Difference based M estimator of generalized semiparametric model with NSD errors

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R E S E A R C H

Open Access

Difference-based M-estimator of generalized

semiparametric model with NSD errors

Fangning Fu

1

_{, Zhen Zeng}

1

_{and Xiangdong Liu}

1*

*_{Correspondence:}

[email protected]

1_{Department of Statistics, Jinan}

University, Guangzhou, P.R. China

Abstract

In this paper, we consider the generalized semiparametric model (GSPM)

yi=h

(

xT_i

β)

+f(ti) +ei, 1≤i≤n,

whereh(·) is a known function,eiare dependent errors. We obtain an estimator of the parametric component

β

for the model by a diﬀerence-based M-estimator. In addition, we prove the asymptotic normality of the proposed estimator and investigate the weak convergence rate of the wavelet estimator off(·). Furthermore, we apply these results to a partially linear model with dependent errors.

MSC: 60F05; 62F12; 62G05

Keywords: Generalized semiparametric model; NSD random variables; M-estimator; Asymptotic normality; Weak convergence rate

1 Introduction

Consider the generalized semiparametric model

yi=h

xT_iβ+f(ti) +ei, 1≤i≤n, (1)

whereyiare scalar response variables,h(·) is a continuously diﬀerentiable known function,

the superscriptT denotes the transpose, xi= (xi1, . . . ,xid)Tare explanatory variables,βis

ad-dimensional unknown parameter,f(·) is an unknown function, and 0≤t1≤t2≤ · · · ≤ tn≤1. Some authors commented that the assumption of independence is a serious

restric-tion (see Huber [1] and Hampel [2]); so for the errorsei, we conﬁne ourselves to negatively

superadditive dependent (NSD) errors. NSD random variables have been introduced by Hu [3] and are widely used in statistics; see [4–12].

The theory of the GSPM is an extension of the classical theory of partially linear mod-els; the component of the generalized parametrich(xT

iβ) for GSPM includes the linear

parametric component xT

iβ, exponential parametric componente

xT_iβ_{, and so on.}

As is well known, the generalized partially linear model and partially linear single-index model (h(·) is an unknown link function) are also derived from the partially linear model. There is a substantial amount of work for generalized partially linear model (see [13–

18] and, for a partially linear single-index model, [19–24]); this research is devoted to

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presenting various methods to obtain estimators ofβ andf(ti) and investigating some

large-sample properties of these estimators.

In this paper, we consider a difference-based estimator method to estimate the unknown parametric componentβ. This difference-based estimator is optimal in the sense that the estimator of the unknown parametric component is asymptotically efficient. For example, Tabakan et al. [25] studied a difference-based ridge in a partially linear model. Wang et al. [26] obtained a difference-based approach to the semiparametric partially linear model. Zhao and You [27] used a difference-based estimator method to estimate the parametric component for partially linear regression models with measurement errors. Duran et al. [28] investigated the difference-based ridge and Liu-type estimators in semiparametric re-gression models. Wu [29] discussed a restricted difference-based Liu estimator in partially linear models. Hu et al. [30] presented a difference-based Huber–Dutter (DHD) estima-tor to obtain the root varianceσ and parameterβfor a partially linear model. However, Most of the results rely on the independence errors. Wu [31] studied the difference-based ridge-type estimator of parameters in a restricted partial linear model with correlated er-rors, but this paper just focuses on estimating the linear component. Zeng and Liu [32] used a difference-based and ordinary least-square method to obtain the estimator of an unknown parametric component, but this paper ignores the fact that a difference-based estimator may cause greater bias in moderately sized samples than other estimators. In-spired by these papers, we propose a difference-based M-estimator (DM) methods for generalized semiparametric model with NSD errors. The M-estimator is a most famous robust estimator, which was introduced by Huber [33]. In addition, onceβis estimated, we can estimatef(·) by a variety of nonparametric techniques. In this paper, the estimator off(·) is obtained by the wavelet method.

The paper has the following structure. In Sect.2, we present the estimation procedure. In Sect.3, we establish the main results. The proofs of the main results are provided in the

Appendix.

2 Estimation method

2.1 Notation

Throughout the paper, Z is the set of integers, N is the set of natural numbers, R is the set of real numbers. A sequence of random variablesηnis said to be of smaller order in

probability than a sequencedn(denoted byηn=oP(dn)) ifηn/dnconverges to 0 in

prob-ability, andηn=OP(dn) ifηn/dnis bounded in probability. Convergence in distribution is

denoted byHn D

→H. For any arbitrary functionh(·),h(·),h(·), andh(·) are the ﬁrst, second, and third derivatives of h(·), respectively.xis the Euclidean norm of x, and

x=max{k∈Z:k≤x}. LetC0,C1,C2,C3,C4be positive constants, and letβ0be the

true parameter. LetΘ={β:|β–β0| ≤C0}.

2.2 Difference-based M-estimation

Lety˜i=

m

q=0dqyi+q,h˜i(β) =

m

q=0dqh(xTi+qβ),f˜(ti) =

m

q=0dqf(ti+q), ande˜i=

m

q=0dqei+q,

whered0,d1, . . . ,dmsatisfy the conditions m

q=0 dq= 0,

m

q=0

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Theny˜i,h˜i(β),˜f(ti), ande˜ican be seen as themth-order diﬀerences ofyi,h(xTiβ),f(ti), and

ei, respectively. Hence, applying the diﬀerencing procedures, model (1) becomes

˜

yi=h˜i(β) +f˜(ti) +e˜i, 1≤i≤n–m. (3)

From Yatchew [34] we find that the application of differencing procedures in model (1) can remove the nonparametric effect in large samples, so we ignore the presence off˜(·). Thus (3) becomes

˜

yi=h˜i(β) +e˜i 1≤i≤n–m. (4)

Letρbe a convex function. Assume thatρhas a continuous derivativeψand there isa

such thatψ(a) = 0. We can propose the diﬀerence-based M-estimator given by minimizing

Q(β) =

n–m

i=1

ρy˜i–h˜i(β) +a

. (5)

Let ad×1 vectorβˆnbe the minimizer of (5) andβˆn∈Θ. Writeh˜i(β) =

m

q=0dq×

h(xT

i+qβ)xi+q,h˜ik(β) =

m

q=0dqh(xTi+qβ)x(i+q)k, with 1≤k≤d,h˜i(β) =

m

q=0dqh(xTi+qβ)× xi+qxTi+q, andh˜i(β)h˜jT(β) =

m

q=0dqh(xTi+qβ)xi+q

m

q=0dqh(xTj+qβ)xTj+q. Then the estimator

satisﬁes

∂Q(βˆn)

∂β = –

n–m

i=1

ψ(eˆ˜i+a)h˜i(βˆn) = 0 (6)

witheˆ˜i=y˜i–h˜i(βˆn). The convexity ofρguarantees the equivalence of (5) and (6) and the

asymptotic uniqueness of the solution; otherwise, it is unimportant.

We estimate the nonparametric functionf(·) by the wavelet method. The formal deﬁni-tion of the wavelet method is the following.

Suppose that there exist a scaling functionφ(·) in the Schwartz spaceSland a

multires-olution analysis{Vm˜}in the concomitant Hilbert spaceL2(R) with the reproducing kernel

Em˜(t,s) given by

Em˜(t,s) = 2m˜E0

2m˜t, 2m˜s= 2m˜

k∈Z

φ2m˜t–kφ2m˜s–k.

LetAi= [si–1,si] denote intervals that partition [0, 1] withti∈Aifor 1≤i≤n. Then the

estimator of the nonparameterf(t) is given by

ˆ fn(t) =

n

i=1

yi– xTiβˆn Ai

Em˜(t,s)ds. (7)

3 Main results

We now list some conditions used to obtain the main results.

(C1) max1≤i≤nxi=O(1), and the eigenvalues ofn–1

n

i=1xixTi are bounded above and away from zero.

(C2) b,bc–d2_{> 0}_{, where}_b₌_E{_ψ₍_η₎_}_,_c₌_E{_η2_ψ₍_η₎_}_,_d₌_E{_ηψ₍_η₎_}_with_η₌_e_˜

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(C3) Eψ(e˜i+a) = 0.

(C4) The functionρis assumed to be convex, not monotone, and possessing bounded derivatives of suﬃciently high order in a neighborhood of the pointxT

iβ0. In

particular,ψ(t)should be continuous and bounded in a neighborhood ofxT iβ0.

(C5) h(·)is assumed to possess bounded derivatives of suﬃciently high order in a neighborhood of pointxT

iβ0.

(C6) f(·)∈Hα_{(Sobolev space) for some}_α_{> 1/2}_.

(C7) f(·)is a Lipschitz function of orderγ > 0.

(C8) φ(·)belongs toSl, which is a Schwartz space forl≥α, is a Lipschitz function of order 1, and has a compact support, in addition to| ˆφ(ξ) – 1|=O(ξ)asξ→0, whereφˆdenotes the Fourier transform ofφ.

(C9) si,1≤i≤n, satisfymax1≤i≤n(si–si–1) =O(n–1), and2m˜ =O(n1/3).

Remark1 Condition (C1) is often imposed in M-estimation theory of regression models. Condition (C2) is used by Silvapullé [35] for HD estimation. In this paper, this condition is also necessary for M-estimation. Condition (C3) is used by Wu [36] and Zeng and Hu [37] witha= 0. We require this in order that the expectation of (5) reaches its minimum at the true valueβ0. For Condition (C4), higher-order derivatives are technically

conve-nient (Taylor expansions), but their existence is hardly essential for the results to hold; see Huber [1]. Condition (C5) is quite mild and can be easily satisﬁed. Conditions (C6)–(C9) are used by Hu et al. [38].

Remark2 The assumption ofψ(a) = 0 and Condition (C4) are serious restrictions, which shows that the M-estimator in our paper is a particular case of the classical M-estimator. However, in our study, these conditions are necessary.

Theorem 3.1 Let{en,n≥1}be a sequence of NSD random variables with Een= 0,and let

for someδ> 0,

sup n≥1E|en|

2+δ_<_∞_. ₍₈₎

Suppose that

sup j≥1

i:|i–j|≥u

cov(ei,ej)→0 as u→ ∞. (9)

Set˜ei=

m

q=0dqei+q,where{dq, 1≤q≤m}are deﬁned in(2).Let{ci, 1≤i≤n–m}be an

array of constants satisfyingmax1≤i≤n–m|ci|=O(1),and suppose thatψ(a) = 0and

Condi-tions(C3)and(C4)hold.Then

(n–m)–1/2τ–1

n–m

i=1

ciψ(e˜i+a) D

→N(0, 1), (10)

provided that

τ2= lim n→∞(n–m)

–1

_n_–_m

i=1

c2_iVarψ(e˜i+a)

+ 2

n–m

i=1

n–m

j=i+1 cicjCov

ψ(e˜i+a),ψ(e˜j+a)

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Theorem 3.2 Let{en,n≥1}be a sequence of NSD random variables with Een= 0

satisfy-ing conditions(8)and(9).Assume that conditions(C1)–(C5)hold.Then

(n–m)–1/2τ_β–1E

∂2Q(β0) ∂β∂βT

(βˆn–β0)

D

→N(0,Id), (11)

provided that

τ_β2= lim n→∞

1

n–m

_n_–_m

i=1 ˜

h_i(β0)h˜iT(β0)Var

ψ(˜ei+a)

+ 2

n–m

i=1

n–m

j=i+1 ˜

h_i(β0)h˜jT(β0)Cov

ψ(˜ei+a),ψ(e˜j+a)

is a positive deﬁnite matrix,where Idis the identity matrix of order d.

Corollary 3.1 Let h(xT_iβ) = xT_iβ,and let{en,n≥1}be a sequence of NSD random

vari-ables with Een= 0 satisfying conditions(8)and(9).Assume that Conditions(C1)–(C4)

hold.Then

(n–m)–1/2τ_β–1E

∂2_Q₍_β 0) ∂β∂βT

(βˆn–β0)

D

→N(0,Id), (12)

provided that

1

n–m

_n_–_m

i=1 ˜

xix˜Ti Var

ψ(e˜i+a)

+ 2

n–m

i=1

n–m

j=i+1 ˜

xix˜Tj Cov

ψ(˜ei+a),ψ(e˜j+a)

is a positive deﬁnite matrix.

Corollary 3.2 Let{en,n≥1}be a sequence of NSD random variables with Een= 0

sat-isfyingCov_|_i_–_j_|_>_m_¯(ei,ej) = 0 withm¯ <∞.Assume that Condition(C1)–(C5)and(8)hold.

Then

(n–m)–1/2τ_β–1E

∂2Q(β0) ∂β∂βT

(βˆn–β0)

D

→N(0,Id),

provided that

1

n–m

_n_–_m

i=1 ˜

h_i(β0)h˜iT(β0)Var

ψ(˜ei+a)

+ 2

¯ m

k=1

n–m–k

i=1 ˜

h_i₊_k(β0)h˜iT(β0)Cov

ψ(˜ei+k+a),ψ(e˜i+a)

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Corollary 3.3 (Zeng and Liu [32]) Letρ(t) =t2_,_h_(xT

iβ) = xTiβ,and let {en,n≥1} be a

sequence of NSD random variables with Een= 0satisfying conditions(8)and(9).Assume

that conditions(C1)–(C2)hold.Then

(n–m)–1/2τ_β–1

n–m

i=1 ˜

xix˜Ti(βˆn–β0)

D

→(0,Id),

provided that

τ_β2= lim n→∞(n–m)

–1

_n_–_m

i=1 ˜

xix˜Ti Var(˜ei) + 2 n–m

i=1

n–m

j=i+1 ˜

xix˜Tj Cov(e˜i,e˜j)

Corollary 3.4 Letρ(t) =t2_,_h_(xT iβ) =ex

T

iβ_,_{and let}{_e_n_,_n≥₁}_{be a sequence of NSD random}

variables with Een= 0satisfying conditions(8)and(9).Assume that conditions(C1)–(C2)

hold.Then

(n–m)–12_τ–1 β

n–m

i=1

_m

q=0 dqe

xT i+qβ0_x

i+q

2

(βˆn–β0)

D

→(0,Id),

provided thatτ2

β=limn→∞(n–m)–1Var(ni=1–me˜i

m

q=0dqex

T i+qβ0_x

i+q)is a positive deﬁnite

matrix.

Theorem 3.3 Under the conditions of Theorem3.2,assume that Conditions(C6)–(C9)

hold.Then

sup

0≤t≤1

fˆn(t) –f(t)=OP

n–γ₊_O

P(τm˜) +OP

n–1/3_M

n

as n→ ∞, (13)

where Mn→ ∞in arbitrary slowly rate,andτm˜ = 2–m˜(α–1/2)if1/2 <α< 3/2,τm˜ =

√ ˜ m2–m˜

ifα= 3/2,andτm_˜ = 2–m˜ ifα> 3/2.

Appendix

A.1 Lemmas

In this section, we present the proofs of the main results. We ﬁrst need some lemmas.

Lemma 1 Under Conditions(C1), (C4),and(C5),suppose that eisatisﬁes(8).Then

∂2_Q₍_β 0) ∂β∂βT –E

∂2_Q₍_β 0) ∂β∂βT =OP

(n–m)12 ₍₁₄₎

and

supn–3/2Rnl(β˜)=sup

n–3/2 ∂

3

∂β∂βT_∂β l

Q(β˜)→0 as n→ ∞, (15)

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Proof We have

∂2

∂β∂βTQ(β0) –E

∂2

∂β∂βTQ(β0)

=

_n_–_m

i=1

ψ(e˜i+a)h˜i(β0)h˜iT(β0) +

n–m

i=1

ψ(e˜i+a)h˜i(β0)

–

E

_n_–_m

i=1

ψ(˜ei+a)h˜i(β0)h˜iT(β0)

+E

_n_–_m

i=1

ψ(˜ei+a)h˜i(β0)

=

_n_–_m

i=1

ψ(e˜i+a)h˜i(β0)h˜iT(β0) –E

_n_–_m

i=1

ψ(e˜i+a)h˜i(β0)h˜iT(β0)

+

_n_–_m

i=1

ψ(e˜i+a)h˜i(β0) –

n–m

i=1

Eψ(˜ei+a)h˜i(β0)

:=I1+I2.

From (8) we have

sup i≥1

j:|i–j|≥u

covψ(ei+a),ψ(ej+a)→0 asu→ ∞.

Therefore, for a ﬁxed smallε, there exists a positive integerδ=δεsuch that

sup i≥1

j:|i–j|≥δ

covψ(ei+a),ψ(ej+a)<ε

for 1≤k1,k2,l1,l2≤d, and thus

n–m

i=1

j:|i–j|≥1 ˜

h_ik₁(β0)h˜il1(β0)h˜

jk2(β0)h˜

jl2(β0) ×Eψ(e˜i+a) –Eψ(e˜i+a)

ψ(e˜j+a) –Eψ(e˜j+a)

=

j:1≤|i–j|<δ

n–m

i=1 ˜

jk2(β0)h˜

+

n–m

i=1

j:|i–j|≥δ ˜

jk2(β0)h˜

≤2δ max

1≤i≤n–m

h˜_i(β0)2

× max

1≤i,j≤n–mE

ψ(e˜i+a) –Eψ(e˜i+a)

n–m

i=1

h˜_i(β0)2

+ max

1≤i≤n–mh˜ i(β0)

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×

n–m

i=1

i:|i–j|≥δ

Eψ(˜ei+a) –Eψ(e˜i+a)

ψ(˜ej+a) –Eψ(e˜j+a)

≤2δ max

1≤i≤n–m

h˜_i(β0)2

× max

1≤i,j≤n–mE

n–m

i=1

h˜_i(β0) 2

+ max

1≤i≤n–m

h˜_i(β0) 4

(n–m)ε.

By Condition (C5) we have that max1≤i≤n{qm=1|dqh(xTi+qβ˜)|}, max1≤i≤n{mq=1|dq ×

h(xT

i+qβ˜)|}, andmax1≤i≤n{

m

q=1|dqh(xTi+qβ˜)|}are bounded by some constantC1. Then

by (C4) it follows that, for same constantM> 0,

(n–m)–2E

_n_–_m

i=1

ψ(e˜i+a)h˜ik(β0)h˜il(β0) –E

_n_–_m

i=1

ψ(˜ei+a)h˜ik (β0)h˜il(β0) 2

= (n–m)–2

_n_–_m

i=1 ˜

h_ik2(β0)h˜il2(β0)E

2

+

n–m

i=1

j:|j–i|≥1 ˜

jk2(β0)h˜

jl2(β0)

×Eψ(e˜i+a) –Eψ(e˜i+a)

≤(n–m)–1 max

1≤i≤n–m

h˜_i(β0) 2

(n–m)–1

n–m

i=1

h˜_i(β0) 2

M

+ 2δ(n–m)–1 max

1≤i≤n–mh˜ i(β0)

2

(n–m)–1

n–m

i=1

h˜_i(β0) 2

M

+ (n–m)–1 max

1≤i≤n–m

h˜_i(β0) 4

ε

≤(2δ+ 1)(n–m)–1 max

1≤i≤n–m

h˜_i(β0) 2

(n–m)–1

n–m

i=1

h˜_i(β0) 2

M

+ (n–m)–1 max

1≤i≤n–m

h˜_i(β0)4ε

= (2δ+ 1)(n–m)–1 _max 1≤i≤n–m

m q=0 dqh

x_iT₊_qβ0

xi+q

2

×(n–m)–1

n–m

i=1 m q=0 dqh

xT_i₊_qβ0

xi+q

2 M

+ (n–m)–1 max

1≤i≤n–m

m q=0 dqh

xT_i₊_qβ0

xi+q

(9)

≤(n–m)–1

(2δ+ 1)C2₁ max

1≤i≤n–m m

q=0

xi+q2(n–m)–1C12

n–m

i=1

m

q=0

xi+q2M

+C4₁

max

1≤i≤n–m m

q=0 xi+q2

2 ε ,

and from (C1) it follows that

(n–m)–1Var(I1) =O(1). (16)

By the Chebyshev inequality it suﬃces to verify thatI1=OP((n–m)

1

2_{). In the same way,}

we easily obtain thatI2=OP((n–m)

1

2_{). Consequently,} ∂2

∂β∂βTQ(β0) –E

∂2

∂β∂βTQ(β0) =OP

(n–m)12_. ₍₁₇₎

Note that, for 1≤l≤d,

∂3 ∂β∂βT_∂β

l

Q(β˜)

=

n–m

i=1 3ψ ˜ yi–

m

q=0

dqh(xi+qβ˜) +a

_m

q=0

dqh(xi+qβ˜) m

q=0

dqh(xi+qβ˜)x(i+q)lxi+qxTi+q

–ψ

˜ yi–

m

q=0

dqh(xi+qβ˜) +a

_m

q=0

dqh(xi+qβ˜)

3

x(i+q)lxi+qxTi+q

–ψ

˜ yi–

m

q=0

dqh(xi+qβ˜) +a

_m

q=0

dqh(xi+qβ˜)x(i+q)lxi+qxTi+q .

By Conditions (C1), (C4), and (C5), for 1≤k,l,s≤dand some constantM> 0, we have

sup n–3/2

n–m

i=1 3ψ ˜ yi–

m

q=0

dqh(xi+qβ˜) +a

×

m

q=0

dqh(xi+qβ˜) m

q=0

dqh(xi+qβ˜)x(i+q)lx(i+q)kx(i+q)s

–ψ

˜ yi–

m

q=0

dqh(xi+qβ˜) +a

_m

q=0

dqh(xi+qβ˜)

3

x(i+q)lx(i+q)kx(i+q)s

–ψ

˜ yi–

m

q=0

dqh(xi+qβ˜) +a

_m

q=0

dqh(xi+qβ˜)x(i+q)lx(i+q)kx(i+q)s

≤(n–m)–1/2MC2₁+C₁3+C1

max

1≤i≤n–m m

q=0

xi+q(n–m)–1 n–m

i=1

m

q=0 xi+q2

→0, n→ ∞.

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Lemma 2 If (C1)–(C5)hold,then √

n(βˆn–β0) =Op(1). (18)

Proof We can prove Lemma2by an argument similar to Lemma 4 of Silvapullé [35], so

we omit the details.

Lemma 3(Zhou and You [39]) If Condition(C8)holds,then

(a1) |E0(t,s)| ≤₍₁₊C_|_t_–k_s_|₎k,|Em˜(t,s)| ≤

2m˜_C

(1+2m˜_|_t_–_s_|₎k (wherek∈N,andC=C(k)is a constant

depending onkonly); (a2) sup₀_≤_s_≤₁|Em˜(t,s)|=O(2m˜); (a3) sup_t₀1|Em_˜(t,s)|ds≤C2;

(a4) ₀1Em˜(t,s)ds→1,n→ ∞.

A.2 Proof of Theorem3.1

By Condition (8) we have

sup n≥1

Ee2_n<∞ and lim x→∞sup_n_≥₁Ee

2

nI

|en|>x

= 0,

from which it follows that

C3:=sup

n>m

(n–m)–1

n–m

i=1

m

q=0

Var(dqei+q) <∞,

and for allε> 0,

(n–m)–1

n–m

i=1

m

q=0

E(dqei+q)2I

|dqei+q| ≥

√

n–mε→0 asn→ ∞.

Then we can ﬁnd a positive number sequence{εn,n≥1}withεn→0 such that

(n–m)–1

n–m

i=1

m

q=0

E(dqei+q)2I

|dqei+q| ≥

√ n–mεn

→0 asn→ ∞.

Now we deﬁne the integers:m0= 0 and forj= 0, 1, 2, . . . ,

m2j+1=min

m:m≥m2j, (n–m)–1 m

i=m2j+1

m

q=0

Var(dqei+q) >√εn ,

m2j+2=m2j+1+

1

εn

+m.

Denote

(11)

wherel=l(n) is the number of blocks of indicesIj. Then

l√εn≤(n–m)–1 l

j=1

i∈Ij

m

q=0

Var(dqei+q)≤(n–m)–1 n–m

i=1

m

q=0

E(dqei+q)2≤C3, (19)

and hence we havel≤C3/√εn. If the remainder term is not zero, then as the construction

ends, we put all the remainder terms into a block denoted byJl. Hence, by the Lagrange

mean value theorem, 1

√ n–m

n–m

i=1

ciψ(˜ei+a)

=√ 1

n–m

i=1

ciψ(e˜i+a) –

1

√ n–m

n–m

i=1 ciψ(a)

=√ 1

n–m

i=1

ciψ(ξi)e˜i, (20)

whereξi=t˜ei+afor somet∈[0, 1].

Moreover, settingai=τ–1ciψ(ξi), we have

1

√ n–m

n–m

i=1

ciψ(˜ei+a)

=√ 1

n–m

i=1 ai˜ei

=√ 1

n–m

l

j=1

i∈Ij

aie˜i+

1

√ n–m

l

j=1

i∈Jj

aie˜i

:=I+J.

By the argument in the proof of Theorem 4.1 in Zeng and Liu [32] we have

(n–m)–1/2

n–m

i=1 ai˜ei

D

→N(0, 1), (21)

which implies

(n–m)–1/2τ–1

n–m

i=1

ciψ(e˜i+a) D

→N(0, 1). (22)

The proof is completed.

Now we will use Theorem3.1to prove Theorem3.2. Expanding _∂β∂Q(βˆn) aboutβ0, we

have

∂

∂βQ(βˆn) = ∂

∂βQ(β0) + ∂2

∂β∂βTQ(β0)(βˆn–β0) +

1 2

Rnl(β˜,βˆn,β0)

(12)

whereβ˜=sβˆn+ (1 –s)β0for somes∈[0, 1], and

Rnl(β˜,βˆn,β0)

1≤l≤d

=(βˆn–β0)TRnl(β˜)(βˆn–β0), (βˆn–β0)TRn2(β˜)(βˆn–β0), . . . ,

(βˆn–β0)TRnd(β˜)(βˆn–β0)

T

.

From (6) we have

∂2

∂β∂βTQ(β0)(βˆn–β0) = –

∂

∂βQ(β0) –

1 2

Rnl(β˜,βˆn,β0)

1≤l≤d (23)

and, by Lemma1and Lemma2,

(n–m)–12

E ∂ 2

∂βT_∂βQ(β0) +OP

(n–m)12₍_βˆ_n_–_β₀₎

= –(n–m)–12

∂

∂βQ(β0) +

1 2

Rnl(β˜,βˆn,β0)

1≤l≤d

= –(n–m)–12 ∂

∂βQ(β0) +oP(1)

= (n–m)–12

_n_–_m

i=1

ψ(e˜i+a)h˜i(β0) +oP(1).

We now show that 1

√ n–mτβ

n–m

i=1

ψ(˜ei+a)h˜i(β0)

D

→N(0,Id). (24)

Letube a 1×dsuch thatu= 1. By the Cramér–Wold theorem it suﬃces to verify that 1

√ n–mτ

n–m

i=1

ψ(e˜i+a)uh˜i(β0)

D

→N(0, 1), (25)

whereτ2₌_lim

n→∞Var((n–m)–1/2ni=1–mψ(e˜i+a)uh˜i(β0)); by the deﬁnition ofτβ2,τ2> 0.

By Theorem3.1, (25) follows fromuh˜_i(β0) =O(1). The proof is completed.

By (7) we have

ˆ

fn(t) –f(t) = n

i=1

yi–h

xT_iβˆn Ai

Em˜(t,s)ds–f(t)

=

n

i=1

hxT_iβ+f(ti) +ei–h

xT_iβˆn Ai

Em˜(t,s)ds–f(t)

=

n

i=1

hxT_iβ–hxT_i βˆn Ai

(13)

+

_n

i=1 f(ti)

Ai

Em˜(t,s)ds–f(t) + n

i=1 ei

Ai

Em˜(t,s)ds

:=I1+I2+I3.

By the argument in the proof of Theorem 3.2 in Hu [30] we have

I2=O

n–γ+O(τm˜) (26)

and

I3=OP

n–13_M_n_. ₍₂₇₎

By Lemma3, (C1), and (C5) we assume that

max

1≤i≤n

h(ξi)xTisup t

n

i=1

Ai

Em˜(t,s)ds≤C4,

whereξi=rxiTβ+ (1 –r)xTi βˆn,r∈[0, 1]. It follows that

I3≤sup

t

n

i=1

hxT_iβ–hxT_i βˆn Ai

Em˜(t,s)ds

≤sup t

n

i=1

h(ξi)xTi (β–βˆn) Ai

Em˜(t,s)ds

≤max

1≤i≤n

h(ξi)xiTsup t

n

i=1

Ai

Em˜(t,s)dsβ–βˆn

≤C4β–βˆn.

By Lemma2we get

I3=Op

n–1/2. (28)

Then Theorem3.2follows from (26), (27), and (28).

Acknowledgements

This authors would like to thank a referee and an Associate Editor for their comments and suggestions. Funding

The research is supported Support by National Natural Science Foundation of China [grant number 71471075]. Availability of data and materials

Not application. Competing interests

The authors declare that they have no competing interests. Authors’ contributions

(14)

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Received: 6 December 2018 Accepted: 4 March 2019

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