Clasificación de referencia: Gloria Corpas Pastor (1996)

Given a n × p data matrix R with sample size n and dimension p, assume that the unbiased precision matrix is Ω0 = arg min

Ω∈Rp×p

L(Ω|R), and the concord estimator, which is biased, is ˆΩ = arg min

Ω∈Rp×p

L(Ω|R) + P (Ω, λ). Define ∆Ω = ∆Ω = Ω0 − ˆΩ, ˆw = vec( ˆΩ),

and ∆w = vec(∆Ω). Thus, the tuning parameter λ controls how much bias ˆΩ contains relative to Ω0, given the data set R. For a general precision matrix variable Ω, let

w = vec(Ω) = (ω11, ω21, · · · , ωp1, ω12, ω22, · · · , ωp2, · · · , ω1p, ω2p, · · · , ωpp)T

Further define f (w0) = L(Ω0|R), apply Taylor expansion to f (w0) up to the second

order: L(Ω0|R) = L( ˆΩ + ∆Ω|R) = f (w0) = f ( ˆw + ∆w) ≈ f ( ˆw) + ∇f ( ˆw)T∆w +1 2∆w T_∇2_{f ( ˆw)∇w} (B.7)

where the gradient vector ∇f (w) =  ∂f ∂ω11 , ∂f ∂ω21 , · · · , ∂f ∂ωp1 , · · · , · · · , ∂f ∂ω1p , ∂f ∂ω2p , · · · , ∂f ∂ωpp T (B.8) is a p2-dimensional column vector, and the Hessian matrix

∇2_{f (w) = vec}−1

is a p2 _{× p}2 _{matrix in which ⊗ represents the Kronecker product. Because Ω}

0 is the

unbiased parameter, it is the critical point that minimize the pseudo-likelihood function L(Ω). That is, L(Ω0|R) ≤ L( ˆΩ(λ|R) for all Concord estimator ˆΩ. Correspondingly,

w0 = vec(Ω0) is a critical point that minimize f (w0). Given a fixed ˆw, we hope to find

∆w such that ˆw + ∆w is a critical point, thus we set d d∆w  f ( ˆw) + ∇f ( ˆw)T∆w + 1 2∆w T_∇2 f ( ˆw)∇w  = 0 (B.10) which results in ∇f ( ˆw) + ∇2f ( ˆw)∆w = 0 ⇒ ∆w = −(∇2f ( ˆw))−1∇f ( ˆw) (B.11) Because ∇L(Ω|R) = dL(Ω|R) dΩ =          ∂L ∂ω11 ∂L ∂ω12 · · · ∂L ∂ω1p ∂L ∂ω21 ∂L ∂ω22 · · · ∂L ∂ω2p · · · · ∂L ∂ωp1 ∂L ∂ωp2 · · · ∂L ∂ωpp          =          ∂f ∂ω11 ∂f ∂ω12 · · · ∂f ∂ω1p ∂f ∂ω21 ∂f ∂ω22 · · · ∂f ∂ω2p · · · · ∂f ∂ωp1 ∂f ∂ωp2 · · · ∂f ∂ωpp          = vec−1(∇f (w)) (B.12) ∇2L(Ω|R) = d 2_L(Ω|R) (dΩ)2 = ∇L(Ω|R) ⊗ ∇L(Ω|R) = vec −1 (∇f (w)) ⊗ vec−1(∇f (w)) = ∇2f (w) (B.13)

We have ∆Ω = vec−1(∆w) = −vec−1  (∇2L( ˆΩ|R))−1vec(∇L( ˆΩ|R))  (B.14) Thus the bias of the Concord estimator ˆΩ under tuning parameter λ is

ˆ ∆ = Ω0− ˆΩ = ∆Ω = −vec−1  (∇2L( ˆΩ|R))−1vec(∇L( ˆΩ|R))  = −vec−1  ∇2L(Ω|R)    Ω= ˆΩ −1 vec  ∇L(Ω|R)  Ω= ˆΩ  (B.15)

Assume that S = ((sij)) = 1_nRTR where sij = _n1RT·iR·j. Based on the results of [40] and

[44], ∂L(Ω|R) ∂ωij = −nΩ−1_(D)+n 2(SΩ + ΩS) = −nΩ −1 (D)+ 1 2(R T RΩ + ΩRTR) (B.16) ∂2_L(Ω|R) ∂ωij∂ωkh = p X i=1 Ω−2_ii (eieTi ⊗ eieTi ) + n 2(S ⊗ I + I ⊗ S) = p X i=1 Ω−2_ii (eieTi ⊗ eieTi ) + 1 2(R T R ⊗ I + I ⊗ RTR) (B.17)

where ei is a column vector with all elements are 0 except the i-th element. Thus the

bias is ˆ ∆ = −vec−1  p X i=1 ˆ Ω−2_ii (eieTi ⊗ eieTi ) + 1 2(R T_{R ⊗ I + I ⊗ R}T_R) −1 · vec  −n ˆΩ−1_(D)+1 2(R T_{R ˆΩ + ˆΩR}T_R)  (B.18)

ˆ

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