WANG XUE-JUN,HU SHU-HE,LING JI-MIN,WEI YUN-FEI AND CHEN ZHU-QIANG

(School of Mathematical Science,Anhui University,Hefei,230601)

Strong Consistency of M Estimator in Linear Model for?φ-mixing Samples*

WANG XUE-JUN,HU SHU-HE,LING JI-MIN,WEI YUN-FEI AND CHEN ZHU-QIANG

(School of Mathematical Science,Anhui University,Hefei,230601)

Communicated by Wang De-hui

The strong consistency of M estimator of regression parameter in linear model for?φ-mixing samples is discussed by using the classic Rosenthal type inequality. We get the strong consistency of M estimator under lower moment condition,which generalizes and improves the corresponding ones for independent sequences.

?φ-mixing sample,M estimator,strong consistency

1 Introduction

Consider the following linear model:

where xi(i=1,2,···,n)is a known p-dimensional vector,β0is an unknown p-dimensional regression parametric vector,and e1,e2,···,enare random errors.Let f be a convex function on R.The M estimator of β0is?βnsatisfying the following equation:

The scope of M estimator is very wide containing the least squares estimator,maximum likelihood estimator etc.Since Huber[1]studied the M estimator of regression parameter in linear model,many authors have shown great interest in this f i eld and obtained many useful results;see,for example,[2–5].

Throughout this paper,we use the following notations:Let

be a p-dimensional vector,and

Denote

Let f be a convex function,ψ-and ψ+denote the left derivative and right derivative of function f,respectively.an=O(bn)denotes that there exists a positive constant C such thatfor all sufficiently large n.C and C(i≥1)are positive constants which

imay be di ff erent in various places.

As for the sufficient condition for the strong consistency of M estimator,Chen and Zhao[2]obtained the following result:

Theorem 1.1[2]Let e1,e2,···,en,···be a sequence of independent random variables with identical distribution,and f be a convex function satisfying the following two conditions:

(1)There exist constants l1＞0 and l2＞0 such that

(2)There exists a constant Δ＞0 such that

If there exists a δ with 0＜δ≤1 such that dn=O(n-δ),then,is the strong consistency estimator of β0.

Yang[3]improved the result of Theorem 1.1 and obtained the following Theorem 1.2 and Theorem 1.3.

Theorem 1.2[3]Let e1,e2,···,en,···be a sequnce of independent random variables with identical distribution,and f be a convex function satisfying the following two conditions:

(1)There exist constants l1＞0 and l2＞0 such that

(2)There exist constants h0＞0,Δ＞0 and 0＜δ≤1 such that

and

Theorem 1.3[3]Let e1,e2,···,en,···be a sequence of independent random variables, and f be a convex function satisfying the following two conditions:

(1)There exist constants l1＞0 and l2＞0 such that

(2)There exist constants h0＞0,Δ＞0,0＜δ≤1 andsuch that

Unfortunately,e1,e2,···,en,···are not independent in most cases.So it is valuable to extend the result for independent samples to the case of dependent samples.Some authors have shown great interests in this fi eld and obtained some valuable results.Wu[4–6]studied the strong consistency of M estimator of regression parameter in linear model for ρ-mixing,φ-mixing and ψ-mixing samples(see[4]),the strong consistency of M estimator of regression parameter in linear model for-mixing samples(see[5]),the strong consistency of M estimator for NA samples(see Theorem 3.3.1 of[6]),and so forth.In this paper,we investigate the strong consistency of M estimator of regression parameter in linear model for a class of-mixing samples and greatly reduce the moment condition of|ψ+(ei±Δ)|.

Let{Xn,n≥1}be a sequence of random variables de fi ned on a fi xed probability space (Ω,F,P).Write

Given σ-algebras B,R in F,let

De fi ne theφ?-mixing coefficients byfi nite subsets S,T?N such that dist(S,T)≥k}, k≥0. Obviously,

De fi nition 1.1 A sequence of random variables{Xn,n≥1}is said to be-mixing if there exists a k∈N such that(k)＜1.

It is easily seen that independent sequence is the special case of-mixing sequence. The concept of-mixing was introduced by Wu and Lin[7].Some authors have studied the concept and got some valuable results,for example,Wu and Lin[7],Wang and Hu[8],Wu[9], and so forth.

The following lemma plays an important role to prove the main result of this paper.The proof is similar to which of Lemma 5.1.1 in[6],so we omit the details.

Lemma 1.1 Let{Xn,n≥1}be a sequence ofφ?-mixing random variables with

Then there exists a positive constant C depending only on(·)and q such that

2 Main Result and Its Proof

Theorem 2.1 Let e1,e2,···,en,···be a sequence ofφ?-mixing random variables,and f be a convex function satisfying the following two conditions:

(1)There exist constants l1＞0 and l2＞0 such that

(2)There exist constants h0＞0,Δ＞0,0＜δ≤1 andsuch that

Proof.Without loss of generality,we assume that β0=0.Let

Then the model(1.1)can be replaced by

and

Take

Let m be a positive integer such thatEach side of the hypercube Dcan bendivided intoequal parts.Thus,the hypercube Dncan be divided intosmall hypercubes denoted as{Bj:1≤j≤Nn},The length of Bjisa nd the center is denoted as bj.Denote

According to the proof of Theorem 3.1 in[2],we can see that

Thus,in order to prove Theorem 2.1,we only need to show

Firstly,we prove(2.8).Denote

By(2.2)we have

By the convexity of f,we can see that

If β∈Dn,then by(2.5)and(2.6)we have

which imply that for β∈Bj,

and

It follows from(2.1)and(2.4)–(2.6)that

and

Therefore,for all n large enough,

By Markov's inequality and(2.13)–(2.15),we have for all n large enough

This completes the proof of(2.8).

Now we prove(2.7).We discuss it for two cases.

1)If 0＜δ＜1,we denote

where

Let

It is easy to see that

which yields that

Take

By Lem ma 1.1 we have

Therefore,

Finally,we prove that

It follows from (2.14)that

which implies that for all n large enough

Since

it follows that

Therefore,

2)If δ=1,we can also get

by using the similar method of the proof of the Theorem in[5]. The proof is completed.

Remark 2.1 We have pointed out that-mixing sequence contains independent sequence as a special case,and thus Theorem 2.1 generalizes the result for independent sequence.On the other hand,the condition(2.2)reduces the moment condition of|ψ+(ei±Δ)|(forwhich greatly improves and extends the results of Chen and Zhao[2]and Yang[3].

References

[1]Huber P J.Robust regression.Ann.Statist.,1973,1:799–821.

[2]Chen X R,Zhao L C.M Method in Linear Model(in Chinese).Shanghai:Science and Technology Press of Shanghai,1996.

[3]Yang S C.Strong consistence of M estimator of regression parametric in linear model(in Chinese).Acta Math.Sinica,2002,45:21–28.

[4]Wu Q Y.Strong consistency of M estimator in linear model for ρ-mixing,φ-mixing,ψ-mixing samples(in Chinese).Math.Appl.,2004,17:393–397.

[5]Wu Q Y.Strong consistency of M estimator in linear model for?ρ-mixing samples(in Chinese). Acta Math.Sinica,2005,25:41–46.

[6]Wu Q Y.Probability Limit Theory for Mixing Sequences(in Chinese).Beijing:Science Press of China,2006.

[7]Wu Q Y,Lin L.Convergence properties of?φ-mixing random sequences(in Chinese).J.Engrg. Math.,2004,21:75–80.

[8]Wang X J,Hu S H.Large deviations for the partial sums of a class of random variable sequences (in Chinese).Math.Appl.,2007,20:541–547.

[9]Wu Q Y.Almost sure convergence for?φ-mixing random variable sequences.Math.Appl.,2008, 21:629–634.

tion:62J05,62F12

A

1674-5647(2013)01-0032-09

*Received date:April 27,2010.

The NSF(11201001,11171001,11126176)of China,the NSF(1208085QA03)of Anhui Province,Provincial Natural Science Research Project(KJ2010A005)of Anhui Colleges,Doctoral Research Start-up Funds Projects of Anhui University and the Students'Innovative Training Project(2012003)of Anhui University.