Journal of Mathematical Research&ExpositionAug.,2008,Vol.28,No.3,pp.645–650DOI:10.3770/j.issn:1000-341X.2008.03.026Empirical Likelihood of Density Function for DependentSeriesJIN Shu Hua(Department of Mathematics,Changchun Taxation College,Jilin130117,China)(E-mail:yzx8170561@)Abstract With the application of the special properties of strongly stationary m−dependent series,this paper is concerned with the empirical likelihood confidence intervals of density func-tion under m−dependent series.The limit distribution of empirical likelihood ratio statistics is given out,and the empirical likelihood confidence intervals of parameters can be constructed.A simulation study is conducted to show thefinite sample performance of the empirical likelihood based method.Keywords m−dependent series;density function;empirical likelihood.Document code AMR(2000)Subject Classification62G07;62E20Chinese Library Classification O211.6Empirical likelihood(EL)method is a nonparametric method of inference introduced by Owen[1−2].It has been studied extensively because of its generality and effectiveness.The EL method has been successfully applied in many areas such as linear regression models[3−5], generalized linear models[6],quantile[7],general estimation equation[8],semiparametric model[9], dependent process[10],survival analysis[11],amongst others,mixture models[12]and so on.More references on the EL can be found in the recent monograph of Owen[13].It should be noted that the above work seems to focus on independent data,but the actual data are usually dependent. Chen[14]discussed the empirical likelihood confidence intervals of density function under i.i.d. samples.As far as the dependent samples are concerned,the situations are more complex.Jiang and Qin[15]discussed the empirical likelihood confidence intervals of density function underϕ-mixing random variables.In this paper,we will use the special properties of strongly stationary m-dependent series to construct the empirical likelihood confidences intervals of density function under m-dependent series.1.Introduction and main resultLet X1,X2,...be strongly stationary m−dependent random samples from an unknown den-sity function f(x).For afixed x∈R,let f(x)=θ1.We refer to[16]for the definitions of646JIN S Hstrongly stationary random series and m−dependent.To state the main results of this paper,we make the following assumptions:(i)For i=1,2,...,m−1,g i(x,y)is the joint density function of(X1,X i+1),and g i(x,y) and f(x)admit continuous r th(r≥2)derivatives in neighbourhoods of the points(x,y)and x, respectively.(ii)K(·)is a kernel,K=K(−u),and K(·)is a kernel of order r,that is:K(y)d y=1, y i K(y)d y=0,i=1,2,...,r−1, y r K(y)d y=k,k=0.For instance,the Gaussian kernel,K(x)=12πe−x25),nh−→∞.Letωi=1h)−θ1,i=1,2,...,n.The empirical log-likelihood ratio ofθ1isl(θ1)=2ni=1log(1+λωi),(1)whereλis the solution of11+λωi=0.(2) Theorem Suppose that Assumptions(i)–(iii)hold,thenl(θ1)−→χ2(1),as n−→∞.2.LemmasLemma1Let X1,X2,...,X n be strongly stationary m-dependent series with E|X1|r<∞. For some r≥2,then E| n i=1X i|r≤Cn r/2E|X i|r.The proof can be found in[17].Lemma2Let X1,X2,...,X n be univariate m-dependent r.v.s.,EX i=0,E|X i|3<∞(i= 1,2,...,n).Thensup x |P(n i=1X iB3n,where B n=Empirical likelihood of density function for dependent series647 In our simulation studies,we generate e′i s i.i.d.from the standard normal distribution N(0,1), and X i=e i+0.5e i−1−0.3e i−2,so X′i s are2−dependent random series.The density functionof X i isf(x)=12π×√2×1.34}.The kernel function K(t)is the Gaussian kernelK(t)=12πe−t2/2.Also,three different bandwidths of h are selected to be(n log n)−1/2,(n log n)−1/3and (n log n)−1/5,respectively.It is easy to check all the conditions(i)–(iii)in the paper are satisfied. The sample size has been chosen to be10,20and50,respectively.The coverage probabilities are calculated for the empirical likelihood method based on500simulated data.The nominal levels are taken to beα=0.10and0.05,respectively.The results are presented in Tables1and 2.From Tables1and2,we see that the empirical likelihood method performs well and the cov-erage accuracies increase as the size n increases.We choose three bandwidths h=(n log n)−1/2, (n log n)−1/3and(n log n)−1/5.Clearly,the bandwidth plays an important role here.However, we shall not address the problem of how tofind the optimal bandwidth.100.62200.75000.8400200.65600.82200.8680500.74000.84000.9040n h=(n log n)−1/2h=(n log n)−1/3h=(n log n)−1/5Table2Coverage probabilities forθ1,α=0.054.Proof of main resultProof of Theorem Put¯ω=1nni=1ω2i,Z n=max1≤i≤n|ωi|.Noting that|ωi|=|1h)−θ1|≤K(x−X i h+θ1≤C648JIN S Hit is clear to see thatZ n =max 1≤i ≤n|ωi |=O(h −1) a.s.(3)Let ρ2(K )=K 2(t )d t .Since S −ES =hn 2n max1≤i ≤nE (ω2i−Eω2i )2=Ch 2h K (x −X 1h4K 4(x −X 1h3E (1h))+C ≤Cnh+Ch 23(ωi −Eωi ).Then EX ni =0,and Eωi =O(h r ).So we getE |X ni |3=h 3n(O(h −3)+O(h −2)O(h r )+O(h −1)O(h 2r )+O(h 3r ))=O(n −1)+O(n −1h r +1)+O(n −1h 2r +2)+O(n −1h 3r +3)=O(n −1)<+∞.Let B 2n =Var( n i =1X ni )= n i =1EX 2ni +2 m −1i =1EX n 1X ni +1.Noting thatEX 2ni=h 23E (ωi −Eωi )2=h 23[Eω21−(Eω1)2]=h 23Eω21−h23O(h 2r )=h23(1n2n23h 2)+O(n −23h 2r +2)=h3θ1K 2(v )d v +O(n −2n 2n2hK (x −X 1hK (x −X 1hK (x −X i +1hK (x −X i +1hK (x −X 1hK (x −X i +1hK (x −uhK (x −vEmpirical likelihood of density function for dependent series649 Letx−uh=v′.ThenI= +∞−∞ +∞−∞K(u′)K(v′)g i(x−hu′,x−hv′)d u′d v′.From Assumption(i),we knowI=g i(x,x)+O(h r).It follows thatE(ω1−Eω1)(ωi+1−Eωi+1)=g i(x,x)−θ21+O(h r).Thereforeni=1EX n1X ni+1=O(n−23hθ1 K2(v)d v+O(n13h2), where n1B nD−→N(0,1),(5) where“D−→”denotes convergence in distribution.Obviously,ni=1X ni=h3ni=1(ωi−Eωi)=n23h r+1),which implies that√2)D−→N(0,1).From Assumption(iii),we see that(nh2r+1)12).(6)By(2)we have|λ|≤¯ωh−Z n¯ω=h¯ω2h1θ1 K2(v)d v+o p(1)=O p(n−12).(7)Letγi=λωi.Then we getmax1≤i≤n|γi|=|λ|Z n=O p(n−12)O p(h−1)=O p((nh)−1nni=1ωiγ2i/(1+γi).650JIN S H Therefore,we may writeλ=hs−1¯ω+τ,where|τ|≤h2+ηi,whereni=1|ηi|≤B|λ|3ni=1|ωi|3=B O p(n−32)O p(h−2)n O p(1)=o p(1).So we havel(θ1)=nhS−1¯ω2−nτ2Sh=n O p(n−2)O p(1)h−1=O p((nh)−1)=o p(1),we get l(θ1)=nhS−1¯ω2+o p(1).This completes the proof.2 References[1]OWEN A B.Empirical likelihood ratio confidence intervals for a single functional[J].Biometrika,1988,75(2):237–249.[2]OWEN A.Empirical likelihood ratio confidence regions[J].Ann.Statist.,1990,18(1):90–120.[3]OWEN A.Empirical likelihood for linear models[J].Ann.Statist.,1991,19(4):1725–1747.[4]CHEN Songxi.On the accuracy of empirical likelihood confidence regions for linear regression model[J].Ann.Inst.Statist.Math.,1993,45(4):621–637.[5]CHEN Songxi.Empirical likelihood confidence intervals for linear regression coefficients[J].J.MultivariateAnal.,1994,49(1):24–40.[6]KOLACZYK E D.Empirical likelihood for generalized linear models[J].Statist.Sinica,1994,4(1):199–218.[7]CHEN Songxi,HALL P.Smoothed empirical likelihood confidence intervals for quantiles[J].Ann.Statist.,1993,21(3):1166–1181.[8]QIN Jing,LAWLESS J.Empirical likelihood and general estimating equations[J].Ann.Statist.,1994,22(1):300–325.[9]SHI Jian,LAU T S.Empirical likelihood for partially linear models[J].J.Multivariate Anal.,2000,72(1):132–148.[10]KITAMURA Y.Empirical likelihood methods with weakly dependent processes[J].Ann.Statist.,1997,25(5):2084–2102.[11]W ANG Qihua,LI Gang.Empirical likelihood semiparametric regression analysis under random censorship[J].J.Multivariate Anal.,2002,83(2):469–486.[12]ZOU F,FINE J P,YANDELL B S.On empirical likelihood for a semiparametric mixture model[J].Biometrika,2002,89(1):61–75.[13]OWEN A B.Empirical Likelihood[M].Chapman&Hall,New York.[14]CHEN Songxi.Empirical likelihood confidence intervals for nonparametric density estimation[J].Biometrika,1996,83(2):329–341.[15]JIANG Bo,QIN Yongsong.Empirical likelihood confidence intervals of density function under dependentsamples[J].J.Anhui University of Technology,2006,23(3):352–355.[16]SUN Zhibin,ZHANG Junjian.Empirical likelihood ratio confidence regions for mean under strongly station-ary m dependent samples[J].Math.Practice Theory,2005,35(7):207–214.[17]SHI Xiquan.Bootstrap estimation for mean under m dependent samples[J].Chinese Sci.Bull.,1986,31(6):404–407.[18]SHERGIN V V.The rate of convergence in the central limit theorem for m-dependent random variables[J].Teor.Veroyatnost.i Primenen.,1979,24(4):781–794.(in Russian)。
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