Page 107 - Contributed Paper Session (CPS) - Volume 2

CPS1443 Shogo H.N. et al.
−1
1 ⊗2
̂
Λ ≔ ∑( (+1)ℎ − ) .

2 ℎ
=0

Note that this adaptive procedure of optimisation has an advantage to less
the computational burden in comparison to the simultaneous quasi-likelihood
function ℍ (, |Λ) proposed in Favetto (2014, 2016) especially when

parameter spaces have high-dimensionality.

3. Result
In the first place, we give some theoretical results for the adaptive
estimators. Let → and → ℒ denote convergence in probability and
convergence in law, respectively. The true value of variance of noise Λ is Λ
⋆
and vech means the half vectorisation of a symmetric matrix . The first
theorem states the consistency of the estimators.

Theorem 1 (Nakakita and Uchida, 2018b). Under the regularity conditions
in Nakakita and Uchida (2018b), Λ → Λ , ̂ → , and → .

̂

⋆
̂
⋆

⋆

The next theorem shows asymptotic normality of the estimators.

Theorem 2 (Nakakita and Uchida, 2018b). Under the regularity conditions
in Nakakita and Uchida (2018b), the following convergence in law holds.
̂
√vech(Λ − Λ )

⋆
⋆
⋆

⋆
ℒ
[ √ (̂ − ) ] → ∼ (+1)/2 + 1 + 2 (, (Λ , , )),

⋆

̂
⋆
√ ( − )

where (Λ , , ) is the same one defined in Nakakita and Uchida
⋆

⋆
⋆
(2018b).

The following theorem gives the result for convergence of moments, which
can be obtained through the framework of quasi-likelihood analysis (QLA)
discussed in Yoshida (2011).

Theorem 3 (Nakakita and Uchida, 2018a). Under the regularity conditions
in Nakakita and Uchida (2018a), for all continuous and at most polynomial
growth function : (+1)/2 + 1 + 2 → , the following convergence holds:
[ (√vech(Λ − Λ ), √ (̂ − ), √ ( − ))] → [()].
⋆
̂
⋆
̂

⋆

Secondly we computationally demonstrate the performance of the
adaptive estimators in comparison with the estimators by the existent method,
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