高维线性模型中的经验似然

J.Sys.Sci.&Math.Scis.27(1)(2007,2),124–133∗(100080)MR(2000)62E20,62F051Y=X Tβ+ε,(1.1) X kε0βT(Y1,X1),(Y2,X2),···,(Y n,X n).(1.1),β,βεββ[1,2],[2]Wilks[3],Owen(2001)(1.1)X k n(1.1),Owen[4],kχ2k.k*(10231030,10071090)2006-10-20.11252Owenβ∈R k 1≤i ≤n ,εi (β)=Y i −X T i β.{ε1,ε2,···,εn }β0{ε1(β0),ε2(β0),···,εn (β0)}β∈R k ,Owen[4]L (β)=sup{{p i }n i =1∈F (β)}nj =1p j ,(2.1)F (β)= {p i ≥0}ni =1:n j =1p j =1,n j =1p j X j εj (β)=0 .[4],LagrangeL (β)=n i =1p i (β),p i (β)=(n (1+λT X i εi (β)))−1,λ∈R kn i =1X i εi (β)(1+λT X i εi (β))=0.(2.2)Tn (β)=−2log n i =1(np i (β)).(2.3)β0,T n = T n (β0).k ,[4]T n d →χ2k .knT n p →∞.k ,V n =n i =1X i X T i .V nk ×kV n ≥0.λ1n =min eig (V n )λkn =max eig (V n ),min eig (·)max eig (·)M 2n =max 1≤i ≤n||X i ||2,||·||kλ1n ≤n i =1||X i ||2≤nM 2nλkn ≤n i =1||X i ||2≤nM 2n .||X 1||=0.(A1)r >0,E (|ε1|2+r )<∞.(A2)0≤δ<c (r ),nM 2n kλ1n=O (n δ),c (r )=r(2(2+r )).(A3)k =k (n )→∞(n →∞),k 32=O (n c (r )−δ)r <1;o (n 16−δ)1(A1)–(A3)H 0:β=β0T n −k √2kd →N (0,1).(2.4)12627kβ3E (ε21)=1.(A.2),1≤nM 2n kλ1n≤C 0n δnC 0>0λ1n ≥n 1−δM n(C 0k )≥n 1−δ||X 1||(C 0k ),n 1−δk −1nV nA =(a ij )k ×kk ×k||·||M ||A ||M =max 1≤i,j ≤k|a ij |,AB =A −I k =(b ij )k ×k ,I kk ×k λ1,λ2,···,λkτ1,τ2,···,τk A BAB =A −1−I k =( b ij )k ×k .min 1≤i ≤k|τi +1|=min 1≤i ≤k|λi |>0.1max 1≤i ≤k|λi |≤k ||A ||M .2a,b ∈R k ,||a ||=||b ||=1,|a T Ab |≤max 1≤i ≤k |λi |.3A|| B ||M ≤max 1≤i ≤kτi (1+τi ) =max 1≤i ≤k(λi −1)λi.BC ,B =C Tdiag(τ1,τ2,···,τk )C .A −1=(I k +B )−1=(I k +C T diag(τ1,···,τk )C )−1=C T diag((1+τ1)−1,(1+τ2)−1,···,(1+τk )−1)C=I k −C Tdiagτ1(1+τ1),τ2(1+τ2),···,τk (1+τk )C. B =−C T diag(τ1(1+τ1),τ2(1+τ2),···,τk (1+τk ))C .e iR ki11≤i ≤k .1≤i,j ≤k ,| b ij |=|e T i C T diagτ1(1+τ1),τ2(1+τ2),···,τk (1+τk )Ce j |.2| b ij |≤max 1≤i ≤k|τi(1+τi )|.|| B ||M ≤max 1≤i ≤k|τi (1+τi )|.1k ||B ||M a.s.→0,||B ||M|| B||M k ||B ||M a.s.→0,1max 1≤i ≤k|τi |≤k ||B ||M a.s.→0,3|| B ||M ≤max 1≤i ≤k|τi (1+τi )|.n|| B||M ≤2k ||B ||M (a.s.).B n =V −12n (n i =1X i X T i ε2i )V −12n −I k =n i =1(V −12n X i X T i V −12n )(ε2i −1)=(b ijn )k ×k ,B n 4(A1)–(A3)||B n ||M a.s.=O (kn −(b (r )−δ)log n ),b (r )=min(12,r(2+r )),b (r )>δ.1≤s,l ≤k ,b sln =e T s B n e l=n i =1(e T s V −12n X i )(X T i V −12n e l )(ε2i −1).1127a (sl )ni=(e T s V −12n X i )(X T i V −12n e l ),b sln =n i =1a (sl )ni (ε2i −1).(kn δ)−1b sln=ni =1(kn δ)−1a (sl )ni (ε2i −1)1≤i ≤n ,(kn δ)−1|a (sl )ni |≤(kn δλ1n )−1||X i ||2≤M 2n(kn δλ1n ).(A2)lim sup n →∞max 1≤s,l ≤k(kn δ)−1n i =1|a (sl )ni |≤lim sup n →∞n1−δM 2n(k λ1n)≤C 0<∞.d 1n =max1≤s,l ≤k,1≤i ≤n(kn δ)−1|a (sl )ni |.(A2)d 1n ≤M 2n(kn δλ1n )=O (n −1).(A1)C rE (|ε21−1|(2+r )2)≤C r (1+E (|ε1|2+r )<∞.Ghosh [5]3max 1≤i ≤n|ε2i −1|a.s.=o (n2(2+r )).[6]1,max 1≤s,l ≤k(kn δ)−1|b sln |a.s.=O (max(n22+rd 1n ,d 121n )log n ).||B n ||M =max 1≤s,l ≤k|b sln |a.s.=O (max(n −r 2+r ,n −12)n δk log n )a.s.=O (kn −(b (r )−δ)log n ).2(A3)||B n ||M a.s.=o (1).1≤i ≤n ,Z i =X i εi ,W n =n i =1X i X T i ε2i=n i =1Z i Z T i ,¯Z n =n −1n i =1X i εi .4[6]1,||V −12n n i =1(Z i −¯Z n )(Z i −¯Z n )TV −12n −I k ||M a.s.=O (kn −(b (r )−δ)log n ).λ1n =min eig (W n ).5(A1)–(A3)lim inf n →∞λ1n λn≥12(a.s.).B n =n i =1X i X T i (ε2i −1)=( b ijn )k ×k .4|| Bn ||M a.s.=O (kλ1n n −(b (r )−δ)log n ).a ∈R k ,||a ||=1,a T B n a =a T W n a −a T V n a .1|a T W n a −a T V n a |≤k || B n ||M a T V n a −k || B n ||M ≤a T W n a .λ1n −k || Bn ||M ≤a T W n a , λ1n ≥λ1n −k || B n ||M . λ1n λ1n≥1−O (kn −(b (r )−δ)log n ).(a.s.)UT n =n i =1V −12n X i εi T n i =1V −12n X i εi=n i =1n j =1X T i V −1n X j εi εj1≤i,j ≤n ,a ij =X T i V −1n X j ,A =(a ij )n ×n=(X 1,X 2,···,X n )T V −1n (X 1,X 2,···,X n ),B =diag(a 11,a 22,···,a nn ).12827T n=ni=1nj=1a ijεiεj.T n T nTn=T n−ni=1a iiε2i=1≤i=j≤na ijεiεj.[7] T n(clean)U n6(A1)–(A3)T n −E( T n)V ar( T n)d→N(0,1).[7]Uni=1a ii=ni=1X T i V−1nX i=trni=1V−1nX i X T i=tr(V−1nV n)=k,tr(·)E( T n)=0,E( T2n)=21≤i=j≤n a2ij=21≤i=j≤nX T i V−1nX j X T j V−1nX i=21≤i,j≤n X T i V−1nX j X T j V−1nX i−2ni=1(X T i V−1nX i)2=2ni=1X T i V−1nX i−2ni=1a2ii=2k−2ni=1a2ii.n i=1a2ii≤λ−21nni=1||X i||4≤nM4nλ21n=O(k2n−(1−2δ)),(A1)–(A3)k−1ni=1a2ii=O(kn−(1−2δ))=o(1).E( T2 n )(2k)→1V ar( T n)=2k(1+o(1)).E=A−B,µ1,µ2,···,µn E max1≤i≤n µ2ik→0,de Jong[7] T n CA−B=E=C T diag(µ1,µ2,···,µn)C.A2=A E2=C T diag(µ21,µ22,···,µ2n)C= A2−AB−BA+B2=A+B2−AB−BA.ni=1µ2i=tr(E2)=tr(A+B2−AB−BA)=tr(A)+tr(B2)−2tr(AB)=ni=1a ii+ni=1a2ii−2ni=1a2ii=ni=1a ii−ni=1a2ii=k−ni=1a2ii.1129ni =1µ2ik≤1.max 1≤i ≤nµ4i k 2≤n i =1µ4ik 2≤max1≤i ≤nµ2ik ,max 1≤i ≤n µ2i k →0⇐⇒ni =1µ4ik 2→0.1≤i ≤n ,µi =µik 12.(A −B )k 12=C T diag( µ1, µ1,···, µn )C =Ek 12.n i =1µ4i =tr(E k 12)4=t r (E 4)k 2=tr((A −B )4)k 2.tr((A −B )4)=tr((A +B 2−AB −BA )2)=tr(A −4AB +4AB 2−4AB 3+2ABAB )=n i =1a ii −4n i =1a 2ii +4n i =1a 3ii −4n i =1a 4ii +2ni =1n j =1a ii a jj a 2ij .k−2n i =1a 2ii =O (n−(1−2δ))=o (1)max 1≤i ≤n|a ii |≤M 2n λ1n=O (kn −(1−δ))=o (1).k −2n i =1a 3ii ≤k −2n i =1a 2ii max 1≤i ≤n|a ii |→0,k−2n i =1a 4ii≤k−2n i =1a 2iimax 1≤i ≤na 2ii →0.k−2n i =1a ii =1k=o (1),n i =1n j =1a ii a jj a 2ij≤ni =1a ii n j =1||X i ||2||X j ||4λ31n ≤knM 6nλ31n .k −2n i =1n j =1a ii a jj a 2ij ≤nM 6n kλ31n=O (k 2n −(2−3δ))=o (1).n i =1µ4ik 2=n i =1µ4i =tr ((A −B )4)k 2→0,max 1≤i ≤n µ2ik→0.E ( T 2n )(2k )→1,max 1≤i ≤nµ2iE ( T 2n )=k E ( T2n )·max 1≤i ≤nµ2ik→0.[7]5.2,T n −E ( T n ) E ( T2n )−(E ( T n ))2d →N (0,1).130277(A1)–(A3)T n −k √2kd →N (0,1).T n −k = T n +n i =1a ii (ε2i −1).ni =1a ii (ε2i −1)k 12=o p (1)[6]1,ni =1a ii (ε2i −1)a.s.=O (kn−(b (r )−δ)log n )a.s.=o (1).6T n −k √2k = T n √2k +o p (1)= T n −E ( T n )Var( Tn )·Var( T n )√2k +o p (1)d→N (0,1).1≤i ≤n ,p i =p i (β0)=(n (1+λT X i εi ))−1=(n (1+λT Z i ))−1,Tn = T n (β0)=−2log n i =1(n p i ) =2n i =1log(1+λT Z i ),λni =1Z i (1+λT Z i )−1=0.λ=ρθ,ρ≥0,θ∈R k ,||θ||=1.β=β08(A1)–(A3)||λ||=O p ((kn )12M nλ1n).Z ∗n =max 1≤i ≤n||Z i ||.[5]3,Z ∗n ≤M n max 1≤i ≤n|εi |a.s.=o (M n n 12+r ).ni =1Z i (1+λT Z i )−1=0,0=n i =1λT Z i (1+λT Z i )=λT n i =1Z i −λTn i =1Z i Z T i (1+λT Z i )λ.λTn i =1Z i =λTn i =1Z i Z Ti (1+λT Z i )−1λ,ρθTni =1Z i =ρ2n i =1θT Z i Z T i θ(1+ρθT Z i ).|θT ni =1Z i |≥ λ1n ρ1+ρZ ∗n =⇒ρ≤||ni =1Z i ||( λ1n −Z ∗n||n i =1Z i ||).1131||n i =1Z i ||7,(V −12n n i =1Z i )T(V −12n n i =1Z i )−k√2kd→N (0,1),(ni =1Z i )T V −1n (ni =1Z i )=O p (k ).λ−1kn ||n i =1Z i ||2≤(ni =1Z i )T V −1n (ni =1Z i )λkn ≤nM 2n ,||ni =1Z i ||=O p ((kn )12M n ).Z ∗n ||ni =1Z i ||=o p (k 12n4+r 2(2+r )M 2n ).5(A1)–(A3),λ−11n Z ∗n||n i =1Z i ||=o p (k 32n −(r2(2+r )−δ))=o p (1).ρ=O p ((kn )12M n λ1n ),||λ||=O p ((kn )12M nλ1n).1ni =1Z i (1+λT Z i )−1=0,n i =1Z i =n i =1Z i Z T i (1+λT Z i )λ=W n λ−n i =1(λT Z i )2Z i(1+λT Z i ).R n =ni =1(λT Z i )2Z i (1+λT Z i )−1,λ=W −1nn i =1Z i +W −1n R n .W −1n R nmax 1≤i ≤n|λT Z i |≤||λ||Z ∗n =o p (k 32n −(r2(2+r )−δ))=o p (1),||R n ||≤n i =1 (λT Z i )2Z i (1+λT Z i ) ≤n i =1||λ||2||Z i ||3|(1+λT Z i )|≤||λ||2M 3n (1−||λ||Z ∗n )n i =1|εi |3n[5]3,n i =1|εi |3a.s.=O (n )r ≥1;o (n 3(2+r ))r <1.58,||W −1n R n ||≤ λ−11n ||R n ||≤ λ−11n ||λ||2M 3n(1−||λ||Z ∗n)n i =1|εi |3=O p (k 4n −1+3δ)r ≥1;o p (k 4n −(1+2r2+r−3δ))r <1.(A3)k 32n −(c (r )−δ)=O (1),||W −1n R n ||=o p (1).|log(1+x )−x +x 22|≤|x |3x >−1|log(1+λT Z i )−λT Z i +(λT Z i )22|≤|λT Z i |3132271≤i ≤nmax 1≤i ≤n|λT Z i |=o p (1).Rn =n i =1|λT Z i |3≤||λ||3M 3nn i =1|εi |3=O p (k 92n −(12−3δ))r ≥1;o p (k 92n −(3r2(2+r )−3δ))r <1.Rn =o p (1). T n =2ni =1log(1+λTZ i )=2n i =1λT Z i −λTZ i Z T iλ2+ Rn =n i =1Z i TW −1nn i =1Z i−R T n W −1n R n +o p (1).|R T n W −1n R n |≤λ−11n||R n ||2≤λ−11n||λ||4M 6n n i =1|εi |32=O p (k 7n −(1−5δ))r ≥1;o p (k 7n −(3r2+r −5δ))r <1.k 32n −(c (r )−δ)=O (1)|R T n W −1n R n |=o p (1).T n =(n i =1Z i )T W −1n (n i =1Z i )+o p (1) = T ∗n+o p (1). Tn T ∗nT ∗n = T ∗n −T n +T n = n i =1Z iT(W −1n −V −1n ) ni =1Z i +T n =:R ∗n +T n .7(T n −k )(2k )12d→N (0,1).R ∗nk 12p→0λ∗kV 12n W −1n V 12n −I kR ∗n(2k )12=V −12n n i =1Z iT(V 12n W −1n V 12n−I k ) V −12n n i =1Z i(2k )12,|R ∗n |(2k )12≤λ∗kn i =1Z i T V −1n n i =1Z i (2k )12=λ∗k (T n −k +k )(2k )12≤λ∗k k 212+λ∗k O p (1).λ∗k k 12p→0.1λ∗k ≤k ||V 12W −1n V 12n −I k ||M ,4||V −12n W n V −12n −I k ||Ma.s.=O (kn −(b (r )−δ)log n )a.s.=o (1).n13,||V 12n W −1n V 12n −I k ||M ≤2k ||V −12n W n V −12n−I k ||M a.s.=O (k 2n −(b (r )−δ)log n ).n ,λ∗k k 12≤2k 32||V 12n W −1n V 12n −I k ||M a.s.=O (k 52n−(b (r )−δ)log n )a.s.=o (1).1133R∗n k12=o p(1).7( T∗n−k)(2k)12d→N(0,1),( T n−k)(2k)12d→N(0,1).[1]Owen A.Empirical likelihood ratio confidence intervals for a single functional.Biometrika,1988,75:237–249.[2]Owen A.Empirical likelihood confidence regions.Ann.Statist.,1990,18:90–120.[3]Owen A.Empirical likelihood.Chapman&Hall,New York,2001.[4]Owen A.Empirical likelihood for linear models.Ann.Statist.,1991,19:1725–1747.[5]Ghosh M.A note on bootstrapping the sample median.Ann.Statist.,1984,12:1130–1135.[6]Shi J and Lau T S.Empirical likelihood for partially linear models.J.Multivariate Analysis,2000,72:132–148.[7]de Jong P.A central limit theorem for generalized quadratic forms.Probab.Theory&RelatedFields,75:261–277.EMPIRICAL LIKELIHOOD FOR HIGHER DIMENSIONALLINEAR MODELSShi Jian(Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing100080) Abstract Empirical likelihood for higher dimensional linear model is studied in this paper.When the dimension number of covariates increases with the sample size,conventional empirical likelihood inference becomes invalid.To this end,some mild regularity conditions are imposed and an asymptotic distribution theory is established for modified statistic.Key words Linear models,empirical likelihood,higher dimensional convariates.。