZj(UiUj)/h)(ZiTZjT,(UiUj)/h)K(Uiu0h)K(Uju0h)}(βα2)(1+o(1))=γnAn(βα2){1+o(1)}.

The proof is completed by using Lemma A.2.

Proof of Lemma 3.1

In view of Lemma A.3, it follows that

[γn1Qn(β,α2)Bn(β,α2)]=γn1[Sn(β,α2)Sn(0,0)]γnA(βα2)=op(1).

The proof follows along the same lines of the proof of Theorem A.3.7. of Hettmansperger and McKean (1998), using a “diagonal subsequencing” argument and convexity.

Proof of Theorem 3.2

By Lemma 3.1, γn1Qn(s1,s2)=Bn(s1,s2)+rn(s1,s2), where rn(s1,s2)p0 uniformly over any bounded set. Note that γn1Qn(s1,s2) is minimized by (β^nT,α^2n)T, and Bn(s1, s2) is minimized by (βnT,α2n)T=γn2A1(Sn1T(0,0),Sn2(0,0))T. We first establish the asymptotic representation by following similar argument as in Hjort and Pollard (1993). For any constant c > 0, define

Tn=inf(s1T,s2)(βnT,α2n)=cBn(s1,s2)Bn(βn,α2n)Rn=sup(s1T,s2)(βnT,α2n)cγn1Qn(s1,s2)Bn(s1,s1),

then Rnp0 as n → ∞. Let (s1T,s2)T be an arbitrary point outside the ball { (s1T,s2)T:(s1T,s2)(βnT,α2n)c}, then we can write (s1T,s2)T=(βnT,α2n)T+l12p+1, where l > c is a positive constant and 1d denotes a unit vector of length d. Write

cl[γn1Qn(s1,s2)γn1Qn(βn,α2n)]=clγn1Qn(s1,s2)+(1cl)γn1Qn(βn,α2n)γn1Qn(βn,α2n).

By the convexity of γn1Qn(s1,s2), we have

cl[γn1Qn(s1,s2)γn1Qn(βn,α2n)]γn1Qn(cl(s1,s2)+(1cl)(βn,α2n))γn1Qn(βn,α2n).

Thus,

cl[γn1Qn(s1,s2)γn1Qn(βn,α2n)]γn1Qn(βn+c12p,α2n+c)γn1Qn(βn,α2n)=Bn(βn+c12p,α2n+c)+rn(βn+c12p,α2n+c)Bn(βn,α2n)rn(βn,α2n)Tn2Rn.

If Rn12Tn, then γn1Qn(s1,s2)>γn1Qn(βn,α2n) for all (s1T,s2)T outside the ball. This implies if Rn12Tn then the minimizer of γn1Qn must be inside the ball. Thus

P((βnT,α2n)T(β^nT,α^2n)Tc)P(Rn12Tn)=P(Rn12λc2)0,

where λ is the smallest eigenvalue of A. Therefore, (β^nT,α^2n)T=(βnT,α2n)T+op(1). This in particular implies the asymptotic representations (9), (11) and (12).

We next show the asymptotic normality of â(u0). From (9), we have

nh(a^(u0)a(u0))=γn2(4τf2(u0)(μ0))1Sn11(0,0)+op(1),
(A.1)

where

Sn11(0,0)=2γn[n(n1)]1ij[I(εi+Δi(u0)εj+Δj(u0))1/2](XiXj)Kh(Uiu0)Kh(Uju0).
(A.2)

By (A.2), let us rewrite γn2Sn11(0,0)=Sna1(0,0)+Sna2(0,0), where

Sna(0,0)=2γn1[n(n1)]1ij[I(εiεj)1/2](XjXi)Kh(Uiu0)Kh(Uju0),Snb(0,0)=2γn1[n(n1)]1ij[I(εi+Δi(u0)εj+Δj(u0))I(εiεj)](XjXi)Kh(Uiu0)Kh(Uju0).

We next prove that

Sna(0,0)N(0,43f3(u0)ν0(u0))indistribution.
(A.3)

Note that we can write Sna(0,0)=n[n(n1)]1ijHn(Di,Dj), where Hn(Di, Dj) =Wn(Di, Dj) + Wn(Dj, Di) with

Wn(Di,Dj)=h3/2[I(εiεj)1/2](XjXi)K(Uiu0h)K(Uju0h).

Similarly to the arguments in the proof of Lemma A.3, it can be shown that E[||Hn(Di, Dj)||2] = o(n). By Lemma A.1, this implies that Sna(0,0)=2n1i=1nrn(Di)+op(1) since it is easy to check that [r with macron]n = 0. We have

rn(Di)=E[Hn(Di,Dj)Di]=2h3/2[G(εi)1/2]K(Uiu0h)E{(XiXj)K(Uju0h)Xi,Ui,εi}=2h1/2[G(εi)1/2]K(Uiu0h)[(K(t)f(u0+th)dt)XiE(XjUj=u0+th)K(t)f(u0+th)dt].

Furthermore,

E[rn(Di)rn(Di)T]=13h1E{K2(Uiu0h)[(K(t)f(u0+th)dt)XiE(XjUj=u0+th)K(t)f(u0+th)dt][(K(t)f(u0+th)dt)XiTE(XjTUj=u0+th)K(t)f(u0+th)dt]}13f3(u0)ν0(u0).

To prove the asymptotic normality of Sna(0, 0), it is sufficient to check the Lindeberg-Feller condition: ∀ ε > 0, n1i=1nE{rn(Di)rn(Di)TI(rn(Di)>εn)}0. This can be easily verified by applying the dominated convergence theorem. Next we show that

Snb(0,0)=2h2γn[τf2(u0)μ2(u0)a(u0)+o(1)]+op(1).
(A.4)

We may write Snb(0,0)=[n(n1)]1ijHn(Di,Dj), where Hn(Di,Dj)=Wn(Di,Dj)+Wn(Dj,Di) with

Wn(Di,Dj)=nh1γn[I(εi+Δi(u0)εj+Δj(u0))I(εiεj)](XjXi)K(Uiu0h)K(Uju0h).

Note that

Δj(u0)Δi(u0)=12[(Uju0)2XjT(Uiu0)2XiT]a(u0)+12[(Uju0)2(Uiu0)2]a0(u0)+o((Uiu0)2)+o((Uju0)2).

By applying Lemma A.1, it can be shown that Snb(0,0)=E[Hn(Di,Dj)]+op(1). It follows by using the same arguments as those in the proof of Lemma A.2 that

E[Hn(Di,Dj)]=2nh1γnE{[G(ε+Δj(u0)Δi(u0))G(ε)]g(ε)dε(XjXi)K(Uiu0h)K(Uju0h)}=2nh1γn[τ+O(h)]E[(Δj(u0)Δi(u0))(XjXi)K(Uiu0h)K(Uju0h)](1+o(1))=2h2γn[τf2(u0)μ2(u0)a(u0)+o(1)].

This proves (A.4). By combining (A.3) and (A.4) and using the approximation given in (A.1), we obtain (10).

Proof of Theorem 3.3

A result of Hodges and Lehmann (1956) indicates that the ARE has a lower bound 0.8644/5 = 0.8896, with this lower bound being attained at the density f(t)=3205(5x2)I(x5).

Proof of Theorem 3.4

Let

Vn(α1,ξ1,ξ2,ξ3)=(nh)1i=1n|εiγnα1ξ1(Uiu0)ξ2TXiξ3T(Uiu0)Xi+Δi(u0)|K(Uiu0h),

where α1=γn1(α1a0(u0)), ξ1 [set membership] R, ξ2 [set membership] Rp and ξ3 [set membership] Rp. The subgradient of Vn(α1,ξ1,ξ2,ξ3) with respect to α1 is

Sn(α1,ξ1,ξ2,ξ3)=2γnnhi=1n[I(εiγnα1+ξ1(Uiu0)+ξ2TXi+ξ3T(Uiu0)XiΔi(u0))1/2]K(Uiu0h).

We have Sn(0,0,0,0)=2γn(nh)1i=1n[I(εiΔi(u0))1/2]K(Uiu0h), which is the same as the Sn0 defined in (18). Let Un(α1,ξ1,ξ2,ξ3)=γn1[Sn(α1,ξ1,ξ2,ξ3)Sn(0,0,0,0)], then

Un(α1,ξ1,ξ2,ξ3)=2(nh)1i=1n[I(εiγnα1+ξ1(Uiu0)+ξ2TXi+ξ3T(Uiu0)XiΔi(u0))I(εiΔi(u0))]K(Uiu0h).

For any positive constants ci, i = 1, 2, 3 and ∀ ξ1, ξ2, ξ3 such that ||ξ1|| ≤ c1h−1γn, ||ξ2|| ≤ c2γn and ||ξ3|| ≤ c3h−1γn, we have

Un(α1,ξ1,ξ2,ξ3)=2γng(0)f(u0)α1+op(1).
(A.5)

This can be proved by directly checking the mean and variance. More specifically,

E[Un(α1,ξ1,ξ2,ξ3)]=2h1E{[G(γnα1+ξ1(Uiu0)+ξ2TXi+ξ3T(Uiu0)XiΔi(u0))G(Δi(u0))]K(Uiu0h)}=2h1g(0)E{[γnα1+ξ1(Uiu0)+ξ2TXi+ξ3T(Uiu0)Xi]K(Uiu0h)}(1+O(h))=2γng(0)f(u0)α1(1+O(h)).

And

Var[Un(α1,ξ1,ξ2,ξ3)]4n1h2E{[I(εiγnα1+ξ1(Uiu0)+ξ2TXi+ξ3T(Uiu0)XiΔi(u0))I(εiΔi(u0))]2K2(Uiu0h)}4n1h2E{K2(Uiu0h)}=O(n1h1)=o(1).

By (A.5) and similar proof as that for Lemma 3.1, we have

γn1Vn(α1,ξ1,ξ2,ξ3)=Vn(α1)+op(1),
(A.6)

where Vn(α1)=γn1Sn(0,0,0,0)α1+γng(0)f(u0)α12+γn1Vn(0,0,0,0). Because the function Vn(α1,ξ1,ξ2,ξ3) is convex in its arguments, (A.6) can be strengthened to uniform convergence (convexity lemma, see Pollard 1991), i.e.,

supα1C,ξ1c1h1γnξ2c2γn,ξ3c3h1γnγn1Vn(α1,ξ1,ξ2,ξ3)Vn(α1)=op(1),

where C is a compact set in R. By Theorem 3.2, α^2α0(u0)=Op(h1γn), â(u0) − a(u0) = Op(γn) and â′ (u0) − a′(u0) = Op(h−1γn), we thus have

supα1C|γn1Vn(α1,α^2a0(u0),a^(u0)a(u0),a^(u0)a(u0))Vn(α1)|=op(1).

Note that Vn(α1,α^2a0(u0),a^(u0)a(u0),a^(u0)a(u0))=Qn0(α1,α^2,β^),Sn(0,0,0,0)=Sn0, where Qn0 and Sn0 are defined in Section 2.4. The quadratic function Vn(α1) is minimized by α1n=12γn2[g(0)f(u0)]1Sn0. Similar argument as that for Theorem 3.2 shows that α^1n=α1n+op(1). Thus we have (19). We can write γn2Sn0=T1n+T2n, where

T1n=2γn1nhi=1n[I(εi0)1/2]K(Uiu0h),T2n=2γn1nhi=1n[I(εiΔi(u0))I(εi0)]K(Uiu0h).

By the Lindeberg-Feller central limit theorem, T1nN (0, f (u0)ν0/3) in distribution. By checking mean and variance, we have

T2n=h2γng(0)f(u0)a0(u0)μ2(1+o(1))+op(1).

Combining the above results and using (19), the proof is completed.

To prove Theorem 3.5, we first extend Lemma A.1 to almost sure convergence.

Lemma A.4

If E[||Hn(Di, Dj)||2] = O(h−2), then Un − Ûn = o(1) almost surely and Un = [r with macron]n + o(1) a.s.

Proof

The proof of Powell, Stock and Stoker (1989) for Lemma A.1 suggests that E[||UnÛn||2] = O(n−2h−2). By Theorem 1.3.5 of Serfling (1980), i=1nE[UnU^n2]=O(n1h2)<. This implies that UnÛn = o(1) almost surely. The second result follows by an application of the strong law of large numbers to Ûn.

Proof of Theorem 3.5

Let β* and α2 be defined the same as before. We introduce the reparametrized objective function Q¯n(β,α2). Let S¯n(β,α2)=(S¯n1T(β,α2),S¯n2(β,α2))T denote the gradient function of Q¯n(β,α2), which is defined similarly as in Section 2.2. We first show that S¯n(β,α2) has a similar local linear approximation as stated in Lemma A.3. To make the proof concise, we prove this for S¯n1(β,α2), where

S¯n1(β,α2)=2γn[n(n1)]1ij(Vi+Vj)[I(εiγnα2(Uiu0)/hγnβTZi+Δi(u0)εjγnα2(Uju0)/hγnβTZj+Δj(u0))1/2](ZiZj)Kh(Uiu0)Kh(Uju0).

Let Un=γn1[S¯n1(β,α2)S¯n1(0,0)]=[n(n1)]1ij(Vi+Vj)Mn(Di,Dj,β,α2), where Mn(Di,Dj,β,α2)=12[mn(Di,Dj,β,α2)+mn(Dj,Di,β,α2)] and

mn(Di,Dj,β,α2)=2[I(εiγnα2(Uiu0)/hγnβTZi+Δi(u0)εjγnα2(Uju0)/hγnβTZj+Δj(u0))1/2](ZiZj)Kh(Uiu0)Kh(Uju0).

Note that Un=2n1i=1nVi[(n1)1j=1,jinMn(Di,Dj,β,α2)]. Conditional on {Di}i=1n, this is a weighted average of Vi. Note that

E(Un{Di}i=1n)=[n(n1)]1ijMn(Di,Dj,β,α2).Var(Un{Di}i=1n)=n2i=1n[(n1)1j=1,jinMn(di,dj,β,α2)]2.

By Lemma A.4, it can be shown that [n(n1)]1ijMn(Di,Dj,β,α2)=γnAβ+o(1) almost surely, where A* = 4τf2(u0)diag(Ip2Ip) [multiply sign in circle] Σ(μ0). It is also easy to check that n2i=1n[(n1)1j=1,jinMn(Di,Dj,β,α2)]2=o(1) almost surely. Thus for almost surely every sequence {Di}i=1n, Un=γnA*β*+op(1), where op(1) is in the probability space generated by {Vi}i=1n. The proofs of Lemma 3.1 and the asymptotic representation in Theorem 3.2 can be similarly carried out to show that for almost surely every sequence {Di}i=1n,

nh[a¯n(u0)a(u0)]=γn2(4τf2(u0)(μ0))1S¯na(0,0)+op(1),
(A.7)

where op(1) is in the probability space generated by {Vi}i=1n, and

S¯na(0,0)=2γn[n(n1)]1ij(Vi+Vj)[I(εi+Δi(u0)εj+Δj(u0))1/2](XiXj)Kh(Uiu0)Kh(Uju0).

The approximation (A.1) can be strengthened to almost surely convergence, i.e.,

nh[a^n(u0)a(u0)]=γn2(4τf2(u0)(μ0))1Sna(0,0)+o(1)a.s..
(A.8)

Combining (17) and (A.7), we have that for almost surely every sequence {Di}i=1n,

nh[a¯n(u0)a^n(u0)]=γn2(4τf2(u0)(μ0))1[S¯na(0,0)Sna(0,0)]+op(1).

Note that

γn2[S¯na(0,0)Sna(0,0)]=2γn1[n(n1)]1ij[(Vi1/2)+(Vj1/2)][I(εi+Δi(u0)εj+Δj(u0))1/2](XiXj)Kh(Uiu0)Kh(Uju0)=4γn1n1i=1n(Vi1/2){(n1)1j=1,jin[I(εi+Δi(u0)εj+Δj(u0))1/2](XiXj)Kh(Uiu0)Kh(Uju0)}.

And E{γn2[S¯na(0,0)Sna(0,0)]{Di}i=1n}=0. We have

Var{γn2[S¯na(0,0)Sna(0,0)]{Di}i=1n}=16γn2n2(n1)2i=1n{j=1,jin[I(εi+Δi(u0)εj+Δj(u0))1/2)](XiXj)Kh(Uiu0)Kh(Uju0)}2=W1+W2,

where

W1=16γn2n2(n1)2h4i=1nj=1,jin[I(εi+Δi(u0)εj+Δj(u0))1/2]2(XiXj)(XiXj)TK2((Uiu0)/h)K2((Uju0)/h),W2=16γn2n2(n1)2h4i=1nj1inj2i,j1n[I(εi+Δi(u0)εj1+Δj1(u0))1/2][I(εi+Δi(u0)εj2+Δj2(u0))1/2](XiXj1)(XiXj2)TK2((Uiu0)/h)K((Uj1u0)/h)K((Uj2u0)/h).

Lemma A.4 can be used to show that W1 = o(1) almost surely; and a minor extension of Lemma A.4 to third-order U-statistic can be used to show that W2=43f3(u0)ν0(u0)+o(1) almost surely. The asymptotic normality of γn2[S¯na(0,0)Sna(0,0)] follows by showing that the condition of Lindeberg-Feller central limit theorem for triangular arrays holds almost surely. We have, for almost surely every sequence {Di}i=1n,

γn2[S¯na(0,0)Sna(0,0)]N(0,43f3(u0)ν0(u0))

in distribution. This completes the proof.

Footnotes

1Lan Wang is Assistant Professor, School of Statistics, University of Minnesota, Minneapolis, MN 55455. Email: ude.nmu.tats@nal. Bo Kai is a graduate student, Department of Statistics, The Pennsylvania State University, University Park, PA 16802. Email: ude.usp@iakob. Runze Li is Professor, Department of Statistics and The Methodology Center, The Pennsylvania State University, University Park, PA 16802-2111. Email: ude.usp.tats@ilr. Wang’s research is supported by National Science Foundation grant DMS-0706842. Kai’s research is supported by National Science Foundation grants DMS 0348869 as a research assistant. Li’s research is supported by NIDA, NIH grants R21 DA024260 and P50 DA10075. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIDA or the NIH.

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