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(Department of Mathematics and Physics, Henan University of Urban Construction, Pingdingshan 467000, China)

Empirical Likelihood Statistical Inference for Partially Linear Model with Restricted Condition

LIUChang-sheng*,LIYong-xian

(DepartmentofMathematicsandPhysics,HenanUniversityofUrbanConstruction,Pingdingshan467000,China)

Inthispaper,weapplytheempiricallikelihoodmethodtopartiallylinearmodelwithparameterlinearrestrictedhypothesis.Forthesakeoftestinghypothesis,anempiricallog-likelihoodratioteststatisticbasedonthedifferenceofthenullandalternativehypothesesisconstructed.Furthermore,thelimitingdistributionoftheteststatisticsisprovedtobeastandardChi-squareddistribution.Numericalsimulationconfirmstheadvantageoftheproposedmethod.

empiricallikelihood;restrictedcondition;partiallylinearmodel;hypothesistest;Chi-squaredistribution

1 Empirical likelihood estimation on parameter

For the need of constructing the test statistic, we first develop estimating approach for model (1) under the null hypothesis in this section. That is, we estimate the unknown quantities in model (1) with the restricted condition Aβ=b.Thenmodel(1)canbewrittenas

(3)

whereKh(·) =K(·/h)/h,K(·) is a kernel function andh=hnis a sequence of positive numbers tending to zero, called bandwidth. Simple calculation yields that

(4)

For 1≤i≤n, let

In order to construct the empirical likelihood ratio function, we now introduce one auxiliary random vectorZi(β),

(5)

1.1 Empirical likelihood estimation on parameter without restriction

Next we discuss profile empirical likelihood estimation without restriction conditionsAβ=b. Whenβis true parameter,E(Zi(β))=0. Thus, by the idea of Owen[1], an empirical likelihood-ratio forβcan similarly be defined as follows:

(6)

wherep=(p1,…,pn) is a probability vector.

Ifβis true parameter, a unique maximum forpin (6) exists. By the Lagrange multiplier method, the supremum occurs at

(7)

whereλ(β) is the solution to

(8)

By (6) and (7), we can get

(9)

In the following, we define the profile empirical likelihood estimator without any restriction conditions

(10)

whereZi(β) andλ(β) satisfy (5) and (8), respectively.

1.2 Empirical likelihood estimation on parameter with restrictionAβ=b

(11)

whereηis ak×1 vector that contains the Lagrange multipliers. By differentiating functionF(β,η) with respect toβandη, we obtain the following equations:

(12)

and

(13)

2 Test statistic and its properties

In order to formulate the main results, we need the following assumptions. These assumptions are quite mild and can be easily satisfied.

Lethj(Ti)=E(Xij|Ti),Vi=Xi-E(Xi|Ti), 1≤i≤n, 1≤j≤p.

Assumption1 E(e|X,T)=0andE(|e|4|X,T)<∞.

Assumption3 g(·)andhj(·)areofoneorderLipschitzcontinuousfunctions.

Assumption 5 The kernel functionK(·) is a bounded symmetric density function with compact support and satisfies ∫K(u)du=1,∫uK(u)du=0 and ∫u2K(u)du<∞.

Assumption 6 The density functionsf(t) ofTis bounded away from zero and have bounded continuous second partial derivatives. Namely, 0

Under the above assumptions, we can get the following result, proved in Section 4.

Theorem3Underthenullhypothesisoftestingproblem(1.2)andtheassumptions1-6,wehave

3 Simulation studies

In this section, we present the result of some simulations to illustrate our methods. In our simulations, the data are generated from the following model:

yi=xi1β1+xi2β2+g(ti)+εi,i=1,…,n,

(14)

Tab.1 The rejection frequencies for H0:β1-β2=0↔H1:β1-β1=c with α=0.05

We summarize our findings as follows. When the null hypothesis is true (that is,c=0), the rejection frequencies (estimated sizes) of both our proposed test basedTnand the restricted least-squares approach test basedWnare quite good and close to their nominal levels 0.05 under different error distributions. Under the alternative hypothesis, the rejection rate seems very robust to the variation of the type of error distribution. With the increasing ofc, the test power of our proposed test is slightly better than the test based on the residual sum of squares.

4 Proof of the main results

In the sequel, letCdenote positive constant whose value may vary at each occurrence.

Lemma 1 Suppose that Assumptions 1-6 hold.

whereG0(·)=g(·) andGl(·)=hl(·)(j=1,…,p).

ProofTheproofissimilartoLemmaA.1inLiang[9]etal.

Lemma2SupposethatAssumptions1-6hold.Wecanobtain

ProofTheproofissimilartoLemmaA.2inLiang[9]etal.

Lemma3SupposethatAssumptions1-6hold.ifβ0istruevalueofβ, We can obtain

ProofFromthedefinitionofZi(β), we have

Lemma4SupposethatAssumptions1-6hold.Ifβ0istruevalueofβ,Wehavemax1≤i≤n‖Zi(β0)‖=op(n1/2).

ProofAsimilarproofcanbefoundinLiang[10]etal.

Lemma5SupposethatAssumptions1-6hold.Ifβ0isthetruevalueofβinmodel(3),satisfying(7)and(8),thenwehave

ProofApplyingtheTaylorexpansion,from(8)andLemma1～4,weobtainthat

(15)

In view of Lemma 1～4, we have

This completes the proof.

TheproofofTheorem1

(16)

(17)

(18)

where

(19)

We can also get

(20)

This completes the proof.

TheproofofTheorem2issimilarasthatofTheorem1andthusisleftforthereaders.

TheproofofTheorem3

ProofBy(10)andapplyingtheTaylorexpansion,wehave

(21)

where

Similarly, we can also get

(22)

with |r2n|=op(1).

From (21) and (22), we can get

I1+I2+op(1).

Op(n-1)·Op(n1/2)·op(n1/2)=op(1).

(23)

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(编辑 HWJ)

2016-03-27

河南省科技计划项目资助(112300410191)

O

A

1000-2537(2017)04-0075-08

具有限制条件的部分线性模型的经验似然推断

刘常胜*,李永献

(河南城建学院数理系, 中国 平顶山 467000)

本文将经验似然方法应用到具有限制假设条件的部分线性模型中. 为了检验假设条件, 构造基于零假设和对立假设条件下的极大经验对数似然比估计值的差值统计量. 而且在零假设下证明该统计量的极限分布为标准的χ2分布. 数值模拟表明所提出的检验统计量的优势.

经验似然; 限制条件; 部分线性模型; 假设检验; χ2分布

10.7612/j.issn.1000-2537.2017.04.013

*通讯作者,E-mail：csliu@hncj.edu.cn