Asymptotic property of two stage estimator under missing response data

LUO Shuanghua,ZHANG Yafei

(School of Science,Xi′an Polytechnic University, Xi′an 710048,China)

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Asymptotic property of two stage estimator under missing response data

LUO Shuanghua,ZHANG Yafei

(School of Science,Xi′an Polytechnic University, Xi′an 710048,China)

Abstract:Under missing response data, the semiparametric regression model Y=X′β+g(T)+ε is considered to establish the two stage estimators ofn(t) and 2 of β, g(t) and σ2. Then the mean of Y is derived by the imputed every missing Yi. It is shown that these estimators have asymptotic normality andn(t) has the better convergence rate.

Key words:semiparametric regression model; two stage estimator; missing response data; asymptotic normality; best convergence rate

0Introduction

In this paper, we consider the following semiparametric regression model

(1)

In the semiparametric regression analysis setting up, the basic inference begins by considering the random sample

(Xi,Yi,Ti,δi),

(2)

By a purely semiparametric approach to discussing the missing data (2), the MAR assumption would require that there exists a chance mechanism denoted byp(Xi,Ti), such that

P(δ=1|Xi,Yi,Ti)=P(δ=1|Xi,Ti)=p(Xi,Ti)

(3)

holds almost surely. In practice, (3) is a common assumption for statistical analysis with missing data and is reasonable in many practical applications, see reference[14].

1The two stage estimator

In this section we define the estimators that we will analyze in this paper. We describe how to estimate the regression function.

Letα=Eg(Ti),ei=g(Ti)-α+εi,i=1,…,n, the model (1) turn into following

(4)

Where thee1,…,enare independent identically distributed random variables with Ee1=0 and 0<σ2=Ee12=Eε12+Var(g(Ti))=σ02+σ12<∞. The model (4) can be changed into the following form

(5)

Inordertoobtainthesolutionofthefollowingleastsquaresproblem(5),wehavetofindαandβtominimize

Wn=(Yn-Xnβ-1nα)′Qn(Yn-Xnβ-1nβ).

(6)

Byoptimizationtheory,wehavethat

andthus

(7)

(8)

Now,wedefinedthenonparametricestimatorofg(t)that

(9)

(10)

Usingthegeneralizedleastsquaresforthemodel(10),wecanfindβtominimize

(11)

andobtainestimatorofβthat

Sowenowcanobtaintheestimationofθ=E(Y).Theregressionimputationestimatorofθcanbedenotedby

(12)

Thus,wehavethepropensityscoreweightedestimator

(13)

(14)

withW(·,·)istheweightingfunctionandhnisthebandwidthsequence.

2The asymptotic properties and consistencies

We explore the asymptotic distribution and consistency of the all estimators. The following notation and assumptions are needed.

(ⅰ) TheT1,T2…Tnare independent identically distributed random variables and the {Ti} is independent of the {ei}.

(ⅱ) The rank(Xn)=p

(ⅳ) E[g(T1)]2<∞.

(ⅴ) Existence 0

(ⅵ) The probability density function ofTiisr(t) and

In what follows the main results will be established for the asymptotic distribution and consistency of the semiparametric regression model.

Theorem 1Under conditions (ⅰ)～(ⅴ), we have that

Theorem 4Under conditions (i)～(vi), we have

(15)

3Sketches of the proofs

In this section, we will give the proof of Theorem 1～3.The following lemmas are needed for our technical proofs.

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ProofSimilar to the theorem in reference[15].

Proof the Theorem 1Similar to the Lemma 2.1 in reference [16].

Proof the Theorem 2

(2)Firstly,weprovetheconclusion(ⅰ)of(2).

ByLemma1inreference[17],inordertoobtaintheproofof(i),weonlyprove

and

ItfollowsfromLinderbergtheoremthat

Theorem 3

(1) LetWn(t)=(Wn1(t),…,Wnn(t))′, whereti∈Cf∧{Ti,f(ti)>0}.Since

Let

(16)

whenn>p.TheCauchy-Schwarzinequalityyields

whenn≥p.Therefore,

(17)

(16)and(17)showJ2→0,a.s.Thiscompletestheproof(1)ofTheorem3.

(2)Itisnotdifficulttoobtain

and

Ithasbeenprovedthat

(18)

and

(19)

Usingthesamemethodoftheprooffor(1)ofTheorem3,itfollowsfromLemma3inreference[17]that

(20)

Bytheconditions(Ⅴ)weknowthat

and

(21)

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編輯、校對：師瑯

DOI：10.13338/j.issn.1006-8341.2016.02.011

Received date：2015-10-30

Foundation item:The National Natural Science Foundations(11201362);the Science Foundation of the Education Department of Shaanxi Province(14JK1305);the Natural Science Foundations of Shaanxi Province(2016JM1009)

Corresponding author:LUO Shuanghua(1976—),female,native of Suining city,Sichuan province,research area is quantile regression,missing data analysis and processing nonparametric estimations.E-mail: iwantflyluo@163.com

CLC number:O 212.7

Document code:A

缺失響應(yīng)數(shù)據(jù)下二階段估計(jì)的漸近性質(zhì)

羅雙華，張亞飛

(西安工程大學(xué) 理學(xué)院,陜西 西安 710048)

摘要:在缺失響應(yīng)數(shù)據(jù)下考慮半?yún)?shù)回歸模型Y=X′β+g(T)+ε,建立該模型參數(shù)β,g(t)和σ2的二階段估計(jì)n(t)和2,并通過對每個缺失響應(yīng)數(shù)據(jù)Yi進(jìn)行插值,得到了響應(yīng)數(shù)據(jù)的均值．研究表明,這些參數(shù)的估計(jì)具有漸近正態(tài)性,并且n(t)具有較好的收斂速度．

關(guān)鍵詞:半?yún)?shù)回歸模型;二階段估計(jì);缺失響應(yīng)數(shù)據(jù);漸近正態(tài)性;最佳收斂速度

Article ID:1006-8341(2016)02-0197-07

Citation format：LUO Shuanghua,ZHANG Yafei.Asymptotic property of two stage estimator under missing response data[J].紡織高?；A(chǔ)科學(xué)學(xué)報(bào),2016,29(2):197-203.

LUO Shuanghua,ZHANG Yafei.Asymptotic property of two stage estimator under missing response data[J].Basic Sciences Journal of Textile Universities,2016,29(2):197-203.