Page 154 - Contributed Paper Session (CPS) - Volume 1

CPS1216 Teppei O.
⁄
⁄
(, , ) = log det (, ) − 1 2 −1 −1 2 ),
ℓ ((()())

( , , , , ) = ( , , , , ) + ( , , , ).

2
1

2
1

1
2

̃
̃
̃
̃
Let ∑ = ∑ () = ∑( −1 , −1 , ), ∑ ,† = ∑ −1 ,† and ̃ ,† = (∑ ,† , ). We

∗

denote
1

̃ ̃
△ = ( , ):= − ∑ tr (( ̃ −1 ( ) − ̃ −1 ( )) ( − ̃ ,† ))
2
1

1

2

2

for , ∈ clos ().
1
2

Proposition 3.1. Assume [B1], [B2] and [V]. Let { } = 1,2, ∈ ℕ be clos(Λ)-
,
valued random variables. Then
⁄
−1 4 ( ( 1, , ̂ )− ( 2, , ̂ ))

⁄
⁄
= −1 4 △ ( 1, , 2, ) − 1 4 ((∑( 1, ), ∑ † ) − (∑( 2, ), ∑ † ))
2
1 −1 4
⁄

+ ∑(−1) ∑ (̃ −1 , ∑ ̃ ( , ) , ∑ ̃ ,† , ∗ ) + (1). (3.4)

2
=1

The third term in the right-hand side of (3.4) is bias term which does not
appear in the specified model. Due to the third term, we cannot ensure that
⁄
−1 4 ) is (1).
(̂ − ∗
In the following, we consider removing the bias. First, we consider an
estimator ( , ) of ∑ −1 ,† by using the function g appearing in Jacod et al.
[2], that is, ∶ [0,1] → ℝ is continuous, piecewise , (0) = (1) = 0, and
1
1
1
1
∫ () > 0. Let = ( ( + 1)), Ψ = ∫ () , Ψ = ∫ () .

2
′
2
⁄
1

1

0
0
0
For example, let () = ⋀(1 − ) on0 ≤ ≤ 1, then Ψ = 1 12 and Ψ = 1.
⁄
1
2

1
−1

Let ̂ = (̂ , ⋯ , ̂ ), ̂ = ̂ −1 ( ) 1 { ̂ , >0} , and let , be a ×

,

matrix satisfying.

ℓ ̂

[ , ] = ΤΨ {(∑ ) (∑ , ) − , Ψ 1 {=} }.
,
2

1
=1 =1
Then we define a bias-corrected quasi-likelihood function () by
̌

1
̌
() = (, ̂ ) + ∑ (̂ , ∑ (), , , ̂ ).

2
⁄
⁄

By setting ̂ ,,† = ((̂ ̂ ) 1 2 [ , ] ) and ̂ , = ((̂ ̂ ) 1 2 ()] ) , ̌ ()

[∑
is simplified as
1

−1
̌ () = − ∑ { ( ()( − , ))

2

√ ⁄
⁄
+ ℓ −1 2 (( ̂ , () + ̂ , , †) ̂ , () −1 2 )}. (3.5)

4
Let ̌ = argmax ̌ (). Then we obtain optimal rate convergence for ̌ .

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