On Bartlett Correction of Empirical Likelihood in the
Presence of Nuisance Parameters
BY SONG XI CHEN
Department of Statistics, Iowa State University, Ames, Iowa 50011-1210, USA
songchen@iastate.edu
AND
HENGJIAN CUI
Department of Statistics and Financial Mathematics
Beijing Normal University, 100875, China
hjcui@bnu.eud.cn
SUMMARY
Lazar & Mykland (1999) showed that an empirical likelihood defined by two estimating
equations with a nuisance parameter need not be Bartlett correctable. This paper shows that
Bartlett correction of empirical likelihood in the presence of a nuisance parameter depends
critically on the way the nuisance parameter is removed when formulating the likelihood for
the parameter of interest. We establish in the broad framework of estimating functions that
the empirical likelihood is still Bartlett-correctable if the nuisance parameter is profiled out
given the value of the parameter of interest.
Some key words: Bartlett correction; Empirical likelihood; Estimation equation; Nuisance
parameter.
1. INTRODUCTION
Since its introduction by Owen (1988, 1990), empirical likelihood has become an useful
tool for conducting nonparametric or semiparametric inference. Empirical likelihood has
been shown in a wide range of situations as outlined in Owen (2001) to admit limiting chisquared distributions, which is a nonparametric version of the Wilks theorem in the context
of parametric likelihood. Another key property of empirical likelihood which also resembles
1
that of a parametric likelihood is Bartlett correction. Bartlett-correctability is a secondorder property which implies that a simple mean adjustment to the likelihood ratio leads to
its distributional approximation to the limiting chi-squared distribution being improved by
one order of magnitude.
That the empirical likelihood is Bartlett-correctable has been established for a range of
situations; see for example Hall and La Scala (1990) for the case of the mean parameter,
DiCiccio et al. (1991) for smooth functions of means, Chen & Hall (1993) for quantiles,
Chen (1993, 1994) for linear regression and Cui & Yuan (2001) for quantiles in the presence
of auxiliary information. Jing & Wood (1996) showed that the exponentially tilted empirical
likelihood for the mean is not Bartlett-correctable. Indeed, Corcoran (1998) showed that
Kullback-Leibler divergence is the unique member of a large class of divergence measures
that produces Bartlett-correctable empirical likelihood statistics. However, Lazar & Mykland
(1999) showed that in some circumstances, where the empirical likelihood is defined by two
estimating equations and when a nuisance parameter is present, even the use of KullbackLeibler divergence can fail to guarantee Bartlett correctability.
In contrast to the result of Lazar & Mykland (1999), Chen (1994) had earlier proved that
empirical likelihood is Bartlett-correctable in the context of simple linear regression when
one coefficient is treated as a nuisance parameter. It appears that the result obtained by
Lazar & Mykland (1999) is due to absence of a regular Edgeworth expansion for the signed
square root of the empirical likelihood ratio.
In the present paper, we confirm that the result of Chen (1994) holds in general. We
consider the Bartlett property in a broader situation where there are r estimating equations
and the dimension of the nuisance parameter is p, with p < r, which is within the framework
of the empirical likelihood for generalised estimating equations introduced in Qin & Lawless
(1994). It is found that, if the nuisance parameter is profiled out given the parameter of
interest, the empirical likelihood is still Bartlett-correctable. This indicates that the Bartlett
correctability of the empirical likelihood is dependent on the method of nuisance parameter
2
removal when one formulating the likelihood for the parameter of interest, rather than on any
fundamental differences between estimating equations and the smooth function of means.
It is expected that a corresponding result holds for parametric likelihood as well, namely
that the Bartlett correction property only holds in general when the nuisance parameter is
‘profiled out’.
The paper is organized as follows. In Section 2 we establish the empirical likelihood
defined on a set of generalized estimating equations in the presence of nuisance parameters.
The Bartlett correction is established in Section 3. All the algebraic manipulations required
to establish the Bartlett correction is given in the Appendix.
2. EMPIRICAL LIKELIHOOD WITH NUISANCE PARAMETERS
Consider a random vector X with unknown distribution function F which depends on a
r-dimensional parameter (θ, ψ) ∈ Rr−p × Rp . Here the interest is on the parameter θ while
treating ψ as a p-dimensional nuisance parameter. We assume that the parameter (θ, ψ) is
defined by r (r > p) functionally unbiased estimating equations g j (x, θ, ψ), j = 1, 2, · · · , r
such that E{g j (X1 , θ0, ψ0 )} = 0 where (θ0, ψ0 ) is the true parameter value. In particularly,
we define
g(X, θ, ψ) = g 1 (X, θ, ψ), g 2 (X, θ, ψ), · · · , g r (X, θ, ψ)
T
.
Assume that {X1 , X2 , · · · , Xn } is an independent and identically distributed sample drawn
from F . Let V = Cov{g(Xi , θ0, ψ0)} and we assume the following regularity conditions:
(i) V is a r × r positive definite matrix and the rank of E{∂g(X, θ0, ψ0)/∂ψ} is p; (1)
(ii) For any j, 1 ≤ j ≤ p, all the fourth order partial derivatives of g j (x, θ0, ψ) with
respect to ψ are continuous in a neighborhood of θ0 and are bounded by some
integrable function G(x) in the neighborhood;
(iii) Eg(X, θ0 , ψ0)15 < ∞ and the characteristic function of g(X, θ0 , ψ0) satisfies
the Cramér condition: lim sup|t|→∞ |E[exp{itT g(X, θ0 , ψ0)}]| < 1.
3
To simplify derivations, let us first rotate the original estimating functions by defining
wi (θ, ψ) =: T V −1/2g(Xi , θ, ψ),
where T is a r × r orthogonal matrix such that
T V −1/2E
∂g(X, θ
0 , ψ0 )
T
U= Λ 0
∂ψ
r×p
U = (ukl )p×p is an orthogonal matrix and Λ = diag(λ1 , · · · , λp ) is a non-singular p × p
diagonal matrix. Furthermore, let us define Ω = ω kl
p×p
= UΛ−1 .
Let p1 , · · · , pn be non-negative weights allocated to the observations. The empirical likelihood for the parameter (θ, ψ) is
L(θ, ψ) =
subject to
pi = 1 and the constraints
n
pi
i=1
pi wi (θ, ψ) = 0. Let (θ, ψ) = −2 log{nn L(θ, ψ)}
be the log empirical likelihood ratio. Standard derivations in the empirical likelihood show
that
(θ, ψ) = 2
n
log{1 + λT (θ, ψ)wi(θ, ψ)},
i=1
where λ(θ, ψ) satisfies:
n−1
n
i=1
1+
wi (θ, ψ)
T
λ (θ, ψ)wi (θ, ψ)
= 0.
(2)
To obtain the empirical likelihood ratio at θ0 , we need to profile out the nuisance parameter ψ. To simplify notation, let us write wi (ψ) = wi (θ0 , ψ) and let ψ̃ =: ψ̃(θ0 ) be the minima
of (θ0 , ψ) given θ = θ0 and λ̃ = λ(θ0 , ψ̃) be the solution of (2) at (θ0, ψ̃). Let (θ̂, ψ̂) be the
maximum empirical likelihood estimate of parameter (θ, ψ). Since the number of estimating
functions equal to the dimension of parameter (θ, ψ), then (θ̂, ψ̂) = 0. This means that the
log empirical likelihood ratio for θ0 is just
r(θ0 ) =: (θ0 , ψ̃(θ)) = 2
n
i=1
4
log{1 + λ̃T wi (ψ̃)}.
In order to develop an expansion for r(θ0 ), we need to derive expansions for λ̃ and ψ̃
first. We notice from Qin and Lawless (1994) that (λ̃, ψ̃) are the solutions of
Q1n (λ, ψ) = n−1
Q2n (λ, ψ) = n−1
n
i=1
n
i=1
wi (ψ)
=0
1 + λT wi (ψ)
(3)
(∂wi(ψ)/∂ψ)T λ
= 0.
1 + λT wi (ψ)
(4)
Let η = (λT , ψ T )T , η0 = (0, ψ0 ),
⎛
⎞
⎜
Q1n (η) ⎟
Q(η) = ⎝
Q2n (η)
⎠
⎛
and S = E
⎞
∂Q(0, ψ0) ⎜ −I S12 ⎟
=⎝
⎠,
∂η
S21 0
T
. To facilitate easy expressions, we standardize Q to
where S21 = U(Λ, 0) and S12 = S21
Γ(η) = S −1 Q(η). Let wij (ψ) and Γj (η) denote respectively the j-th component of wi(ψ) and
Γ(η). The following α − A system of notations was first used by DiCiccio, Hall and Romano
(1988):
αj1 ...jk = E{wj1 (ψ0 )...wjk (ψ0)}
Aj1 ...jk = n−1
n
wj1 (ψ0 )...wjk (ψ0) − αj1 ...jk .
i=1
We also need to define
β j,j1 ...jk = E
∂ k Γj (0, ψ
0)
∂ηj1 ...∂ηjk
,
B j,j1 ...jk =
∂ k Γj (0, ψ0)
− β j,j1 ...jk
∂ηj1 ...∂ηjk
and
∂ n wip (ψ0) ∂ lwij (ψ0) ∂ m wik (ψ0)
...
∂ψ j1 ...∂ψ jl ∂ψ k1 ...∂ψ km ∂ψ p1...∂ψ pn
n
1
∂ n wip (ψ0)
∂ lwij (ψ0 ) ∂ m wik (ψ0 )
=
...
n i=1 ∂ψ j1 ...∂ψ jl ∂ψ k1 ...∂ψ km ∂ψ p1...∂ψ pn
γ j,j1 ...jl ;k,k1 ...km ;...;p,p1...pn = E
C j,j1 ...jl ;k,k1 ...km ;...;p,p1...pn
− γ j,j1 ...jl ;k,k1 ...km ;...;p,p1...pn .
3. EXPANSIONS TO THE LIKELIHOOD RATIO
5
Since
0 = Γj (ψ̃(θ0 ), λ̃) = B j + β j,k (η̃ k − η0k ) + B j,k (η̃ k − η0k )
+
1 j,kl k
β (η̃
2
− η0k )(η̃ l − η0l ) + 12 B j,kl (η̃ k − η0k )(η̃ l − η0l )
+
1 j,klm k
β
(η̃
6
+
1 j,klmn k
β
(η̃
24
− η0k )(η̃ l − η0l )(η̃ m − η0m )(η̃ n − η0n )
+
1
B j,klmn (η̃ k
24
− η0k )(η̃ l − η0l )(η̃ m − η0m )(η̃ n − η0n ) + Op (n−5/2 ).
− η0k )(η̃ l − η0l )(η̃ m − η0m ) + 16 B j,klm (η̃ k − η0k )(η̃ l − η0l )(η̃ m − η0m )
Here and throughout the paper, we use the tensor notation where if a superscript is repeated
a summation over that superscript is understood. After inverting the above expansion we
have for j ∈ {1, · · · , r + p},
η̃ j − η0j = −B j + B j,k B k − 12 β j,kl B k B l − B j,k B k,lB l + 12 β k,lmB j,k B l B m
+ β j,kl B k,m B mB l − 12 β j,kl β k,mn B m B n B l − 12 B j,kl B k B l + 16 β j,klm B k B lB m + Op (n−2 )
where j, k, l, m, ∈ {1, 2, · · · , r + p}. This implies that for j ∈ {1, · · · , r}
λ̃j = −B j + B j,q B q − 12 β j,uq B u B q − B j,u B u,q B q + 12 β u,qsB j,u B q B s + β j,uq B u,s B s B q
−
1 j,uq u,st s t q
β β B BB
2
− 12 B j,uq B u B q + 16 β j,uqs B u B q B s + Op (n−2 )
(5)
where q, s, t, u ∈ {1, · · · , r + p}, and for k ∈ {1, · · · , p}
ψ̃ k = −B r+k + B r+k,q B q − 12 β r+k,uq B u B q − B r+k,u B u,q B q + 12 β u,qs B r+k,u B q B s
+ β r+k,uq B u,s B s B q − 12 β r+k,uq β u,st B s B tB q − 12 B r+k,uq B u B q + 16 β r+k,uqs B u B q B s
+ Op (n−2 ).
(6)
Derivations given in the appendix show that for a ∈ {1, · · · , r − p}
n−1 (θ0 ) = Ap+a Ap+a − Ap+a
p+b
Ap+aAp+b − 2ω kl C p+a,k Ap+a Al
+ 2γ p+a;p+b,k ω kl Ap+a Ap+b Al + 23 αp+a
p+b p+c
Ap+aAp+b Ap+c
+ γ p+a,kl ω km ω ln Ap+aAm An + Aji B i,q B q B j [2, i, j] − B j,u B j,q B u B q
6
− 2C j,k B j,q B r+k B q + γ j,kl B r+k B r+l B j,q B q + 2γ j,kl B j B r+l B r+k,q B q
− 2γ j;i,l (B j B i B r+l,q B q + B r+l B i B j,q B q [2, j, i]) + 2αjih B j B i B h,q B q
+ ( 12 β j,uq β r+k,st γ j,k − 14 β j,uq β j,st )B u B q B s B t − 12 γ j,kl β j,uq B u B q B r+l B r+k
+ (γ i;j,k β i,uq + γ j;i,k β i,uq − γ j,kl β r+l,pq )B u B q B j B r+k + 2γ j;i;h,k B j B i B h B r+k
− (γ j;i,lk + γ j,l;i,k )B j B i B r+l B r+k + 13 γ j,klm B j B r+k B r+l B r+m
−
1 jihg j i h g
α B BB B
2
+ (γ j;i,l β r+l,uq − αjih β h,uq )B j B i B u B q
− C j,kl B j B r+k B r+l + 2C j;i,l B j B iB r+l − 23 Ajih B j B iB h + Op (n−5/2).
(7)
Let R = R1 + R2 + R3 be a signed root decomposition of n−1 (θ0 ) such that
n−1 (θ0 ) = Rq Rq + O(n−5/2 )
where Rj = Op (n−j/2 ) for j = 1, 2 and 3. Clearly, R1 and R2 can be determined from the
terms of Op (n−1 ) and Op (n−3/2) respectively in (7). Specifically, for a, b, c, d, e ∈ {1, · · · , r−p}
and l, k, m, n, o, v, m, n ∈ {1, · · · , p}
Rq1 = Rq2 = 0 for q ≤ p,
Rp+a
= − 12 Ap+a
2
+
p+b
Rp+a
= Ap+a
1
and
Ap+b − ω kl C p+a,k Al + γ p+a;p+b,k ω kl Ap+b Al
1 p+a p+b p+c p+b p+c
α
A A
3
+ 12 γ p+a,kl ω km ω ln Am An .
p+a
from (7) and expressing all the remaining
After removing terms induced by Rp+a
2 R2
p+a
p+a
terms in terms of As and Cs, we have Rq3 = 0 for q ≤ p and Rp+a
= Rp+a
3
31 + R32 + R33
where
Rp+a
=
31
−
3 p+a p+c p+c p+b p+b
A
A
A
8
+ ω ml C p+b,m Al
1 ml nl p+a,m p+b,n p+b
ω ω C
C
A
2
− αl
p+a p+b
p+a
Ap+b + 12 ω lm C p+b,l Ap+a
p+b
Am
+ ω ml ω kn C l,k C p+a,m An + ω lm C p+a;p+b,l Ap+b Am
ω nl C p+c,n Ap+b Ap+c − αp+a
−
5 p+a p+b p+c p+c p+d p+b p+d
α
A
A A
6
+
1 p+a p+b p+c p+b p+c
A
A A
3
+ αl
p+b p+c
ω mn C p+c,m Ap+b An
+ 12 ω km ω ln C p+a,kl Am An
p+a p+b
7
ω kl γ p+c;p+d,k + 49 αp+a
p+b p+e p+e p+c p+d
α
−
1 kl ml p+a;p+b,k p+c;p+d,m
ω ω γ
γ
2
− 14 αp+a
p+b p+c p+d
Ap+b Ap+c Ap+d ,
Rp+a
= − 12 γ m,kl ω kn ω lo ω vm C p+a,v An Ao − 14 γ p+b,kl ω kn ω lm Ap+a
32
+ γ p+b,kl ω lo ω kn ω vn C p+a,v Ap+b Ao − γ p+b,kl ω lo ω kn An
p+a
p+b
Am An
Ap+b Ao
− γ p+a,kl ω ln ω mo ω kv C v,m Ap+a An Ao + γ p+a;p+b,lω ln ω on C p+c,o Ap+b Ap+c
− γ p+a;p+b,l ω ln An
p+c
Ap+bAp+c − γ p+a;p+b,r+m ω mn ω lo C o,m Ap+b An
− (γ m;p+a,l + γ p+a;m,l )ω ln ω om C p+b,o Ap+b An − (γ p+a;p+b,l + γ p+a;p+c,l )ω ln ω ko C p+b,k An Ao
− ( 12 γ p+c;p+a,l + γ p+a;p+c,l )ω ln Ap+b
Rp+a
=
33
1 kl p+a p+b p+c p+c,k p+b l
ω α
C
A A
3
p+c
Ap+b An
and
+ 12 ω m m ω n n ω ov ω lk γ k,m n γ p+a,ol
+ γ p+b,m n γ p+a;p+b,o − 13 γ p+a,m n o AmAn Av
+ ωm mωn n
1 p+a p+b p+c p+c,m n
α
γ
3
+
1 p+a,m ;p+c p+b;p+c,n
γ
γ
2
−
1 ol kl p+a,m o p+b,n k
ω ω γ
γ
2
+ ω n n ω ol αl
+ ω lo γ p+b,n l (γ o,m ;p+a + γ o;p+a,m ) + 12 ω lo γ o,m n γ p+a;p+b,l
− 12 γ p+a;p+b,m n − 12 γ p+a,m ;p+b,n Ap+b Am An
p+a p+b p+c,on
γ
+ 12 γ p+c,m ;p+a (γ p+c;p+b,n + γ p+b;p+c,n )
+ αp+a
p+b p+d 2 p+d;p+c,n
(3γ
+ γ p+c;p+d,n )
− γ p+a;p+b;p+c,n − ω ol γ p+a;p+b,l γ p+c,on Ap+b Ap+c An .
4. BARTLETT CORRECTABILITY
The key in checking if the empirical likelihood is Bartlett correctable or not is to examine
if the third and the fourth order joint cumulants of R are at the orders of n−3 and n−4
respectively. This is the path taken by DiCiccio, Hall and Romano (1991), Jing and Wood
(1996) and Lazar and Mykland (1999). A formal establishment of the Bartlett correction
can be made by developing Edgeworth expansions for the empirical likelihood ratio under
condition (1).
The joint third-order cumulants of R is
cum(Rp+a , Rp+b , Rp+c )
= E(Rp+a Rp+b Rp+c ) − E(Rp+a )E(Rp+b Rp+c )[3] + 2E(Rp+a )E(Rp+b )E(Rp+c )[3]
8
p+b p+c
p+a p+b p+c
p+a
p+b p+c
−3
= E(Rp+a
1 R1 R1 ) + E(R2 R1 R1 )[3] − E(R2 )E(R1 R1 )[3] + O(n ).
Note that
p+e
−1 de
E(Rp+d
1 R1 ) = n δ ,
p+d p+e
−2 p+a
E(Rp+a
1 R1 R1 ) = n α
−1
E(Rp+a
− 16 αp+a
2 ) = n
p+b p+b
p+d p+e
− ω kl γ p+a,k;l + 12 ω km ω lm γ p+a,kl + O(n−2 ).
p+d p+e
By working out E(Rp+a
2 R1 R1 ) it may be shown that
p+d p+e
p+a
p+d p+e
p+a p+d p+e
1
−3
E(Rp+a
2 R1 R1 ) = E(R2 )E(R1 R1 ) − 3 E(R1 R1 R1 ) + O(n )
(8)
which readily implies that
cum(Rp+a , Rp+b , Rp+c ) = O(n−3 ).
(9)
The joint fourth-order cumulants of R is
cum(Rp+a , Rp+b , Rp+c , Rp+d )
= E(Rp+a Rp+b Rp+c Rp+d ) − E(Rp+a Rp+b )E(Rp+c Rp+d )[3] − E(Rp+a )E(Rp+b Rp+c Rp+d )[4]
+ 2E(Rp+a )E(Rp+b )E(Rp+c Rp+d )[6] − 6E(Rp+a )E(Rp+b )E(Rp+c )E(Rp+d )
p+b p+c p+d
p+a p+b p+c p+d
p+a p+b p+c p+d
= E(Rp+a
1 R1 R1 R1 ) + E(R2 R1 R1 R1 )[4] + E(R3 R1 R1 R1 )[4]
(10)
p+b p+c p+d
p+a p+b
p+c p+d
+ E(Rp+a
2 R2 R1 R1 )[6] − E(R1 R1 )E(R1 R1 )[3]
p+b
p+c p+d
p+a p+b
p+c p+d
− E(Rp+a
2 R1 )E(R1 R1 )[12] − E(R3 R1 )E(R1 R1 )[12]
p+b
p+c p+d
p+a
p+b p+c p+c
− E(Rp+a
2 R2 )E(R1 R1 )[6] − E(R2 )E(R1 R1 R1 )[4]
p+b p+c p+d
p+a
p+b
p+c p+d
−4
− E(Rp+a
2 )E(R2 R1 R1 )[12] + 2E(R2 )E(R2 )E(R1 R1 )[6] + O(n ).
From (8), we have
p+b p+c p+d
p+b p+c p+d
p+b
p+c p+d
−4
E(Rp+a
2 ){E(R1 R1 R1 )[4]+E(R2 R1 R1 )[12]−2E(R2 )E(R1 R1 )[6]} = O(n )
which means the sum of the last three terms in (10) is O(n−4 ). We examine in the following
the other terms in (10).
9
Let
t1 = αp+a
p+b p+c p+d
t3 = αp+a
p+b p+c
+ αp+b
t4 = αp+a
, t2 = δ ab δ cd + δ acδ bd + δ ad δ bc ,
αp+d
p+e p+e
p+c p+d p+a p+e p+e
α
p+b p+e
αp+c
p+d p+e
+ αp+a
p+b p+d
αp+c
p+e p+e
+ αp+a
p+c p+d p+b p+e p+e
p+c p+e p+b p+d p+e
+ αp+a
p+d p+e p+b p+c p+e
α
and
+ αp+a
α
α
.
It is easy to check that
p+b p+c p+d
p+a p+b
p+c p+d
−3
−4
E(Rp+a
1 R1 R1 R1 ) − E(R1 R1 )E(R1 R1 )[3] = n (t1 − t2 ) + O(n ).
(11)
Derivations given in the appendix show that
p+b p+c p+d
p+a p+b
p+c p+d
E(Rp+a
2 R1 R1 R1 )[4] − E(R2 R1 )E(R1 R1 )[12]
= n−3 [−6t1 + 2t2 − 16 t3 + 23 t4 − {ω kl γ p+a,k;l αp+b
+ 12 {γ p+a,kl ω km ω lm αp+b
p+c p+d
p+c p+d
}[4]
}[4]] + O(n−4 ),
(12)
p+b p+c p+d
p+a p+b
p+c p+d
E(Rp+a
2 R2 R1 R1 )[6] − E(R2 R2 )E(R1 R1 )[6]
= n−3 {3t1 − t2 + 16 t3 − 59 t4 + 13 ω kl (γ p+a,k;l αp+b
− 16 ω km ω lm γ p+a,kl αp+b
p+c p+d
p+c p+d
[2, a, b][6]} + O(n−4 )
)[2, a, b][6]
(13)
and
p+b p+c p+d
p+a p+b
p+c p+d
−3
1
−4
E(Rp+a
3 R1 R1 R1 )[4] − E(R3 R1 )E(R1 R1 )[12] = n (2t1 − 9 t4 ) + O(n ). (14)
Combining (11), (12), (13) and (14), we see all the terms of order n−3 cancel each other
and hence
cum(Rp+a , Rp+b , Rp+c , Rp+d ) = O(n−4 )
(15)
This and (9) mean that the empirical likelihood ratio for θ0 admits Bartlett correction despite
the presence of the nuisance parameters.
10
5. AN EXAMPLE
To connect our finding with that of Lazar and Mykland (1999), we present here an
example which would mean the empirical likelihood was not Bartlett correctable by applying
the results of Lazar and Mykland (1999), but it is Bartlett correctable based on some known
result.
Assume a bivariate random vector X = (Z, Y )T has a distribution F , and {Xi =
(Zi , Yi )}ni=1 be an independent and identically distributed sample drawn from F . Let θ0 =
E(X) = 0 and ψ = E(Y ). The interest here is to infer θ while treating ψ as a nuisance
parameter. There are two estimating equations: g 1 (X, θ, ψ) = Z −θ and g 2 (X, θ, ψ) = Y −ψ.
We further assume that Cov(X) = I2, E(Z 3 ) = 0 and X admits other conditions assumed
in (1).
To be consistent with the notations used in Lazar and Mykland (1999), we let
⎛
U. =
xi ,
T
(yi − ψ), 0
,
U.. =
⎜
⎜
⎜
⎜
⎜
⎝
− x2
−
i
xi (yi − ψ)
−
−
⎞
xi (yi − ψ)
0
(yi − ψ)2
−n
0 ⎟
⎟
⎟
,
−n ⎟
⎟
0
⎠
κr = n−1 E(Ur ), κrs = n−1 E(Urs ), κr,s = n−1 Cov(Ur , Us ),
κr,ts = n−1 Cov(Ur , Uts ) and
κrs,tu = n−1 Cov(Urs , Utu ).
It is easy to see that κr =: EUr = 0,
⎛
(κrs ) =
⎜
⎜
⎜
⎜
⎜
⎝
−1
0
⎞
⎛
⎞
⎛
0 ⎟
⎜
⎜
⎜
⎜
⎜
⎝
1 0 0 ⎟
⎜
⎜
⎜
⎜
⎜
⎝
0
−1 −1
0
−1
0
⎟
⎟
⎟,
⎟
⎠
and (κr,s ) = (κr,s )+ , where
(κr,s ) =
+
0 1 0
0 0 0
⎟
⎟
⎟,
⎟
⎠
(κrs ) = (κrs )−1 =
⎞
−1
0
0
0
−1
0
−1
1
stands for the Moore-Penrose inverse.
i
i
= κi,ακα,rs = κi,rs , βrst
= κi,α κα,rst = κi,rst for i = 1, 2, 3, and
Define βrs
i
Vi
Ur = Vr , Urs = Vrs + βrs
i
i
and Urst = Vrst + βrs
Vit [3] + βrst
Vi .
11
0 ⎟
⎟
⎟
⎟
⎟
⎠
Moreover let v denote the cumulants of V . These notations are fully consistent with Lazar
and Mykland (1999). It is easy to verify that (v rs) = (κrs ), v 11 = v 23 = −v 33 = −1, v 13 = 0
and v111 = −E(x3), and v113 = v133 = v13,13 = v1133 = 0. Hence b0 = b1 = c = 0 in equation
(5) of Lazar and Mykland (1999). Moreover,
ω̃4 =
37 −1
n v111v111v 11v 11v 11 + O(n−2 )
18
= − 37
n−1 {E(x3)}2 + O(n−2 ), ρ4 = − 37
{E(x3 )}2 = 0.
18
18
This means that the fourth order cumulants of R were not at the order of n−4 , and thus the
empirical likelihood would not be Bartlett correctable.
However, we note that the empirical likelihood ratio statistic
(θ0 ) = 2
n
log{1 + λ̂1 g 1 (Xi , 0, ψ̂) + λ̂2 g 2 (Xi , 0, ψ̂)}
i=1
where λ̂1 , λ̂2 , ψ̂ are the solutions of
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
n
xi
= 0,
2
i + λ (yi − ψ)
i=1 1 +
n
yi − ψ
= 0 and
1
2
i=1 1 + λ xi + λ (yi − ψ)
n
−λ2
= 0.
1
2
i=1 1 + λ xi + λ (yi − ψ)
λ1 x
From the third equation, λ̂2 = 0, and λ̂1 should satisfy
Therefore, (θ0 ) = 2
n
xi
i=1
1 + λ̂1 xi
n
i=1
n
= 0 and
ψ̂ = i=1
n
i=1
yi /(1 + λ̂1 xi )
1/(1 + λ̂1 xi )
.
log(1 + λ̂1 xi ). This is essentially the empirical likelihood for the
mean, and is known to be Bartlett correctable.
6. DISCUSSION
The Bartlett factor for the empirical likelihood ratio (θ0 ) can be derived by deriving the
first two cumulants of R.
p+a
−1 p+a
Since E[Rp+a
+ O(n−2 ) where μp+a = − 16 αp+a
1 ] = 0 and E[R2 ] = n μ
ω kl γ p+a,k;l + 12 γ p+a,kl ω km ω lm , we have cum(Rp+a ) = n−1 μp+a + O(n−2 ).
12
p+b p+b
−
Derivations given in the appendix show that
cum(Rp+a , Rp+e ) = n−1 δ ae + n−2 Δae + O(n−3 )
(16)
where
Δae =
1 p+a p+e p+b p+b
α
2
− 13 αp+a
p+b p+c
αp+e
p+b p+c
− ω kl γ p+a,k;l;p+e [2, a, e] − 12 ω km ω ln γ p+a,kl αmn
+ ω ml [γ p+b;p+b,m αl
p+a p+e
−
p+e
1 p+a p+e p+c p+b p+b p+c
α
α
36
[2, a, e]
+ 12 (γ p+b;p+e,m − γ p+e;p+b,m )αl
p+a p+b
][2, a, e]
− ω ml ω nl γ p+a;p+b,n γ p+e;p+b,m + ω kl ω mn γ p+a,k;l γ p+e,m;n
+ 2ω ml γ p+e,m;l;p+a [2, a, e] − ω ml ω nl γ p+a,m;p+e,n − 13 ω mn γ p+b,m;n αp+a
+ ω kl [ 23 γ p+e,k;l αp+a
p+b p+b
− γ p+a;p+e,k αl
p+b p+b
p+e p+b
][2, a, e]
+
1 p+b,kl km lm p+a p+e p+b
γ
ω ω α
6
−
1 p+a,kl lm ov km p+e,o;v
γ
ω ω ω γ
][2, a, e] − 12 ω km ω ln ω k m ω l n γ p+a,kl γ p+e,k l .
2
+ ω kv ω ln ω mn γ p+a,m;v γ p+e,kl [2, a, e]
Let cα be the upper α quantile of the χ2r−p distribution with density function gr−p . Then,
√
by developing an Edgeworth expansion for nR under conditions (1), it may be shown that
P {(θ0 ) < cα} = α − n−1 Bc cα gr−p (cα ) + O(n−2 )
and
P {l(θ0 ) < cα (1 + n−1 Bc )} = α + O(n−2 ),
where Bc = (r − p)−1 {μT μ +
d
a=1
Δaa} is the Bartlett factor.
ACKNOWLEDGEMENT
The authors would like to thank Professor Nicole Lazar for informative discussions. The
project is supported by an Academic Research Grant of the National University of Singapore
(R-155-000-018-112) and the NSFC (10071009) of China.
13
REFERENCES
CHEN S. X. (1993). On the coverage accuracy of empirical likelihood regions for linear
regression model. Ann. Inst. Statist. Math. 45, 621-637.
CHEN, S. X. (1994). Empirical likelihood confidence intervals for linear regression coefficients. J. Multivariate Anal. 49, 24-40.
CHEN, S. X. & HALL, P. (1993). Smoothed empirical likelihood confidence intervals for
quantiles. Ann. Statist. 21, 1166-1181.
CUI H. J. & YUAN X. J. (2001). Smoothed empirical likelihood confidence interval for
quantile in the partial symmetric auxiliary information. J. Sys. Sci. and Math. Scis.
21(2), 172-181.
DICICCIO, T. J., HALL, P. & ROMANO, J. P. (1991). Empirical likelihood is Bartlett correctable. Ann. Statist. 19, 1053-1061.
DICICCIO, T. J. & ROMANO, J. P. (1989). On adjustments based on the signed root of the
empirical likelihood ratio statistic. Biometrika 76, 447-56.
JING, B. Y. and WOOD, A. T. A. (1996). Exponential empirical likelihood is not Bartlett
correctable. Ann. Statist. 24 365-369.
LAZAR, N. A. & MYKLAND, P. A. (1999). Empirical likelihood in the presence of nuisance
parameters. Biometrika 86, 203-211.
OWEN, A. B. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18,
90-120.
OWEN, A. B. (1991). Empirical likelihood for linear models. Ann. Statist. 19, 1725-1747.
QIN J. & LAWLESS, J. (1994). Empirical likelihood and general estimation equations. Ann.
Statist. 22, 300-325.
APPENDIX
Basic formulae
We start with providing some basic formulae which will be used throughout the appendix.
14
From the definition of the S matrix in Section 2, it can be easily checked that
⎛
S
−1
T
−1 T
T
−1
⎜ −I + S12 (S12 S12 ) S12 S12 (S12 S12 )
⎟
=⎝
⎛
⎞
T
T
(S12
S12)−1 S12
T
(S12
S12)−1
⎠
=
⎜
⎜
⎜
⎜
⎜
⎝
⎞
0
0
0 −Ir−p
Ω
0
T
Ω
0
ΩΩT
⎟
⎟
⎟
⎟.
⎟
⎠
From the definitions of B and A,
⎛
B=
⎜
⎜
⎜
⎜
⎜
⎝
B
..
.
1
Br
⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎞
⎜
S −1 ⎝
A ⎟
0
⎠
⎛
=
⎜
⎜
⎜
⎜
⎜
⎝
⎞
0
⎟
⎟
⎟
−A2 ⎟
⎟
⎠
ΩA1
where AT = (A1, · · · Ar )T =: (AT1 , AT2 )T . Here A1 = (A1, · · · , Ap)T and A2 = (Ap+1 , · · · , Ar )T
constitute a partition of vector A. The special form of S −1 given early means that for positive
integers k and a
B k = 0 for k ≤ p; B p+a = −Ap+a for a ≤ r − p, and B r+k = ω kl Al for k ≤ p.
(A.1)
Let B1 = (B 1, · · · , B r )τ and B2 = (B r+1 , · · · , B r+p )τ . Since SB = (Aτ , 0τp×1 )τ which means
that −B1 + S12 B2 = A. As S12 = (γ j,k )r×p and from (A.1) we have
γ j,k B r+k = Aj I(j ≤ p)
(A.2)
where I(·) is the indicator function.
Since
⎛
⎜
(B u,q )r+p×r+p = S −1 ⎝
S21(B j,k )r×p = (C k,m )τp×r
⎞
−(A ) (C ) ⎟
ij
(C i,l)T
i,l
0
⎠,
(A.3)
and S21(B j,r+a )r×p = 0.
As S21 = (γ j,k )τ , these mean
γ j,k B j,l = C l,k for l ≤ r and k ≤ p and γ j,k B j,r+a = 0.
15
(A.4)
Furthermore, (A.3) also implies the following which bridges B s,t with Ajm and C j,m :
⎛
⎞
k,l
⎜ (B )
⎜
⎜
⎜
⎜
⎝
(B k,r+l ) ⎟
⎟
⎟
(B p+a,l) (B p+a,p+b) (B p+a,r+l) ⎟
⎟
⎠
(B r+k,l ) (B r+k,p+b ) (B r+k,r+l )
⎛
=
(B k,p+b )
(ω
⎜
⎜
⎜
⎜
⎜
⎝
mk
C
l,m
)
⎞
(ω
(Ap+a,l)
mk
C
p+b,m
)
0
−(C p+a,l)
(Ap+a,p+b )
(ω km (ω nm C l,n − Aml )) (ω km (ω nm C p+b,n − Am,p+b )) (ω km C m,l)
⎟
⎟
⎟
⎟.
⎟
⎠
It can be also shown that for a fixed h ∈ {1, · · · , r}
⎛
⎞
⎜ 0
⎜
⎜
⎛
=
⎜
⎜
⎜
⎜
⎜
⎝
⎛
⎜
⎜
⎜
⎜
⎜
⎝
Ω
0
0
0
∂ 2 Q1
∂λh∂ψ
∂ 2 Q2
∂λh∂ψ
⎞
⎟
⎠
⎞
T
0
Ω
0 −Ir−p
Ω
∂ 2 Q1
∂λh∂λ
⎟
⎜
⎟×E⎝ 2
⎟
∂ Q2
⎠
∂λh∂λ
T
ΩΩ
(β s,th ) = ⎜ 0 −Ir−p
⎜
⎝
⎛
ΩT ⎟
⎟
0
0
ΩΩT
0
klh
(2α
⎟
⎟
⎟
⎟×
⎟
⎠
⎞
kp+bh
)
(2α
(2αp+alh )
−(γ
)
(2αp+ap+bh )
−(γ h;k,m + γ k;h,m )T −(γ h;p+a,m + γ p+a;h,m )T
(β s,tr+k ) =
⎜
⎜
⎜
⎜
⎜
⎝
⎛
=
⎜
⎜
⎜
⎜
⎜
⎝
⎞
0
0
0 −Ir−p
Ω
0
0
0
0 −Ir−p
Ω
0
T
Ω
0
ΩΩT
T
Ω
0
ΩΩT
⎛
⎟
⎟
⎜
⎟
⎟×E⎝
⎟
⎠
⎞
⎟
⎟
⎟
⎟×
⎟
⎠
16
∂ 2 Q1
∂ψk ∂λ
∂ 2 Q1
∂ψk ∂ψ
∂ 2 Q2
∂ψk ∂λ
∂ 2 Q2
∂ψk ∂ψ
+γ
k;h,m
)
⎟
⎟
h;p+a,m
p+a;h,m ⎟
;
−(γ
+γ
) ⎟
⎟
⎠
(γ h,kl )
Similarly for a fixed k ∈ {1, · · · , p}
⎛
h;k,m
⎞
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎝
⎞
−(γ l,k;m + γ l;m,k )
−(γ l,k;p+a + γ l;p+a,k )
(γ l,mk ) ⎟
⎟
⎟
.
−(γ p+a,k;m + γ p+a;m,k ) −(γ p+a,k;p+b + γ p+a;p+b,k ) (γ p+a,mk ) ⎟
⎟
(γ l,mk )T
(γ p+a,mk )T
0
⎠
It may be shown from the above equation that
β l,p+a
p+c
= −ω ol (γ p+c;p+a,o + γ p+a;p+c,o ),
β l,r+m
β p+a,p+b
p+c
= −2αp+a
= γ p+c;p+a,m + γ p+a;p+c,m ,
β l,p+a
r+n
= ω ol γ p+a;on ,
β p+a,p+b
r+n
= γ p+a,n;p+b + γ p+a;p+b,n ,
β r+k,p+a
p+c
= 2ω ko αo
β r+k,p+a
r+n
= ω ko ω mo γ p+a,mn − ω ko (γ o,n;p+a + γ o;p+a,n ) and
β r+k,r+m
p+c
= ω ko ω no γ p+c,nm − ω ko (γ p+c;o,m + γ o,p+c;m ).
p+b p+c
,
β p+a,r+m
p+c
β l,r+m r+n = 0,
p+a p+c
p+c
β r+k,r+m
β p+a,r+m
r+n
= ω ol γ p+c;om
r+n
= ω ko γ o,mn ,
= −γ p+a,mn
− ω ko ω no (γ p+c;p+a,n + γ p+a;p+c,n ),
Derivation of (7)
Now we are ready to expand
n−1 l(θ0) = n−1
n
i=1
[λ̃T wi (ψ̃) − 12 {λ̃T wi (ψ̃)}2 + 13 {λ̃T wi (ψ̃)}3 − 14 {λ̃T wi (ψ̃)}4] + Op (n−5/2 )
By substituting expansions for λ̃ and ψ̃ given in (5) and (6) into the above equation, we
have for a, b, c, d, e ∈ {1, 2, · · · , r − p}, f, g, h, i, j ∈ {1, 2, · · · , r}, k, l, m, n, o ∈ {1, 2, · · · , p}
and q, s, t, u ∈ {1, 2, · · · , r + p}:
n−1 l(θ0) = −2B j Aj − B j B j + 2B j,q B q (Aj + B j ) − β j,uq B u B q (Aj + B j )
− 2B j,u B u,q B q (Aj + B j ) + β u,qs B j,u B q B s (Aj + B j ) − β j,uq β u,st B q B s B t(Aj + B j )
− B j,uq B u B q (Aj + B j ) + 13 β j,uqs B u B q B s (Aj + B j ) + 2β j,uq B u,s B s B q (Aj + B j )
+ 2γ j,k − B j,q B q B r+k +
−
1 u,qs j,u q s r+k
β B B B B
2
+
1 j,uq u q r+k
B B B B
2
1 j,uq u q r+k
β B B B
2
+ B j,u B u,q B q B r+k
− β j,uq B u,s B q B s B r+k + 12 β j,uq β u,st B q B s B tB r+k
− 16 β j,uqs B u B q B s B r+k − 12 β j,uq B u B q B r+k,s B s [2, j, r + k]
17
+ B j,u B u B r+k,q B q + 14 β j,uq β r+k,st B u B q B s B t
+ 2C j,k B j B r+k − B j,q B q B r+k [2, j, r + k] + 12 β j,uq B u B q B r+k [2, j, r + k]
+ γ j,kl {−B j B r+k B r+l + B j,q B r+k B r+l B q [3, j, r + k, r + l]
−
1 j r+k r+l,uq u q
B B β
B B [3, j, r
2
+
1 j,klm j r+k r+l r+m
γ
B B B B
3
+ k, r + l]} − C j,kl B j B r+k B r+l
− B j,u B u B j,q B q − 14 β j,uq β j,stB u B q B s B t
+ β j,uq B u B q B j,s B s − B j B i Aji + B j B i,pB pAji [2, j, i] − 12 β j,uq B u B q B i Aji[2, j, i]
+ 2γ j;i,l B j B i B r+l − B j B i B r+l,q B q + 12 β r+l,uq B j B i B u B q − B r+lB i B j,q B q [2, j, i]
+
1 j,uq u q i r+l
β B B B B [2, j, i]
2
−
2 jih j i h
α B BB
3
+ 2B j B iB r+l C j;i,l − (γ j;i,lk + γ j,l;i,k )B j B i B r+l B r+k
+ 2αjih B j B i B h,q B q − αjih β j,uq B u B q B iB h − 23 Ajih B j B iB h
+ 2γ j;i;h,k B j B iB h B r+k − 12 αjihg B j B iB h B g + Op (n−5/2 )
By using those formulae given at the beginning of the appendix, it is shwon in Chen and
Cui (2002) that
n−1 (θ0 ) = −2Aj B j − B j B j − Aji B j B i + 2C j,k B j B r+k + 2γ j;i,l B j B iB r+l − 23 αjih B j B i B h
− γ j,kl B j B r+k B r+l + Aji B i,q B q B j [2, i, j] − B j,u B j,q B u B q − 2C j,k B j,q B r+k B q
+ γ j,kl B r+k B r+l B j,q B q + 2γ j,kl B j B r+l B r+k,q B q
− 2γ j;i,l (B j B iB r+l,q B q + B r+l B iB j,q B q [2, j, i]) + 2αjih B j B iB h,q B q
+ ( 12 β j,uq β r+k,st γ j,k − 14 β j,uq β j,st )B u B q B sB t
+ (γ i;j,k β i,uq + γ j;i,k β i,uq − γ j,kl β r+l,pq )B u B q B j B r+k − 12 γ j,kl β j,uq B u B q B r+l B r+k
− (γ j;i,lk + γ j,l;i,k )B j B i B r+l B r+k + 13 γ j,klm B j B r+k B r+lB r+m
+ 2γ j;i;h,k B j B i B h B r+k − 12 αjihg B j B i B h B g + (γ j;i,lβ r+l,uq − αjih β h,uq )B j B iB u B q
− C j,klB j B r+k B r+l + 2C j;i,l B j B i B r+l − 23 Ajih B j B i B h + Op (n−5/2).
(A.5)
It can be shown from (A.1), (A.2) and (A.4) that −2Aj B j − B j B j = Ap+aAp+a and
−Aji B j B i + 2C j,k B j B r+k + 2γ j;i,l B j B iB r+l − 23 αjih B j B iB h − γ j,kl B j B r+k B r+l
= −Ap+a
p+b
Ap+a Ap+b − 2ω kl C p+a,k Ap+a Al + 2γ p+a;p+b,k ω kl Ap+a Ap+b Al
18
+
2 p+a p+b p+c p+a p+b p+c
α
A A A
3
+ γ p+a,kl ω km ω ln Ap+aAm An .
These and (A.5) readily imply (7).
Derivations of (12).
p+b p+c p+d
p+a p+b
p+c p+d
The fives terms involved in E(Rp+a
2 R1 R1 R1 )[4] − E(R2 R1 )E(R1 R1 )[12] are
respectively
p+b
− 12 {E(Ap+a
p+b
Ap+b Ap+b Ap+c Ap+d )[4] − E(Ap+a
Ap+b Ap+b )E(Ap+c Ap+d )[12]}
= n−3 (−6t1 + 2t2 − 12 t3 − 2t4 ) + O(n−4 );
− ω kl {E(C p+a,k AlAp+b Ap+c Ap+d )[4] − E(C p+a,k Al Ap+b )E(Ap+c Ap+d )[12]}
= −n−3 {ω kl (γ p+a,k;l αp+b
+ γ p+a,k;p+d αl
p+b p+c
p+c p+d
+ γ p+a,k;p+b αl
p+c p+d
+ γ p+a,k;p+c αl
p+b p+d
)}[4] + O(n−4 );
γ p+a;p+b ,k ω kl {E(Ap+b AlAp+b Ap+c Ap+d )[4] − E(Ap+b AlAp+b )E(Ap+c Ap+d )[12]}
= n−3 {ω kl (γ p+a;p+b,k αl
p+c p+d
+ γ p+a;p+c,k αl
p+b p+d
1 p+a p+b p+c
α
{E(Ap+b Ap+c Ap+b Ap+c Ap+d )[4]
3
+ γ p+a;p+d,k αl
p+b p+c
)}[4] + O(n−4 );
− E(Ap+b Ap+c Ap+b )E(Ap+c Ap+d )[12]}
= n−3 ( 13 t3 + 83 t4) + O(n−4 ) and
1 p+a,kl km ln
γ
ω ω {E(Am An Ap+b Ap+c Ap+d )[4]
2
= n−3 ( 12 γ p+a,kl ω km ω lm αp+b
p+c p+d
− E(Am An Ap+b )E(Ap+c Ap+d )[12]}
[4]) + O(n−4 ).
Combining these five terms give (12).
Derivations of (13).
p+b
Since Rp+a
= Ap+a , the terms involved is closely related to 15 terms in Rp+a
1
2 R2 . The
terms involved are
E(Ap+a
p+b
Ap+b
p+c
Ap+b Ap+c Ap+c Ap+d )[6] − E(Ap+ap+b Ap+bp+c Ap+b Ap+c )
×E(Ap+c Ap+d )[6] = n−3 (12t1 − 3t2 + 6t3 + 13t4 ) + O(n−4 );
19
ω mn {E(C p+a,k C p+b,m Al An Ap+c Ap+d )[6] − E(C p+a,k C p+b,m AlAn )E(Ap+c Ap+d )[6]}
= n−3 {ω ml (γ p+b,m;p+d γ p+a,k;p+c + γ p+a,k;p+d γ p+b,m;p+c )[6]} + O(n−4 );
ω mn γ p+a;p+b ,k γ p+b;p+c ,m {E(Ap+b Ap+c Al An Ap+c Ap+d )[6] − E(Ap+b Ap+c AlAn )
×E(Ap+c Ap+d )[6] = n−3 {ω ml (γ p+a;p+c,k γ p+b;p+d,m + γ p+a;p+d,k γ p+b;p+c,m )[6]} + O(n−4 );
1 p+a p+b p+c p+bp+d p+e
α
α
{E(Ap+b Ap+c Ap+d Ap+e Ap+c Ap+d )[6]
9
×E(Ap+c Ap+d )[6]} = n−3 ( 23 t3 +
16
t )
9 4
− E(Ap+b Ap+c Ap+d Ap+e )
+ O(n−4 );
E(Am An Am An Ap+c Ap+d )[6] − E(Am An Am An )E(Ap+c Ap+d )[6]} = O(n−4 );
ω kl {E(C p+a,k Ap+b
p+b
Ap+b AlAp+c Ap+d )[6] − E(C p+a,k Ap+b
p+b
= n−3 {ω kl (γ p+a,k;p+c αlp+bp+d + γ p+a,k;p+d αlp+bp+c + 2γ p+a,k;l αp+b
ω kl γ p+a;p+c ,k {E(Ap+b
p+b
Ap+b Ap+c Al Ap+c Ap+d )[6] − E(Ap+b
×E(Ap+c Ap+d )[6]} = n−3 {ω kl (γ p+a;p+d,k αl
− 16 αp+a
p+b p+c
p+b p+c
Ap+b Al )E(Ap+c Ap+d )[6]}
p+c p+d
p+b
+ γ p+a;p+c,k αl
)[6]} + O(n−4 );
Ap+b Ap+c Al)
p+b p+d
)[6]} + O(n−4 );
{E(Ap+bp+d Ap+b Ap+c Ap+d Ap+c Ap+d )[2, a, b][6]
−E(Ap+bp+d Ap+b Ap+c Ap+d )E(Ap+c Ap+d )[2, a, b][6]} = n−3 (−2t3 −
ω km ω ln γ p+a,kl {E(Ap+b
p+b
Ap+b Am An Ap+c Ap+d )[6] − E(Ap+b
p+b
16
t)
3 4
+ O(n−4 );
Ap+b AmAn )
×E(Ap+c Ap+d )[6]}
= n−3 (2ω km ω lm γ p+a,kl αp+b
p+c p+d
[6]) + O(n−4 );
ω kl ω mn γ p+a;p+b ,m {E(C p+b,k Ap+b AlAn Ap+c Ap+d )[6] − E(C p+b,k Ap+b AlAn )E(Ap+c Ap+d )[6]}
= n−3 {ω kl ω ml (γ p+a;p+d,m γ p+b,k;p+c + γ p+a;p+c,m γ p+b,k;p+d )[6]} + O(n−4 );
20
ω kl αp+a
p+b p+c
= n−3 (2ω kl αp+a
{E(C p+b,k Ap+b Ap+c AlAp+c Ap+d )[6] − E(C p+b,k Ap+b Ap+c Al )E(Ap+c Ap+d )[6]
p+c p+d p+b,k;l
γ
[6]) + O(n−4 );
E(C p+b,o AmAn Av Ap+c Ap+d )[6] − E(C p+b,o AmAn Av )E(Ap+c Ap+d )[6] = O(n−4 );
E(Ap+b Ap+c Ap+d AlAp+c Ap+d )[6] − E(Ap+b Ap+c Ap+d Al )E(Ap+c Ap+d )[6] = O(n−4 );
E(Ap+b AmAn Av Ap+c Ap+d )[6] − E(Ap+b Am An Av )E(Ap+c Ap+d )[6] = O(n−4 );
ω km ω ln αp+a
p+b p+c p+b,kl
γ
{E(Ap+b Ap+c Am An Ap+c Ap+d ) − E(Ap+b Ap+c Am An )
×E(Ap+c Ap+d )} = n−3 (2ω km ω lm αp+a
p+c p+d p+b,kl
γ
) + O(n−4 ).
Derivation of (14).
Due to the fact that E(Al Ap+a) = 0 for l ∈ {1, · · · , p}, there is no need to compute terms
in Rp+a
involving Al . It may be shown that
3
3
{E(Ap+a p+c Ap+c p+b Ap+b Ap+bAp+c Ap+d )[4]
8
− E(Ap+a
p+c
Ap+c p+b Ap+b Ap+b )
×E(Ap+cAp+d )[12]} = n−3 (3t4 ) + O(n−4 );
ω ml {E(C p+b ,m Alp+a Ap+b Ap+b Ap+c Ap+d )[4] − E(C p+b ,m Alp+a Ap+b Ap+b )
×E(Ap+c Ap+d )[12]}
= n−3 [ω ml {(γ p+d,m;p+c + γ p+c,m;p+d )αl
+ (γ p+b,m;p+c + γ p+c,m;p+b )αl
p+a p+d
p+a p+b
+ (γ p+b,m;p+d + γ p+d,m;p+b )αl
p+a p+c
}[4]] + O(n−4 );
ω ml ω nl {E(C p+a,m C p+b ,n Ap+b Ap+b Ap+c Ap+d )[4] − E(C p+a,m C p+b ,n Ap+b Ap+b )
×E(Ap+c Ap+d )[12]}
= n−3 [ω ml ω nl {γ p+a,m;p+c (γ p+b,n;p+d + γ p+d,n;p+b ) + γ p+a,m;p+b (γ p+c,n;p+d + γ p+d,n;p+c )
+γ p+a,m;p+d (γ p+b,n;p+c + γ p+c,n;p+b )}[4]] + O(n−4 );
21
−αp+a
p+b l
ω nl {E(C p+c ,n Ap+b Ap+c Ap+b Ap+c Ap+d )[4] − E(C p+c ,n Ap+b Ap+c Ap+b )
×E(Ap+c Ap+d )[12]}
= n−3 [−ω nl {(γ p+d,n;p+c + γ p+c,n;p+d )αl
+ (γ p+b,n;p+c + γ p+c,n;p+b )αl
− 56 αp+a
p+b p+c
p+a p+d
p+a p+b
+ (γ p+b,n;p+d + γ p+d,n;p+b )αl
p+a p+c
}[4]] + O(n−4 );
{E(Ap+c p+d Ap+b Ap+d Ap+aAp+c Ap+d )[4] − E(Ap+c p+d Ap+b Ap+d Ap+a )
t ) + O(n−4 );
×E(Ap+c Ap+d )[12]} = n−3 (− 20
3 4
1
{E(Ap+a p+b p+c Ap+b Ap+c Ap+bAp+c Ap+d )[4]
3
− E(Ap+a
p+b p+c
Ap+b Ap+c Ap+b )
×E(Ap+c Ap+d )[12]} = n−3 (8t1 ) + O(n−4 );
(αl
p+a p+b
− 14 αp+a
p+b p+e p+e p+c p+d
ω kl γ p+c ;p+d ,k + 49 αp+a
p+b p+c p+d
α
− 12 ω kl ω ml γ p+a;p+b ,k γ p+c ;p+d ,m
) × {E(Ap+b Ap+c Ap+d Ap+b Ap+c Ap+d )[4] − E(Ap+b Ap+c Ap+d Ap+b )
×E(Ap+c Ap+d )[12]}
= n−3 [−6t1 +
32
t
9 4
+ ω kl {(γ p+d,k;p+c + γ p+c,k;p+d )αl
+ (γ p+b,k;p+d + γ p+d,k;p+b )αl
−
p+a p+c
1 kl ml
ω ω {γ p+a;p+b,k (γ p+c;p+d,m
2
p+a p+b
+ (γ p+b,k;p+c + γ p+c,k;p+b )αl
p+a p+d
}[4]
+ γ p+d;p+c,m ) + γ p+a;p+c,k (γ p+b;p+d,m + γ p+d;p+b,m )
+ γ p+a;p+d,k (γ p+b;p+c,m + γ p+c;p+b,m )}[4]] + O(n−4 );
γ p+a;p+b ,l ω ln ω on {E(C p+c ,o Ap+b Ap+c Ap+b Ap+c Ap+d )[4] − E(C p+c ,o Ap+b Ap+c Ap+b )
×E(Ap+cAp+d )[12]}
= n−3 [ω ln ω on {γ p+a;p+b,l(γ p+c,o;p+d + γ p+d,o;p+c ) + γ p+a;p+c,l (γ p+b,o;p+d + γ p+d,o;p+b )
+ γ p+a;p+d,l (γ p+b,o;p+c + γ p+c,o;p+b )}[4]] + O(n−4 );
−γ p+a;p+b ,l ω ln {E(An,p+c Ap+b Ap+c Ap+b Ap+c Ap+d )[4] − E(An,p+c Ap+b Ap+c Ap+b )
×E(Ap+c Ap+d )[12]}
= n−3 {−2ω ln (γ p+a;p+b,l αn
p+c p+d
+ γ p+a;p+c,l αn
+ O(n−4 ).
22
p+b p+d
+ γ p+a;p+d,l αn
p+b p+c
)[4]}
Derivations of (16).
p+e
p+a p+e
The main task in deriving (16) is to work out E(Rp+a
2 R2 ) and E(R1 R3 ) for a, e ∈
p+a
{1, · · · , r − p}. There are 15 terms in Rp+a
2 R2 . Those 15 terms and their corresponding
expectations, denoted in Jiae for i = 1, · · · , 15 are reported below:
(1). 14 Ap+a
p+b
J1ae = 14 [(αp+a
Ap+e
p+c
Ap+b Ap+c ,
p+e p+b p+b
− δ ae) + αp+a
p+b p+b p+e p+c p+c
α
+ αp+a
p+b p+c p+e p+b p+c
α
(2). ω kl ω mn C p+a,k C p+e,m AlAn ,
J2ae = ω kl ω ml γ p+a,k;p+e,m + ω kl ω mn (γ p+a,k;l γ p+e,m;n + γ p+a,k;n γ p+e,m;l );
(3). ω kl ω mn γ p+a;p+b,k γ p+e;p+c,m Ap+b Ap+c Al An ,
J3ae = ω kl ω ml γ p+a;p+b,k γ p+e;p+b,m ;
(4). 19 αp+a
p+b p+c
J4ae = 19 [αp+a
αp+e
p+d p+e
Ap+b Ap+c Ap+d Ap+e ,
p+b p+b p+e p+d p+d
α
+ 2αp+a
p+b p+c
αp+e
p+b p+c
];
(5). 14 ω km ω ln ω k m ω l n γ p+a,kl γ p+e,k l AmAn Am An ,
J5ae = 14 γ p+a,kl γ p+e,k l [ω km ω lm ω k m ω l m + ω km ω ln (ω k m ω l n + ω k n ω l m )]
(6). 12 ω kl C p+a,k Ap+e
p+b
J6ae = 12 ω kl (γ p+a,k;p+b αl
Ap+b Al [2, a, e],
p+e p+b
(7). − 12 ω kl γ p+a;p+c,k Ap+e
J7ae = − 12 ω kl γ p+a;p+b,k αl
(8). − 16 αp+a
p+b p+c
J8ae = − 16 (2αp+a
p+d
αp+e
(9). − 14 ω km ω ln γ p+a,kl Ap+e
[2, a, e];
p+b p+c
+ αp+a
p+b p+b
αp+e
Ap+b Am An [2, a, e],
p+b p+b
[2, a, e];
(10). −ω kl ω mn γ p+a;p+b,m C p+e,k Ap+b Al An [2, a, e],)
ae
= −ω kl ω ml γ p+a;p+b,m γ p+b;p+e,k [2, a, e];
J10
(11). − 13 ω kl αp+a
p+b p+c
)[2, a, e];
Ap+bAp+c Ap+d [2, a, e],
p+b
J9ae = − 14 ω km ω lm γ p+a,kl αp+e
p+b p+b
Ap+b Ap+c Al[2, a, e],
p+e p+b
Ap+e
p+b p+c
p+b
+ γ p+a,k;lαp+e
C p+e,k Ap+b Ap+c Al[2, a, e], )
23
p+d p+d
)[2, a, e];
];
ae
J11
= − 13 ω kl αp+a
p+b p+b p+e,k;l
γ
[2, a, e];
(12). − 12 ω km ω ln ω ov γ p+a,kl C p+e,o Am An Av [2, a, e],
ae
= − 12 γ p+a,kl [ω km ω ln ω on γ p+e,o;m + ω km ω ln ω om γ p+e,o;n + ω lm ω ov ω km γ p+e,o;v ] [2, a, e];
J12
(13).
1 kl p+a;p+b,k p+e p+c p+d p+b p+c p+d l
ω γ
α
A A A A [2, a, e],
3
ae
= 0;
J13
(14).
1 km ln ov p+a,kl p+e;p+b,o p+b m n v
ω ω ω γ
γ
A A A A [2, a, e],
2
ae
= 0;
J14
(15).
1 km ln p+a p+b p+c p+e,kl p+b p+c m n
ω ω α
γ
A A A A [2, a, e],
6
ae
= 16 ω km ω lm αp+a
J15
p+b p+b p+e,kl
γ
[2, a, e];
Note that
J1ae =: J1ae + J4ae + J8ae = 14 (αp+a
7 p+a
− 36
α
p+b p+c p+e p+b p+c
α
p+e p+b p+b
− δ ae ) +
1 p+a p+b p+b p+e p+c p+c
α
α
36
].
J2ae = ω kl ω ml γ p+a,k;p+e,m + ω kl ω mn (γ p+a,k;l γ p+e,m;n + γ p+a,k;n γ p+e,m;l ).
J3ae = ω kl ω ml γ p+a;p+b,k γ p+e;p+b,m .
J5ae = 14 γ p+a,kl γ p+e,k l [ω km ω lm ω k m ω l m + ω km ω ln (ω k m ω l n + ω k n ω l m )]
J6ae = ω kl (γ p+a,k;p+b αlp+ep+b + γ p+a,k;l αp+ep+bp+b )[2, a, e].
J7ae = − 12 ω kl γ p+a;p+b,k αlp+ep+b [2, a, e].
1 km lm p+a,kl p+ep+bp+b
ae
= − 12
ω ω γ
α
[2, a, e]
J9ae + J15
ae
ae
= −ω kl ω ml γ p+e;p+b,m γ p+b;p+a,k [2, a, e]. J11
= − 13 ω kl αp+a
J10
p+b p+b p+e,k;l
γ
[2, a, e].
ae
= − 12 γ p+a,kl [ω km ω ln ω on γ p+e,o;m + ω km ω ln ω om γ p+e,o;n
J12
+ω lm ω ov ω km γ p+e,o;v ][2, a, e].
Combine the above terms, we arrive at
p+e
−2 ae
−3
E(Rp+a
2 R1 )[2, a, e] = n J16 + O(n )
where
ae
= −(αp+a
J16
p+e p+b p+b
− δ ae ) − ω kl γ p+a,k;l;p+e [2, a, e] + ω kl γ p+a;p+b,k αl
24
p+b p+e
[2, a, e]
+
2 p+a
α
3
p+b p+c p+e p+b p+c
α
1
+ ω km ω ln γ p+a,kl αmn
2
p+e
[2, a, e].
p+e
ae
There are 25 terms in Rp+a
3 R1 , whose expectations are denoted by J16+i for i =
1, · · · , 25.
ae
= 34 [(αp+a
J17
p+e p+c p+c
− δ ae ) + αp+a
ae
= ω ml [γ p+e,m;l;p+a + γ p+b,m;p+b αl
J18
ae
= 12 ω lm [γ p+b,l;mαp+a
J19
p+e p+b
p+b p+c
p+a p+e
p+b p+c
+ αp+a
p+e p+c
+ γ p+b,m;p+e αl
p+a p+b
][2, a, e],
+ γ p+b,l;p+e αm
αp+e
p+a p+b
αp+b
p+b p+c
][2, a, e],
ae
J20
= − 12 ω ml ω nl [γ p+a,m;p+e,n + γ p+a,m;p+b γ p+b,n;p+e + γ p+a,m;p+e γ p+b,n;p+b ][2, a, e],
ae
= ω ml ω kn [γ l,k;n γ p+a,m;p+e + γ l,k;p+e γ p+a,m;n ][2, a, e],
J21
ae
= −ω nl [(γ p+e,n;p+b + γ p+b,n;p+e )αl
J22
ae
= −2ω mn γ p+c,m;n αp+a
J23
ae
J24
= − 53 [2αp+a
p+e p+c
ae
= 12 ω km ω lm γ p+a,kl;p+e [2, a, e],
J25
ae
= 2αp+a
J27
p+e p+b p+b
ae
J28
= − 32 αp+a
αp+a
+ 16
9
+2αl
p+a p+e
p+a p+e
][2, a, e],
+ αp+a
p+e p+c
αp+c
p+d p+d
],
ae
J26
= ω lm γ p+a;p+e,l;m [2, a, e],
,
p+e p+b p+b
p+b p+e
+ γ p+c,n;p+c αl
,
p+b p+c p+e p+b p+c
α
p+a p+b
αp+e
+ ω kl (γ p+b;p+e,k + γ p+e;p+b,k )αl
p+b p+e
p+a p+b
[2, a, e]
− 12 ω kl ω ml γ p+a;p+b,k (γ p+b;p+e,m + γ p+e;p+b,m )[2, a, e]
ω kl γ p+c;p+c,k + 89 αp+a
p+e p+e
αp+e
p+c p+c
− 12 ω kl ω ml γ p+a;p+e,k γ p+c;p+c,m [2, a, e],
ae
J29
= − 12 ω kn ω ln ω vm γ m,kl γ p+a,v;p+e [2, a, e],
ae
= ω lo ω kn ω vn γ p+e,kl γ p+a,v;o [2, a, e],
J31
ae
J30
= − 12 γ p+b,kl ω km ω lm αp+a
ae
J32
= −ω lo ω kn γ p+e,kl αon
p+a
p+e p+b
[2, a, e],
ae
= −ω ln ω mn ω kv γ p+a,kl γ v,m;p+e [2, a, e],
J33
ae
= ω ln ω on [γ p+a;p+b,l(γ p+e,o;p+b + γ p+b,o;p+e ) + γ p+a;p+e,l γ p+c,o;p+c ][2, a, e],
J34
ae
J35
= −ω ln [2γ p+a;p+b,l αnp+bp+e + γ p+a;p+e,l αnp+cp+c ][2, a, e],
ae
= −ω mn ω lo γ p+a;p+e,r+m γ o,m;n [2, a, e],
J36
ae
= −ω ln ω om (γ m;p+a,l + γ p+a;m,l )γ p+e,o;n [2, a, e],
J37
ae
= − 12 ω ln (γ p+c;p+a,l + γ p+a;p+c,l )αnp+ep+c [2, a, e],
J38
25
,
],
ae
J39
= −ω ln ω kn (γ p+a;p+b,l + γ p+a;p+c,l )γ p+b,k;p+e [2, a, e],
ae
J40
= 13 ω kl γ p+c,k;l αp+a
ae
= ω m m ω n m [ 13 αp+a
J41
p+e p+c
[2, a, e] and
p+e p+c p+c,m n
+ 12 γ p+c,m ;p+a (γ p+c;p+e,n + γ p+e;p+c,n )
γ
+ 12 γ p+a,m ;p+c γ p+e;p+c,n + ω lo γ p+e,n l (γ o,m ;p+a + γ o;p+a,m ) + 12 ω lo γ o,m n γ p+a;p+e,l
− 12 ω ol ω kl γ p+a,m o γ p+e,n k − 12 γ p+a;p+e,m n − 12 γ p+a,m ;p+e,n ][2, a, e].
In summary, we have:
1
ae
ae
ae
ae
ae
ae
J42
=: J1ae + J16
+ J17
+ J24
+ J27
+ J28
= αp+a p+e p+b p+b − 13 αp+a
2
1 p+a p+b p+b p+e p+c p+c
1 p+a p+e p+c p+b p+b p+c
+ 36 α
α
− 36 α
α
− ω kl γ p+a,k;l;p+e [2, a, e] + 12 ω km ω ln γ p+a,kl αmn
+ ω ml [γ p+b;p+b,m αl
−
p+a p+e
p+e
αp+e
[2, a, e]
+ (γ p+b;p+e,m + 2γ p+e;p+b,m )αl
1 ml nl p+a;p+e,n p+b;p+b,m
ω ω [γ
γ
2
p+b p+c
p+a p+b
][2, a, e]
+ γ p+a;p+b,n (γ p+b;p+e,m + γ p+e;p+b,m )][2, a, e],
ae
ae
ae
ae
ae
ae
ae
J43
=: J18
+ J19
+ J20
+ J22
+ J23
+ J26
= ω ml [2γ p+e,m;l;p+a + ( 12 γ p+b,m;p+e − γ p+e,m;p+b )αl
−
1 ml nl p+a,m;p+e,n
ω ω [γ
2
− ω mn γ p+b,m;n αp+a
p+a p+b
][2, a, e]
+ γ p+a,m;p+b γ p+b,n;p+e + γ p+a,m;p+e γ p+b,n;p+b ][2, a, e].
p+e p+b
,
ae
ae
ae
ae
ae
ae
J44
=: J3ae + J6ae + J7ae + J9ae + J15
+ J10
+ J11
+ J35
+ J38
= ω kl ω ml γ p+a;p+b,k γ p+e;p+b,m + ω kl [( 12 γ p+e,k;p+b − 3γ p+e;p+b,k )αl
+
2 p+e,k;l p+a p+b p+b
γ
α
3
− γ p+a;p+e,k αl
1 km lm p+e,kl p+a
ω ω γ
α
− 12
p+b p+b
p+b p+b
p+a p+b
][2, a, e]
[2, a, e] − ω kl ω ml γ p+e;p+b,m γ p+b;p+a,k [2, a, e],
ae
ae
ae
ae
ae
ae
ae
ae
ae
ae
ae
J45
=: J29
+ J30
+ J31
+ J32
+ J33
+ J34
+ J36
+ J37
+ J39
+ J40
= − 12 ω kn ω ln ω vm γ m,kl γ p+a,v;p+e [2, a, e] − 12 γ p+b,kl ω km ω lm αp+a
p+e p+b
.
+ ω kv ω ln ω mn (γ p+a,m;v − γ v,m;p+a )γ p+e,kl [2, a, e]
− ω lm ω kn γ p+e,kl αmn
p+a
[2, a, e] − ω mn ω lo γ p+a;p+e,r+m γ o,m;n [2, a, e]
+ ω kn ω ln [γ p+a;p+b,k (γ p+e,l;p+b − γ p+e;p+c,l ) + γ p+a;p+e,k γ p+b,l;p+b ][2, a, e]
26
p+b p+c
− ω ln ω om (γ m;p+a,l + γ p+a;m,l )γ p+e,o;n [2, a, e] + 13 ω kl γ p+b,k;l αp+a
p+e p+b
[2, a, e],
ae
ae
ae
ae
ae
J46
=: J2ae + J5ae + J12
+ J21
+ J25
+ J41
= ω kl ω mn (γ p+a,k;l γ p+e,m;n + γ p+a,k;n γ p+e,m;l )
+ [ 14 ω km ω lm ω k n ω l n − 12 ω km ω ln ω k m ω l n ]γ p+a,klγ p+e,k l
−
1 p+a,kl km ln on p+e,o;m
γ
[ω ω ω γ
2
+ ω km ω ln ω om γ p+e,o;n
+ ω lm ω ov ω km γ p+e,o;v ][2, a, e] + 12 ω lo γ o,m n γ p+a;p+e,l ][2, a, e]
+ ω ml ω kn [γ l,k;n γ p+a,m;p+e + γ l,k;p+e γ p+a,m;n ][2, a, e]
+ ω m m ω n m [ 13 αp+a
+
p+e p+b p+b,m n
γ
1 p+a,m ;p+b p+e;p+b,n
γ
γ
2
1 p+a p+b p+b p+e p+c p+c
α
α
36
−
p+e p+b p+b
p+a p+e
− 13 αp+a
p+b p+c
αp+e
p+b p+c
1 p+a p+e p+c p+b p+b p+c
α
α
36
− ω kl γ p+a,k;l;p+e [2, a, e] − 12 ω km ω ln γ p+a,kl αmn
+ ω ml [γ p+b;p+b,m αl
+ ω lo γ p+e,n l (γ o,m ;p+a + γ o;p+a,m ) and
ae
ae
ae
ae
ae
ae
J47
=: J42
+ J43
+ J44
+ J45
+ J46
= 12 αp+a
+
+ 12 γ p+b,m ;p+a (γ p+b;p+e,n + γ p+e;p+b,n )
p+e
[2, a, e]
+ 12 (γ p+b;p+e,m − γ p+e;p+b,m )αl
p+a p+b
][2, a, e]
− ω ml ω nl γ p+a;p+b,n γ p+e;p+b,m + ω kl ω mn γ p+a,k;lγ p+e,m;n
+ 2ω ml γ p+e,m;l;p+a [2, a, e] − ω ml ω nl γ p+a,m;p+e,n − 13 ω mn γ p+b,m;n αp+a
+ ω kl [ 23 γ p+e,k;l αp+a
−
p+b p+b
− γ p+a;p+e,k αl
p+b p+b
p+e p+b
][2, a, e]
1 km lm p+e,kl p+a p+b p+b
ω ω γ
α
[2, a, e] + 16 γ p+b,kl ω km ω lm αp+a p+e p+b .
12
+ ω kv ω ln ω mn γ p+a,m;v γ p+e,kl [2, a, e] − 12 γ p+a,kl ω lm ω ov ω km γ p+e,o;v ][2, a, e]
+ [ 14 ω km ω lm ω k n ω l n − 12 ω km ω ln ω k m ω l n ]γ p+a,klγ p+e,k l .
Then, we get
ae
cum(Rp+a , Rp+e ) = n−1 δ ae + n−2 (J47
− μp+a μp+e ) + O(n−3 ) =: n−1 δ ae + n−2 Δae + O(n−3 )
where
Δae =
1 p+a p+e p+b p+b
α
2
− 13 αp+a
p+b p+c
αp+e
p+b p+c
− ω kl γ p+a,k;l;p+e [2, a, e] − 12 ω km ω ln γ p+a,kl αmn
27
−
p+e
1 p+a p+e p+c p+b p+b p+c
α
α
36
[2, a, e]
+ ω ml [γ p+b;p+b,m αl
p+a p+e
+ 12 (γ p+b;p+e,m − γ p+e;p+b,m )αl
p+a p+b
][2, a, e]
− ω ml ω nl γ p+a;p+b,n γ p+e;p+b,m + ω kl ω mn γ p+a,k;l γ p+e,m;n
+ 2ω ml γ p+e,m;l;p+a [2, a, e] − ω ml ω nl γ p+a,m;p+e,n − 13 ω mn γ p+b,m;n αp+a
+ ω kl [ 23 γ p+e,k;l αp+a
p+b p+b
− γ p+a;p+e,k αl
p+b p+b
p+e p+b
][2, a, e]
+
1 p+b,kl km lm p+a p+e p+b
γ
ω ω α
6
−
1 p+a,kl lm ov km p+e,o;v
γ
ω ω ω γ
][2, a, e] − 12 ω km ω ln ω k m ω l n γ p+a,kl γ p+e,k l .
2
+ ω kv ω ln ω mn γ p+a,m;v γ p+e,kl [2, a, e]
28