Some Research On The Empirical Likelihood Method

The empirical likelihood method, introduced by Owen (1988, 1990), is one of the most important statistical inference methods. It has several nice advantages over other statistical methods: the empirical likelihood regions are shaped completely by the sample, Bartlett correctable, range preserving and transformation respecting. For these reasons, empirical likelihood method has been paid great attention by statisticians and economists and has received widely researches and applications.In this paper, the empirical likelihood method is studied based on Glenn N.L. (2006) and Junjian Zhang (1999, 2006).In the first chapter, the backgrounds and recent researches of the empirical likelihood method are introduced, as well as its definition given by Owen (1988, 1990) and some classical results of empirical likelihood estimates on the population mean.In the second chapter, according to the weighted empirical likelihood method proposed by Glenn N.L. (2006), given the weight vectorω_n=(ω_n1,ω_n2,…,ω_nn),ω_ni＞0, sum from i=1 to nω_ni=1 for i.i.d.samples, the weighted empirical likelihood ratio statistic is constructed and the approximate property of -2log R(μ₀) is discussed under some conditions. A nonparametric Wilks's theorem is obtained for constructing confidence regions for the population mean:Theorem 2.1 LetX₁, X₂,…, X_n be i.i.d, sample sequence in R^p with meanμ₀ and variance matrix∑of full rank. Suppose p_i is the probability mass defined on X_i, 1≤i≤n, P_i≥0 and sum from i=1 to n p_i=1. For any n∈N, given weight vectorω_n=(ω_n1,ω_n2,…,ω_nn)′,for 1≤i≤n, , we haveω_ni＞0 and sum from i=1 to nω_ni=1. If the weight vector satisfies then where(?)means convergence in distribution.Meanwhile, the constrained weighted empirical likelihood method is studied and a corollary similar to Owen (1991) is achieved.Corollary 2.2 Let Z₁, Z₂,…, Z_n be i.i.d, sample vectors in R^p+q, where Z_i=(X′_i, Y′_i)′, X_i∈R^p, Y_i∈R^q, and p, q＞0, 1≤i≤n. Given weight vectorω_n=(ω_n1,ω_n2,…,ω_nn)′, which satisfies the condition in theorem 2.1. If E (‖Z₁‖²)＜∞and is with full rank. Denote E(Z₁)=μ_z0=(μ′_x0,μ′_y0)′and where F_n is the empirical distribution function for Z_i, WR(F)=Π_i=1ⁿ((p_i)/(ω_ni))^nω_ni. Ifμ_x=μ_x0+ o_p(1),μ_y=μ_y0+ o_p(1), then where. MeanwhileIn the third chapter, using the blockwise empirical likelihood introduced by Kitamura (1997) for strongly stationaryρ-mixing sequence {X₁, X₂,…, X_n}, letl=o(n^Υ), 0＜Υ＜1, q=[n/l]-1. Denoteξ_k=X_kl+1+X_kl+2+…+ X_(k+1)l, k=0,1,…, q. let n be a multiple of l .Define With new random variables sequence {Y₁, Y₂,…, Y_q}, statistical inference and confidence intervals are obtained for both of the population mean and M-functional by empirical likelihood method. A nonparametric Wilks's theorem is proved thus empirical likelihood method for i.i.d, samples is extended to the strongly stationaryρ-mixing sequence.Theorem 3.1 Let {X_n, n≥1} be a strongly stationaryρ-mixing real random variables sequence with EX₁=μ₀, Var X₁=σ²＞0, and(?)δ≥2, satisfying (?) E|X_k|^2+δ＜∞. Suppose the mixing rate satisfies∑_n=1^∞ρ(n)＜∞, thus(1) A²=σ²+2∑_j=2^∞Cov(X₁, X_j) is convergent;(2) C_rn= {∫xdF|R(F)≥r, 0＜r＜1, F<n}is a closed interval;(3) (?) P{μ₀∈C_rn}=P{χ₁²≤-2logr}.Theorem 3.2 Let {X_n, n≥1} be a strongly stationaryρ-mixing real random variables sequence with EX₁=μ₀, Var X₁=σ²＞0, and(?)δ≥2, satisfying (?) E|X_k|^2+δ＜∞. Suppose the mixing rate satisfies∑_n=1^∞ρ(2ⁿ)＜∞andσ_n²:=VarS_n�'∞, n�'∞. Then(1) (?) (σ_n²)/n= (?) (VarS_n)/n=(?)²,其中(?)²＞0;(2) C_rn= {∫xdF|R(F)≥r, 0＜r＜1, F<n} is a closed interval;(3) (?) P{μ₀∈C_rn}=P{χ₁²≤-2logr}.Theorem 3.3 Letθ₀ be the only solution of E_Fψ(X,θ₀)=∫ψ(X,θ₀)Fd(X)=0,ψ(X,θ) be a increasing measurable function onX and statisfying(a) sup_l≤k≤n E|ψ(X_k,θ₀)|^2+δ＜∞,δ≥2;(b)∑_n=1^∞ρ(n)＜∞.Then(1) A²=Var(ψ(X₁,θ₀))+2∑_j=2^∞Cov(ψ(X₁,θ₀),ψ(X_j,θ₀))is convergent;(2) C_rn= {∫ψ(x,θ)dF|R(F)≥r, 0＜r＜1, F << F_n} is a closed interval;(3) (?) P{μ₀∈C_rn}=P{χ₁²≤-2logr}, where R(F)=Π_i=1ⁿ np_i,∑_i=1ⁿ p_i=1, p_i≥0.