Empirical Likelihood Ratio Confidence Regions With A Nuisance Parameter For Dependent Samples

Empirical likelihood function is a kind of nonparameter interring method.it was introduced by Owen(1988) to construct confidence regions, lately,some authors apply it to linear model, semiparameter model, regression function, density kernel estimation, bias sample and so on, but they discuss samples that are i. i .d. Cui hengjian([1]) have discussed empirical likelihood ratio confidence regions for a interesting parameter θ₀ with a nuisance parameter μ,supposing θ₀ = EH (X,μ),Xii.i.d.In this paper, we generalized estimation equation EH (X,θ₀ ,μ)=0,and{X_i|i≥1}be strongly stationaryφ-mixing random variables([2]).first we choose a consistence estimation μ) (x₁,…,x_n ) =μof μ,then we make use of empirical likelihood ratio method to construct approximate confidence regions for θ₀. Let H ( x,s,t)is continuous function. and note that(?)_μδ={ t | t∈R,|t-μ| ≤δ},for some δ>0. For general estimation equation EH (X,θ₀ ,μ)=0,as μis unknown,we want to construct confidence regions for θ₀,first choose a consistence estimation μ) (x₁,…,x_n )of μ,then empirical likelihood ratio function is defined as Hence , define empirical likelihood ratio where s ∈R¹,satisfying Condition A (1) θ0is the unique solution of equation EH ( X,θ,μ)=0 ; (2) H ( X,θ0 ,μ)is nondegenrate and <∞∈ΘrtE ( sup|H(X,θ0,t)|)μδ,for some r >2; ∑=n=+i(3 )1n 1 H2 (Xi ,θ0,μ?)Var(H(X,θ0,μ))οp(1); (4) 1 (,,?)1(,0,)(1)10n 1 HXnHXPnniiini∑θμ= ∑θμ+ο==; (5) {X i | i≥1}is strongly stationary φ-mixing random variables, and φ(i)is nonincreasing, ∑∞=<∞1()iφi; Theorem 1 Under the assumption of condition A,then as n →∞, l (θ0 )→dσA22χ(21). where ( (,,))2((,0,),(1,0,))1012 θμθμiθμiA VarHXCovHXHX+∞== +∑, σ2 = Var( H(X,θ0,μ)). As A 2and σ2are unknown,the above result could not be used in practice.We will use the blockwise empirical likelihood to overcome this shortcoming of the ordinary empirical likelihood. Put m = [n α], g=???2nm???,for 0 <α<12.for sake of simplicity,suppose that n = 2mg. (,,)(,,),,1,2,,(,,?)(,,?)(,,?)(,,?)2(1)10(21)0'212(21)10202(1)10(21)0ξθμθμξηηθμθμξθμθμiimimiiiiiimimiimimHXHXigYmYmHXHXHXHX?+???+?+?=++====++=++LLLL igYmYmHXHXiiiiiimim,,1,2,,(,,)(,,)''2''21(21)1020'LL====++??+ξηηθμθμ For {X i | i≥1}is strongly stationary random variables and H ( x,s,t)is continuous function,'Yi (i = 1,L,2g)has the same distribute function G ,and G2 gis their empirical distribute function. Then empirical likelihood ratio function is defined as = ???∏∑=≥∑=???= = =gigigiRgPi PiPiPimYi212121''''0' (θ)sup2|1,0,0. Hence , define empirical likelihood ratio ∑== ?=g+ilRmYi210'0' (θ)2log(θ)2log(1λ). Where λ∈R1, satisfying 021( )121' = ∑+==gi iimYmYK λgλ. Condition B (1) θ0is the unique solution of equation EH ( X,θ0 ,μ)=0; (2) H ( X,θ0 ,μ)is nondegerate and βE ( ts∈uΘμpδ|H(X,θ0 ,t)|)< ∞,for some β> 1 ?42α; (3) S 2 = S2'+οp(1),where ∑∑=== =giigiSgmYi SgmY212'21'222 21 ,21(); (4) ∑∑=== n+iipni1n 1 H(Xi ,θ0 ,μ?)1n1 H(X,θ0,μ)ο(1n); (5){X i , i≥1}is strongly stationaryφ-mixing random variable, φ(n)is nonincreasing, ∑<∞∞=()121iiφ; Theorem 2 Under the assumption of Condition B,then as n →∞, 20(1)l ' (θ)→dχ. It follows from Theorem2 that when n trend to infinite ,a 1 ?αconfidence regions for θ0is give for: P (θ0 |l'(θ0)≤Cα)≈1?α, where C αis the (1 ? α)-th quantile of the standard 2χ(1)distribute.