Asymptotic normality of Huber Dutter estimators in a linear EV model with AR(1) processes

R E S E A R C H

Open Access

Asymptotic normality of Huber-Dutter

estimators in a linear EV model with AR(1)

processes

Hongchang Hu

_{and Xiong Pan}

*_{Correspondence:}

[email protected] 1_{School of Mathematics and}

Statistics, Hubei Normal University, Huangshi, 435002, China Full list of author information is available at the end of the article

Abstract

The paper studies a linear errors-in-variables model with first order autoregressive processes. The Huber-Dutter (HD) estimators of unknown parameters are given, and the asymptotic normality of the HD estimators is investigated. Finally, a simple example is given to illustrate our estimation method.

MSC: 60F05; 60G10; 62F35; 62M10; 60G42

Keywords: linear errors-in-variables model; Huber-Dutter estimator; autoregressive processes; asymptotic normality

1 Introduction

Consider the following linear errors-in-variables (EV) model: ⎧

⎨ ⎩

yt=xTtβ+εt,

Xt=xt+ζt, t= , , . . . ,n,

(.)

where the superscriptT denotes the transpose throughout the paper,{yt,t= , , . . . ,n} are scalar response variables,{Xt= (Xt,Xt, . . . ,Xtd)T,t= , , . . . ,n}and{xt= (xt,xt, . . . , xtd)T_{,t= , , . . . ,n}are observable and unobservable random variables, respectively,β=} (β, . . . ,βd)T is a vector ofd unknown parameters,{ζt}are independent and identically distributed (i.i.d.) measurement errors withEζt =  andVar(ζt) =σζId,{ζt}and{εt}are independent,{(εt,ζtT)T}and{xt}are independent, and{εt,t= , , . . . ,n}are the first order autoregressive (AR()) processes

ε=η, εt=aεt–+ηt, t= , , . . . ,n, (.)

where{ηt,t= , , . . . ,n}are i.i.d. random errors with zero mean and finite varianceσ> , and –∞<a<∞is a one-dimensional unknown parameter. A common assumption is that the ratio of the error variancesλ=σ

σ_ζ is known. This is assumed throughout this paper and all variables are assumed scaled so thatλ= .

The linear errors-in-variables model (.) with AR() processes (.) includes three im-portant special models: () an ordinary linear regression model with AR() processes (whenζt = , seee.g., Hu [], Maller [], Pere [], and Fuller []); () an ordinary linear

errors-in-variables model (whena= , seee.g., Miao and Liu [], Miaoet al.[, ], Liu and Chen [], Cui [], Cui and Chen [], Cheng and Van Ness []); () autoregressive processes (whenβ= , seee.g., Hamilton [], Brockwell and Davis [], and Fuller []). The independence assumption for the errors is not always valid in applications, especially for sequentially collected economic data, which often exhibit evident dependence in the errors. Recently, linear errors-in-variables models with serially correlated errors have at-tracted increasing attention from statisticians; see, for example, Baran [], Fanet al.[], Miaoet al.[], among others.

It is well known that in the EV model, the ordinary least-squares (OLS) estimators are biased and inconsistent and that orthogonal regression is better in that case Fuller []. However, both methods are very sensitive to outliers in the data and some robust alter-natives have been proposed. Brown [] and Ketellapper and Ronner [] applied robust ordinary regression techniques in the EV model. Zamar [] proposed robust orthogonal regressionM-estimators and showed that it outperformed the robust ordinary regression. Cheng and Van Ness [] generalized the proposal of Zamar by defining robust orthogonal GeneralizedM-estimators which had bounded influence function in the simple case. He and Liang [] proposed a regression quantile approach in the EV model which allowed for heavier tailed errors distribution than the gaussian distribution. Fekri and Ruiz-Gazen [] proposed robust weighted orthogonal regression.

Over the last  years, several estimators in linear regression models that posses robust-ness have been proposed, such as Wu [], Silvapullé [], Hampelet al.[], Huber and Ronchetti [], Li [], Babu [], Cheng and Van Ness [], Salibian-Barrera and Zamar [], Wu and Wang [], Zhou and Wu [], and so on. It is well known that HD esti-mate approach is one of important robust techniques. Recently, some authors applied HD estimate approach to regression models. For example, Silvapullé [] established asymp-totic normality of HD estimators for the linear regression model with i.i.d. errors. Hu [] investigated asymptotic normality of HD estimators for the linear regression model with AR() errors. Tonget al.[] considered consistency and normality of HD estimators for the partial linear regression model. However, nobody used the HD method to investigate the models (.)-(.).

The paper discusses the models (.)-(.) with a robust approach, which has been sug-gested by Huber and Dutter. We extend some results of Hu [], Silvapullé [],etc.to the EV regression model with AR() errors. The organization of the paper is as follows. In Section  estimators ofβ,aandσare given by HD method. Under general conditions, the asymptotic normality of the HD estimators is investigated in Section . The theoret-ical proofs of main results are presented in Section , a simple example is given in Sec-tion .

2 Estimation method

By (.), we have

εt= t

j=

thusεtis measurable with respect to theσ-fieldHgenerated byη,η, . . . ,ηt,Eεt=  and

Var(εt) =

σ(–_–a_at), if|a| = ,

σt, if|a|= . (.)

Furthermore,

n(a,σ) = n

t= Eε_t–=

σ(an–a_(–+(n_a–)(–₎ a)), if|a| = ,



σn(n– ), if|a|= ,

= ⎧ ⎪ ⎨ ⎪ ⎩

O(n), if|a|< ,

O(an_{), if|a|> ,}

O(n), if|a|= .

(.)

Lety= ,x= . From (.),

εt=yt–

X_tT–ζ_tT β.

By the above equation and (.), we obtain

ηt=εt–aεt–

=yt–XtTβ+ζtTβ–a

yt––XTt–β+ζtT–β . (.)

Thus we could consider HD estimators by minimizing

QβT,σ,a = n

t= ρ

yt–XtTβ+ζtTβ–a(yt––XtT–β+ζtT–β) σ

σ+Anσ, (.)

whereρ≥ is convex,ρ() = ,ρ_|t|(t)→kas|t| → ∞for somek> , and{An}is a suitably chosen sequence of constants.

Remark  Since

Var

yt–XtTβ–a(yt––XTt–β)

 + ( +a_)βT_β

=σ,

using the method of HD, we also obtain HD estimators by minimizing

˜

QβT,σ,a = n

t= ρ

yt–X√tTβ–a(yt––XTt–β)

+(+a_)βT_β

σ

σ+Anσ.

We will investigate its estimators in the future because there are some difficulties. For example, there is very sophisticated calculation, and it is difficult to investigate the asymp-totic properties of these unknown parametric estimators because

yt–XtTβ–a

yt––XtT–β =ηt–

ζ_tT–aζ_tT_– β

Let us introduce some notation: ×(d+ ) vectorθ = (βT_{,σ,a) and its estimatorθ}ˆ n= (βˆ_nT,σˆn,aˆn). For an arbitrary functionf,fandfare the first and second derivatives of

f, respectively.xis the Euclidean norm ofxandε= . (M)ijand (U)iare the (i,j)th component of matrixMand theith component of vectorU, respectively.

The HD estimators forθ are obtained by solving the estimating equations by equating to  the derivatives

∂Q ∂βT = –

n

t= ψ

yt–XtTβ+ζtTβ–a(yt––XtT–β+ζtT–β) σ

Xt–aXt–– (ζt–aζt–)

= – n

t= ψ

εt–aεt– σ

Xt–aXt–– (ζt–aζt–) , (.)

∂Q ∂σ = –

n

t=

ψ

εt–aεt– σ

εt–aεt–

σ –ρ

εt–aεt– σ

+An

= –

_n

t= χ

εt–aεt– σ

–An

, (.)

and

∂Q ∂a = –

n

t= ψ

εt–aεt– σ

εt–, (.)

whereψ=ρandχ(u) =uψ(u) –ρ(u) =_ux dψ(x).

The corresponding estimators, if they exist (see Proposition .), will satisfy

n

t= ψ

_ˆ

εt–aˆnεˆt– ˆ σn

Xt–aˆnXt–– (ζt–aˆnζt–) = , (.)

n

t= χ

ˆ

εt–aˆnεˆt– ˆ σn

=An, (.)

and

n

t= ψ

_ˆ

εt–aˆnεˆt– ˆ σn

ˆ

εt–=  (.)

withεˆt=yt–xTtβˆn.

Although{ζt}are unknown in (.)-(.), but we easily estimate it by the method of Fuller [] in practice.

In what follows, it will be assumed thatn≥d+  andAn> . Without loss of generality, we may assumek= . (Its definition has been given between (.) and Remark .) Therefore

ψ is bounded and increases from – to +. It will also be assumed thatχis bounded.

Case . Leta= ,ζt= . The estimating equations (.)-(.) may be written as

n

t= ψ

yt–XtTβˆn

ˆ σn

Xt= , n

t= χ

yt–XtTβˆn

ˆ σn

=An, (.)

which are the same as Silvapullé’s []. Case . Ifρ(u) =|u|q_{( <q= ),A}

n= , andζt= , then the above equations (.)-(.) may be rewritten as

n

t=

εˆt–aˆnεˆt– ˆ σn

q–εˆt–aˆnεˆt– ˆ σn

(Xt–aˆnXt–) = , (.)

n

t=

εˆt–aˆnεˆt– ˆ σn

q= , (.)

and

n

t=

εˆt–aˆnεˆt– ˆ σn

q–εˆt–aˆnεˆt– ˆ σn ˆ

εt–=  (.)

withεˆt=yt–XtTβˆn.

Leta=  andρ(u) =|u|q_{( <q= ),A}

n= . We rewrite (.)-(.) by

n

t=

yt–XtTβˆn

ˆ σn

q–yt–XtTβˆn

ˆ σn

= , n

t=

yt–XTt βˆn

ˆ σn

q= . (.)

Furthermore, ifσ is a constant, then the estimator ofβsatisfies the following equation:

n

t=

_yt–XT t βˆn

q–

yt–XtTβˆn = , (.)

which isLqor the maximum likelihood estimate equation for the parameterβin a linear regression model withq-norm distribution errors. Many authors have investigated (.), such as Arcones [] and Zeckhauser and Thompson [], Ronner [, ], and so on.

Case . Letζt= . The estimating equations (.)-(.) may be written as

n

t= ψ

_ˆ

εt–aˆnεˆt– ˆ σn

(Xt–aˆnXt–) = , (.)

n

t= χ

_ˆ

εt–aˆnεˆt– ˆ σn

=An, (.)

and

n

t= ψ

ˆ

εt–aˆnεˆt– ˆ σn

ˆ

εt–= , (.)

From Proposition  in Silvapullé [] and pp. in Huber and Ronchetti [], the exis-tence results of the HD estimators may be given by the following.

Proposition . Suppose thatρis continuous and,for some A> ,ν<  –AV–_,where ν is the largest jump in the error distribution, V =χ(–∞)∧χ(∞), then the equation E{ψ(ε–_σμ),χ(ε–_σμ) –A}=  has a solution(μ(A),σ(A)) withσ(A) > .Especially, when A=limn→∞n–An,where Anis defined in(.),we denote it by(μ,σ)withσ> .

3 Main results

To obtain our results, we start with some assumptions.

(A) maxn

t=Xt<∞.

(A) lim_n→∞n(n–A_n–A) = for some <n–A_n,A<min{χ(∞),χ(–∞)}.

(A) The functionψis continuous. (A) Eψ(ηt

σ) = for anyσ> .b=Eψ( ηt

σ) > ,c=E{ψ( ηt

σ) ηt

σ},r=E{ψ( ηt

σ)ηt},br≥c,

Var(ψ(ηt

σ)) <∞,Var(ψ( ηt

σ)ηt) <∞,Var(ψ( ηt σ)η



t) <∞,Eηt <∞and

E(ψ(ηt_σ))_<∞.

(A) For anya∈(–∞,∞),Xn(a) =

n

t=(Xt–aXt–)(Xt–aXt–)Tis positive definite for sufficiently largen.

Remark  The condition (A) is often imposed in the estimation theory of regression models. The condition (A) is used by Tonget al.[]. In addition, by (A) and (A), we can obtain conditionsn–_maxn

t=Xt→ andlimn→∞(n–An–A) = , which are used by Silvapullé []. The conditions (A) and (A) exceptEη

t <∞andE(ψ( ηt

σ))

_<∞are

used by Silvapullé []. The condition (A) is used by Maller [], Hu [],etc.Therefore, our conditions are quite mild and can easily be satisfied.

For ease of exposition, we shall introduce the following notations which will be used later in the paper. Define

=

_˜

θ:| ˜θ–θ| ≤Cn–_,

=

_˜

θ:| ˜β–β| ≤Cn–_,| ˜_σ–σ| ≤_Cn–_,|˜_a–a| ≤_Cn–_,

=

_˜

θ:| ˜β–β| ≤Cn–_,| ˜_σ–σ| ≤_Cn–_,|˜_a–a| ≤_Ca–n_,

Sn(θ) =∂Q

∂θ =

∂Q ∂βT,

∂Q ∂σ,

∂Q ∂a

=Sn(θ)_β,σ,∂Q

∂a

, (.)

and

Fn(θ) = ∂

_Q

∂θ ∂θT = ⎛ ⎜ ⎜ ⎝

∂Q ∂βT_∂β

∂Q ∂βT_∂σ

∂Q ∂βT_∂a

∗ ∂Q

∂σ

∂Q ∂σ ∂a

∗ ∗ ∂Q

∂a

⎞ ⎟ ⎟

where the∗indicates that the elements are filled in by symmetry, and

∂_Q ∂βT_∂β =



σ

n

t= ψ

εt–aεt– σ

·Xt–aXt–– (ζt–aζt–) Xt–aXt–– (ζt–aζt–) T

=Xn(a,ω), (.)

∂Q ∂βT_∂σ =



σ

n

t= ψ

εt–aεt– σ

(εt–aεt–)

Xt–aXt–– (ζt–aζt–)

T

, (.)

∂_Q ∂βT_∂a=

n

t= ψ

εt–aεt– σ

εt– σ

Xt–aXt–– (ζt–aζt–) T

+ n

t= ψ

εt–aεt– σ

(Xt––ζt–)T, (.)

∂Q ∂σ =



σ

n

t= ψ

εt–aεt– σ

(εt–aεt–), (.)

∂_Q ∂σ ∂a=



σ

n

t= ψ

εt–aεt– σ

(εt–aεt–)εt–, (.)

and

∂Q ∂a =



σ

n

t= ψ

εt–aεt– σ

ε_t–. (.)

Theorem . Suppose that conditions(A)-(A)hold.Then,as n→ ∞:

() For|a|< andθ∈,we have

(θˆn–θ)Dn(θ)Var–



_Sn(θ) →_{DN(,Id+), (.)}

whereDn(θ) =E(Fn(θ))andVar(Sn(θ))defined in Lemma.. () For|a|= andθ∈, (.)holds.

() For|a|> andθ∈, (.)holds.

From the above theorem, we may obtain the following corollaries. Here we omit their proofs.

Corollary . Ifβ= and conditions(A)-(A)hold,then

n_(σˆ

n–σ)→DN

,σ



r Var

χ

η σ

(.)

and

 

n(a,σ)(aˆn–a)→DN

,σ



bEψ 

η σ

Corollary . If a= and conditions(A)-(A)hold,then

–Sn(θ)

β,σ

Var–_S

n(θ)

β,σ →DN(,Id+), n→ ∞. (.)

Remark  Corollary . is similar to Theorem  of Silvapullé [].

Corollary . Letσbe a constant.If conditions(A)-(A)hold,then

(βˆn–β)

Xn(a) +n +a σ_ζId

  →

DN

,σ



bEψ 

η σ

Id

(.)

and

 

n(a,σ)(aˆn–a)→DN

,σ



bEψ 

η σ

, n→ ∞. (.)

Corollary . Letζt= .If conditions(A)-(A)hold,then Theorem.holds.

Remark  For|a|< , Corollary . is the same as Theorem . of Hu []. Therefore, we extend the corresponding results of Hu [] to linear EV models.

If we do not consider the dependency on the parametersβˆnandσˆn, then we will obtain Theorem ..

Theorem . Let

[θ]β,σ=

βT,σ , Dn(θ) =diagDn(θ)_β,σ, b

σn(a,σ)

.

Suppose that conditions(A)-(A)hold.Then () for anya∈(–∞,∞),

n–_[θˆ_{n–θ]β,σDn(θ)}

β,σ →DN(,), n→ ∞, (.)

where

=diag

–_n(a,σ)Xn(a) +n

 +a σ_ζId E

ψ

η σ

,Var

χ

η σ

;

() for anya∈(–∞,∞),

 

n(a,σ)(aˆn–a)→DN

,σ



bEψ 

η σ

. (.)

4 Proofs of main results

Lemma . If(A)and(A)hold,then the matrix Dn(θ)is positive definite with E(Sn(θ)) = for sufficiently large n and

VarSn(θ)

= ⎛ ⎜ ⎝

Eψ₍η

σ)(Xn(a) +n( +a

_)σ

ζId) Cov(_∂β∂QT, ∂Q

∂σ) 

∗ nVar(χ(η

σ )) 

  Eψ(η

σ)n(a,σ) ⎞ ⎟ ⎠

=diagVarSn(θ) _β,σ,Eψ

η σ

n(a,σ)

, (.)

where

Cov

∂Q ∂βT,

∂Q ∂σ

=E

ψ

η σ

χ

η σ

n

t=

(Xt–aXt–)T.

Furthermore,Var(Sn(θ))is a positive definite matrix.

Proof Note thatEψ(ηt

σ) = ,E(εt) = , andεt,ζt, andηt+are independent. By (.)-(.), we easily obtain

Dn(θ) = ⎛ ⎜ ⎝ b σXn(a) +

nb σ ( +a

_)σ

ζId _σc

n

t=(Xt–aXt–) 

∗ nr

σ 

  b

σn(a,σ) ⎞ ⎟

⎠. (.)

It is easy to show that

Dn(θ) =

_σbXn(a) +nb

σ

 +a σ_ζId > ,

Dn(θ) =

b σXn(a) +

nb σ

 +a _σ

ζId σc

n

t=(Xt–aXt–)

∗ nr

σ

=b

σXn(a) + nb

σ

 +a σ_ζId ·nr

σ – c bσ

n

t=

(Xt–aXt–)T

·Xn(a) +n

 +a σ_ζId

–n

t=

(Xt–aXt–)

> ,

(.)

and

_Dn(θ)> .

Thus the matrixDn(θ) is positive definite. By (.), we have

E

∂Q ∂βT

= –

n

t= Eψ

ηt

σ

By (.) and Proposition ., we have

E

∂Q ∂σ

= –

n

t= Eχ

ηt

σ

+An→. (.)

Note thatεt–andηtare independent; by (.) andE(εt) = , we have

E

∂Q ∂a

= –

n

t= E

ψ

ηt

σ

εt–

= –

n

t= E

ψ

ηt

σ

E(εt–) = . (.)

Hence, from (.)-(.),

ESn(θ) =

, – n

t= Eχ

ηt

σ

+An, 

→.

By (.), we have

Var

∂Q ∂βT

=Var

ψ

ηt

σ

n

t=

(Xt–aXt–)(Xt–aXt–)T+

 +a σ_ζId

=E

ψ

η σ



Xn(a) +n +a σ_ζId . (.)

By (.), we have

Var

∂Q ∂σ

=

n

t= Var

χ

ηt

σ

=nVar

χ

η σ

. (.)

Note that{ψ(ηt_σ)εt–,Ht}is a martingale difference sequence with

Var

ψ

ηt

σ

εt–

=E

ψ

ηt

σ



Eε_t–,

so we have

Var

∂Q ∂a

=

n

t= Var

ψ

ηt

σ

εt–

= n

t= E

ψ

ηt

σ



Eε_t_–=E

ψ

η σ



n(a,σ). (.)

By (.) and (.), we have

Cov

∂Q ∂βT,

∂Q ∂σ

=E

_n

t= ψ

ηt

σ

Xt–aXt–– (ζt–aζt–) T

·

n

t=

χ

ηt

σ

–Eχ

ηt

σ

= n t= E ψ ηt σ χ ηt σ –Eχ ηt σ

(Xt–aXt–)T

=E ψ η σ χ η σ n t=

(Xt–aXt–)T. (.)

By (.), (.), and noting thatζt,εt–, andηtare independent, we have

Cov

∂Q ∂βT,

∂Q ∂a =E ∂Q ∂βT,

∂Q ∂a =E _n t= ψ ηt σ

Xt–aXt–– (ζt–aζt–)

Tn

t= ψ ηt σ

εt–

= n t= E ψ ηt σ

εt–

Xt–aXt–– (ζt–aζt–)

T

+  n

t>k

E ψ ηt σ ψ ηk σ

εk–

= n t= Eψ ηt σ

Eεt–(Xt–aXt–)T

+  n

t>k

E ψ ηt σ ψ ηk σ

Eεk–

= . (.)

By (.) and (.), we have

Cov ∂Q ∂a, ∂Q ∂σ =E _n t= ψ ηt σ

εt–

n t= χ ηt σ –Eχ ηt σ = n t= E ψ ηt σ χ ηt σ –Eχ ηt σ

Eεt–

+  n

t>k

E χ ηt σ –Eχ ηt σ ψ ηk σ

Eεk–

= . (.)

Hence, (.) follows immediately from (.)-(.).

Similarly to the proof of Dn(θ), we easily prove that the matrixVar(Sn(θ)) is positive definite. Thus, we complete the proof of Lemma ..

Lemma . Assume that(A)and(A)hold.Then:

() for|a|< ,we have

Fn(θ) –Dn(θ) =Op

() for|a|= ,we have

Fn(θ) –Dn(θ) = ⎛ ⎜ ⎝

Op(n) O_p(n) Op(n)

∗ Op(n



_{) Op(n}₎

∗ ∗ Op(n

 )

⎞ ⎟

⎠, (.)

where the∗indicates that the elements are filled in by symmetry; () for|a|> ,we have

Fn(θ) –Dn(θ) = ⎛ ⎜ ⎝

Op(n_{) Op(n}_{) Op(a}n₎

∗ Op(n



) _Op(_n)

∗ ∗ Op(an₎

⎞ ⎟

⎠. (.)

Proof By (.) and (.), we obtain

n–Xn(a,ω) –bXn(a) –nb +a σ_ζId

= 

nσ

n

t=

ψ

εt–aεt– σ

–b

(Xt–aXt–)(Xt–aXt–)T

+ 

nσ

n

t=

ψ

εt–aεt– σ

ζtζtT–bσζId

+ a



nσ

n

t=

ψ

εt–aεt– σ

ζt–ζtT––bσζId

– a

nσ

n

t= ψ

εt–aεt– σ

ζtζtT––



nσ

n

t= ψ

εt–aεt– σ

(Xt–aXt–)ζtT

+ a

nσ

n

t= ψ

εt–aεt– σ

(Xt–aXt–)ζtT–

=U+U+U+U+U+U. (.)

Note that{ψ(ηt_σ),t= , , . . . ,n}are i.i.d. random variables with finite varianceVar(ψ(ηt_σ)), we have

Var

n–

n

t=

ψ

ηt

σ

–Eψ

ηt

σ

=n–

n

t= Var

ψ

ηt

σ

=n–Var

ψ

η σ

=On– . (.)

By the Chebyshev inequality and (.), we have

(U)ij= 

nσ

n

t=

ψ

ηt

σ

–Eψ

ηt

σ

(Xt–aXt–)i(Xt–aXt–)Tj

≤ 

σ maxXt–aXt– _·n–

n

t=

ψ

ηt

σ

–Eψ

ηt

σ

=Op

Similarly, we obtain

(Ui)ij=Op

n– _{. (.)}

By (.), (.), and (.), we have

Xn(θ,ω) –bXn(a) –nb +a σ_ζId=Op

n _{. (.)}

By (.) and (.), we easily obtain

∂Q ∂βT_∂σ –

c σ

n

t=

(Xt–aXt–)T

= 

σ

n

t=

ψ

ηt

σ

ηt

σ –E

ψ

ηt

σ

ηt

σ

(Xt–aXt–)T

– 

σ

n

t= ψ

ηt

σ

ηt

σ(ζt–aζt–)

T

=Op

n _{. (.)}

By (.) and (.), we obtain

n–

∂_Q ∂βT_∂a– 

=n–

n

t= ψ

ηt

σ

εt–

σ (Xt–aXt–)

T

–n–

n

t= ψ

ηt

σ

εt–

σ (ζt–aζt–)

T

+n–

n

t= ψ

ηt

σ

X_tT_––n–

n

t= ψ

ηt

σ

ζ_tT_–

=n–_{(U+U+U+U). (.)}

Note that{ψ(ηt_σ)εt–

σ ,Ht}is a martingale difference sequence with

Var

ψ

ηt

σ

εt– σ

=E

ψ

ηt

σ



E

ε_t– σ

,

so we have

Var

n–

n

t= ψ

ηt

σ

εt–

σ (Xt–aXt–)

T

ij

=n–

n

t= Var

ψ

ηt

σ

εt– σ

(Xt–aXt–)i(Xt–aXt–)Tj

≤ 

σmaxXt–aXt– _E

ψ

η σ



n–

n

t= Eε_t_–

By the Chebyshev inequality and (.), we have

(U)i=Op

 

n(a,σ) . (.)

Similarly, we have

(U)i=Op

 

n(a,σ) . (.)

It is easy to show that

(U)i=Op

n _{, U}

=Op

n _{. (.)}

By (.) and (.)-(.), we have

∂Q

∂βT_∂a–  =Op

 

n(a,σ) . (.)

By (.) and (.), we obtain

∂_Q ∂σ –

nd σ =



σ

_n

t= ψ

ηt

σ

η_t –nd

= 

σ

n

t=

ψ

ηt

σ

η_t–E

ψ

ηt

σ

η_t

=Op

n _{. (.)}

Note that{ψ(ηt_σ)ηtεt–,Ht}is a martingale difference sequence withVar(ψ(ηtσ)ηtεt–) =

Var(ψ(ηt_σ)ηt)Var(εt–), and by (.) and (.), we obtain

∂Q ∂σ ∂a–  =



σ

n

t= ψ

ηt

σ

ηtεt–=Op

n _{. (.)}

By (.) and (.), we obtain

n–

∂Q ∂a –

b σ

n(a,σ)

= 

nσ

_n

t= ψ

ηt

σ

ε_t––bn(a,σ)

= 

nσ

n

t=

ψ

ηt

σ

–E

ψ

ηt

σ

ε_t–

+ b

n(a,σ)σ n

t=

ε_t_––Eε_t–

=T+T. (.)

Since we easily prove that{ψ(ηt

σ) –E(ψ( ηt σ))ε



t–,Ht}is a martingale difference sequence,

Var(T) =  (nσ)

n

t= E

ψ

ηt

σ

–E

ψ

ηt

σ



Eε_t_–

= 

(nσ)Var

ψ

η σ

n

t=

By (.), we obtain

Eε_t_– ≤

t–

j=

a(t––j)Eη_j +σ

t–

j= a(t––j)

t–

k= a(t––k)

=

Eη a

(t–)_(–a–(t–)₎

–a– + σ(a

(t–)_(–a–(t–)₎

–a– ), if|a| = ,

Eη_(t– ) +σ(t– ), if|a|= . (.)

Thus

n

t= Eε_t_–

≤

Eη( a

n_–

(–a–₎ –_an_–) + 

σ

(–a–₎(a n_–

–a––

(an–)

a–_(–a–₎+na–), |a| = ,



Eηn(n– ) +σ (n–)n(n–), |a|= .

(.)

That is,

n

t=

Eε_t_– = ⎧ ⎪ ⎨ ⎪ ⎩

O(n), if|a|< ,

O(an_{), if|a|> ,}

O(n), if|a|= .

(.)

By Chebyshev inequality and (.)-(.), we have

T= ⎧ ⎪ ⎨ ⎪ ⎩

Op(n–_{), if}|a|_{< ,}

Op(n–_an_{), if|a|> ,}

Op(n_{), if}|a|_{= .}

(.)

Similarly to the proof of (.), we easily obtain

T= ⎧ ⎪ ⎨ ⎪ ⎩

Op(n–_{), if}|a|_{< ,}

Op(), if|a|> ,

Op(n–_{), if}|a|_{= .}

(.)

Hence, by (.), (.), and (.), we have

∂Q ∂a –

b σn(θ) =

⎧ ⎪ ⎨ ⎪ ⎩

Op(n), if|a|< ,

Op(an), if|a|> ,

Op(n_{), if}|a|_{= .}

(.)

Thus Lemma . follows from (.), (.), (.)-(.), and (.).

Lemma . Assume that(A), (A),and(A)hold,and Eη

t <∞,E(ψ( ηt

σ))

_<∞and

θ∈.Then,as n→ ∞:

() for|a|< ,we have

Rnl(θ) =

∂ ∂θl

∂

∂θT_∂θQ(θ) =

∂ ∂θl

Fn(θ) =Op

() for|a|= ,we have

Rnl(θ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎝

Op(n_{) Op(n}_{) Op(n)}

∗ Op(n_{) Op(n)}

∗ ∗ Op(n)

⎞ ⎟ ⎠

(d+)×(d+)

.. . ⎛

⎜ ⎝

Op(n) Op(n) Op(n)

∗ Op(n_{) Op(n)}

∗ ∗ Op(n)

⎞ ⎟ ⎠

(d+)×(d+)

⎛ ⎜ ⎝

Op(n_{) Op(n}_{) Op(n)}

∗ Op(n_{) Op(n)}

∗ ∗ Op(n

 )

⎞ ⎟ ⎠

(d+)×(d+)

⎛ ⎜ ⎝

Op(n



) O_p(n) O_p(n)

∗ Op(n_{) Op(n}₎

∗ ∗ Op(n₎

⎞ ⎟ ⎠

(d+)×(d+)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(d+)×

;

() for|a|> ,we have

Rnl(θ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎝

Op(n_{) Op(n}_{) Op(a}n₎

∗ Op(n



) O_p(an)

∗ ∗ Op(an)

⎞ ⎟ ⎠

(d+)×(d+)

.. . ⎛

⎜ ⎝

Op(n_{) Op(n}_{) Op(a}n₎

∗ Op(n



) O_p(an)

∗ ∗ Op(an)

⎞ ⎟ ⎠

(d+)×(d+)

⎛ ⎜ ⎝

Op(n_{) Op(n}_{) Op(a}n₎

∗ Op(n



) O_p(an)

∗ ∗ Op(an) ⎞ ⎟ ⎠

(d+)×(d+)

⎛ ⎜ ⎝

Op(an_{) Op(a}n_{) O} p(an)

∗ Op(an) Op(an)

∗ ∗ Op(an) ⎞ ⎟ ⎠

(d+)×(d+)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(d+)×

.

Proof Let

∂ ∂θl

Fn(θ) = ⎛ ⎜ ⎜ ⎝

∂Q ∂θl∂βT∂β

∂Q ∂θl∂βT∂σ

∂Q ∂θl∂βT∂a

∗ ∂Q

∂θl∂σ

∂Q ∂θl∂σ ∂a

∗ ∗ ∂Q

∂θl∂a ⎞ ⎟ ⎟

⎠. (.)

Case .l= , , . . . ,d.

∂_Q ∂θl∂βT∂β

= –  σ n t= ψ

εt–aεt– σ

Xtl–aXt–,l– (ζtl–aζt–,l)

·Xt–aXt–– (ζt–aζt–) Xt–aXt–– (ζt–aζt–)

T

∂Q ∂θl∂βT∂σ

= – 

σ

n

t=

ψ

εt–aεt– σ

+ψ

εt–aεt– σ

εt–aεt– σ

·Xtl–aXt–,l– (ζtl–aζt–,l) Xt–aXt–– (ζt–aζt–)

T

, (.)

∂_Q ∂θl∂βT∂a

= –

σ

n

t= ψ

εt–aεt– σ

εt– σ

·Xtl–aXt–,l– (ζtl–aζt–,l) Xt–aXt–– (ζt–aζt–) T

– 

σ

n

t= ψ

εt–aεt– σ

(Xt–,l–ζt–,l)

Xt–aXt–– (ζt–aζt–)

T

+Xtl–aXt–,l– (ζtl–aζt–,l) (Xt––ζt–)T

, (.)

∂Q ∂θl∂σ

= – 

σ

n

t= ψ

εt–aεt– σ

Xtl–aXt–,l– (ζtl–aζt–,l) (εt–aεt–)

–

σ

n

t= ψ

εt–aεt– σ

Xtl–aXt–,l– (ζtl–aζt–,l) (εt–aεt–), (.)

∂_Q ∂θl∂σ ∂a

= – 

σ

n

t= ψ

εt–aεt– σ

εt–aεt– σ εt–

Xtl–aXt–,l– (ζtl–aζt–,l)

– 

σ

n

t= ψ

εt–aεt– σ

εt–

Xtl–aXt–,l– (ζtl–aζt–,l)

– 

σ

n

t= ψ

εt–aεt– σ

(εt–aεt–)(Xt–,l–ζt–,l), (.)

and

∂Q ∂θl∂a

= – 

σ

n

t= ψ

εt–aεt– σ

ε_t_–Xtl–aXt–,l– (ζtl–aζt–,l)

– 

σ

n

t= ψ

εt–aεt– σ

εt–(Xt–,l–ζt–,l). (.)

Case .l=d+ .

∂_Q ∂σ ∂βT_∂β = –



σ

n

t=

ψ

εt–aεt– σ

εt–aεt–

σ +ψ

εt–aεt– σ

·Xt–aXt–– (ζt–aζt–) Xt–aXt–– (ζt–aζt–) T, (.)

∂Q ∂σ ∂βT_∂σ = –



σ

n

t=

ψ

εt–aεt– σ

+ψ

εt–aεt– σ

εt–aεt– σ

·(εt–aεt–)

Xt–aXt–– (ζt–aζt–)

T

, (.)

∂_Q ∂σ ∂βT_∂a= –



σ

n

t= ψ

εt–aεt– σ

εt–aεt– σ

εt– σ

Xt–aXt–– (ζt–aζt–)

– 

σ

n

t= ψ

εt–aεt– σ

·

εt– σ

Xt–aXt–– (ζt–aζt–)

T

+εt–aεt–

σ

, (.)

∂Q ∂σ =



σ

n

t=

ψ

εt–aεt– σ

+ψ

εt–aεt– σ

εt–aεt– σ

(εt–aεt–), (.)

∂Q ∂σ_∂a= –



σ

n

t= ψ

εt–aεt– σ

(εt–aεt–)εt–

– 

σ

n

t= ψ

εt–aεt– σ

(εt–aεt–)εt–, (.)

and

∂_Q ∂σ ∂a = –



σ

n

t=

ψ

εt–aεt– σ

εt–aεt–

σ +ψ

εt–aεt– σ

ε_t–. (.)

Case .l=d+ .

∂_Q ∂a∂βT_∂β = –



σ

n

t= ψ

εt–aεt– σ

·εt– σ

Xt–aXt–– (ζt–aζt–) Xt–aXt–– (ζt–aζt–)

T

– 

σ

n

t= ψ

εt–aεt– σ

(Xt––ζt–)

Xt–aXt–– (ζt–aζt–) T

– 

σ

n

t= ψ

εt–aεt– σ

Xt–aXt–– (ζt–aζt–) (Xt––ζt–)T, (.)

∂Q ∂a∂βT_∂σ = –



σ

n

t= ψ

εt–aεt– σ

·(εt–aεt–)(Xt––ζt–)T+εt–

Xt–aXt–– (ζt–aζt–)

T

– 

σ

n

t= ψ

εt–aεt– σ

εt–

σ (εt–aεt–)

·Xt–aXt–– (ζt–aζt–)T, (.)

∂Q ∂a∂βT_∂a= –



σ

n

t= ψ

εt–aεt– σ

ε_t_–Xt–aXt–– (ζt–aζt–)

T

, (.)

∂Q ∂a∂σ = –



σ

n

t= ψ

εt–aεt– σ

εt–(εt–aεt–)

– 

σ

n

t= ψ

εt–aεt– σ

∂Q ∂a∂σ ∂a= –



σ

n

t=

ψ

εt–aεt– σ

εt–aεt–

σ +ψ

εt–aεt– σ

ε_t–, (.)

and

∂Q ∂a = –



σ

n

t= ψ

εt–aεt– σ

ε_t_–. (.)

Similarly to the proof of Lemma ., we easily obtain Lemma .. Here we omit it.

Lemma . (Prakasa Rao []) If {Xn,n≥} are independent random variables with

EXn= ,and s–(+n δ)nj=E|Xj|+δ→for someδ> ,then

s–_n

n

j=

Xj→N(, ),

where s

n=

n

j=EXj.

Lemma .(Hall and Heyde []) Let{Sni,Fni, ≤i≤kn,n≥}be a zero-mean, square-integrable martingale array with differences Xni,and letηbe an a.s.finite random

vari-able.Suppose that_iE{X

niI(|Xni|>ε)|Fn,i–} →p,for allε→,and

iE{Xni|Fn,i–} →p

η.Then

Snkn=

i

Xni→DZ,

where the r.v.Z has characteristic function E{exp(–_η_t_)}.

Proof of Theorem. Expanding_∂θ∂Q(θˆn) aboutθ, we have

∂

∂θQ(θˆn) = ∂

∂θQ(θ) + (θˆn–θ) ∂

∂θT_∂θQ(θ) +  

_˜

Rnl(θ¯,θˆn,θ)_{≤l≤d+}, (.)

whereθ¯=sθˆn+ ( –s)θfor some ≤s≤ and

_˜

Rnl(θ¯,θˆn,θ)

≤l≤d+=

(θˆn–θ)Rn(θ¯)(θˆn–θ)T, . . . , (θˆn–θ)Rn,d+(θ¯)(θˆn–θ)T

.

By (.)-(.) and (.), (.), we have

 =Sn(θ) + (θˆn–θ)Fn(θ) +  

_˜

Rnl(θ¯,θˆn,θ)_{≤l≤d+}. (.)

By (.), we have

(θˆn–θ)Dn(θ) = –Sn(θ) – (θˆn–θ)

Dn(θ) –Fn(θ) – 

_˜

()|a|< . By Lemma .(), Lemma .(), (.)-(.), andθ∈, we have

n–_(θˆ_{n–θ)Dn(θ) = –n}–_{Sn(θ) +op()}

=n–

_n

t= ψ

ηt

σ

Xt–aXt–– (ζt–aζt–)

T ,

n

t= χ

ηt

σ

–An,

n

t= ψ

ηt

σ

εt–

+op()

=n–

_n

t= ψ

ηt

σ

Xt–aXt–– (ζt–aζt–)

T ,

n

t=

χ

ηt

σ

–A

,

n

t= ψ

ηt

σ

εt–

+op(). (.)

Note thatVar(Sn(θ)) =O(n), so we have

(θˆn–θ)Dn(θ)Var–

 _S

n(θ) = –Sn(θ)Var–

 _S

n(θ) +op()

=

–Sn(θ)

β,σ

Var–_S

n(θ) β,σ,

n

t= ψ

ηt

σ

εt–

Eψ

η σ

n(a,σ) –_

+op(). (.)

By Lemma  of Silvapullé [], we easily obtain

–Sn(θ)

β,σ

Var–_S

n(θ) β,σ→DN(,Id+), n→ ∞. (.)

In the following, we will prove that

n

t= ψ

ηt

σ

εt–

Eψ

η σ

n(a,σ) –_

→N(, ), n→ ∞. (.)

Note that{ψ(ηt_σ)εt–(Eψ(η_σ)n(a,σ))–



,H_t}is a martingale differences sequence, so we

will verify the Lindeberg conditions for their convergence to normality. From (.), we have

 –a

n

t=

ε_t–+ε_n–ε_= n

t=

ε_t –aε_t_–

= n

t=

(εt–aεt–)(εt+aεt–)

= n

t=

ηt(ηt+ aεt–)

= n

t= η_t + a

n

t=

ByVar(n_t=ηtεt–) =σn(a,σ) =σnand Chebyshev inequality, we have

n

t=

ηtεt–=Op

n _{. (.)}

Obviously,n–n

t=ηt →pEη. By (.) andmax≤t≤n

ε_t

n =op(), we have

n

t=

ε_t–=Op(n). (.)

By (.), we have

n

t= E

Eψ

η σ

n(a,σ) –

ψ

ηt

σ

ε_t_–Ht–

=

Eψ

η σ

n(a,σ)

–n

t= ε_t_–E

ψ

ηt

σ

=n(a,σ) – n

t=

ε_t–=  +op(). (.)

For given δ> , there is a set whose probability approaches  as n→ ∞on which

max≤t≤n|√εt_n| ≤δ. In this event, for anyc> ,

n

t= E

ψ

ηt

σ

ε_t_– n I

ψ

ηt

σ

εt– √

n

>c

Ht–

= n

t=

∞

c

ydPψ

ηt

σ

ε_√t– n

≤yHt–

= n

t= ε

t– n

∞

c ε_√t–

n

ydPψ

ηt

σ

≤yHt–

≤

n

t= ε_t_–

n

∞

c δ

ydPψ

ηt

σ

≤yHt–

=oδ n

t= ε_t_–

n =oδOp()→, n→ ∞. (.)

Hereoδ→ asδ→. This verifies the Lindeberg conditions, hence (.) follows from Lemma ..

Note that –[Sn(θ)]β,σ[Var–



(S_n(_θ))]β_,σ are asymptotic independent of

n

t= ψ

ηt

σ

εt–

Eψ

η σ

n(a,σ) –_

.