Page 153 - Contributed Paper Session (CPS) - Volume 1

CPS1216 Teppei O.
estimators of the parameter in the diffusion coefficients, and the maximum-
likelihoodtype estimator ̂ attains the optimal rate. On the other hand, we will

1 4
see that ̂ cannot attain the rate ⁄ in the misspecified setting due to an

asymptotic bias term. We can attain optimal rate if we construct a maximum-
likelihood-type estimator ̂ by using a bias-modified quasi-log-likelihood

function.

For a vector = ( ,··· , ) we denote = ( ) . We

1

…
1 1 ,⋯, 1 =1

assume that ⊂ ℝ satisfies Sobolev’s inequality; that is, for any > ,
⁄

there exists > 0 such that sup ∈Λ |()| ≤ ∑ =0,1 (∫ | ()| ) 1 for

Λ
any ∈ (). This is the case when Λ has a Lipshitz boundary. See Adams
1
and Fournier [1] for more details.
To obtain optimal rate convergence, we need strengthened versions of the
assumptions [A1] and [A2].

2
[B1] [A1] is satisfied, sup ( [[ − |ℱ ] ] | − |⁄ 2 ) < ∞, and ∑ and
0≤<≤
∑ exist and are continous on [0, ] × × (). Moreover, there

exists a locally bounded function (, ) such that
| (, , ) − (, , )| ≤ (, )| − |.

for any ∈ [0, ], , ∈ and ∈ .

[B2] There exists positive-valued stochastic processes { } ∈[0,],1≤≤ such

that for any > 0 and
⁄

> 0, ( 1− ) ⋁( −1− −1 ) ⋁ ( −3 5+ ) → 0,

[sup (| − | ⁄ | − | )] < ∞, [sup (| | + 1 | | )] < ∞

⁄
≠ ,
and

is finite for any 1 ≤ ≤ , where the second supremum is taken over
all sequences { } , { } ⊂ [0, ] such that < and
′′
′
′
′′

sup (ℓ ( − )) < ∞.
′′
′

We can see that [A2] holds under [B2].
Here, we see the bias of the quasi-log-likelihood function .

⁄
Let () = (( 1 2 ) ). For a 2 × 2 positive definite matrix B = (Bij)ij

and a 2 × 2 symmetric matrix C = (Cij)ij, set
−1
(, , , ) = ( (, ) (, ))

⁄
⁄
⁄
ℓ (()( − )()(()())
− (1 2) 1 2 −1 −1 2 ),

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