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CPS1443 Shogo H.N. et al.
high-frequency financial data analysis. Here we define our observation
{ } for all = 0, … , ,
ℎ =0,…,

1
ℎ = ℎ + Λ 2 ℎ

where Λ is a × -dimensional positive semi-definite matrix and { }
ℎ =0,…,

is an i.i.d. sequence of -valued random variables such that [ ] = =
0
[0, … ,0] and Var( ) = where indicates the transpose and is the

0
× -dimensional identity matrix. Hence for all = 0, … , , is defined as
ℎ
the summation of the true value of the latent process at ℎ and the random

noise with mean and variance Λ. The parametric inference for a diffusion
parameter and/or a drift parameter based on this noised observation
sequence also has drawn the interest of researchers (e.g., see Favetto, 2014,
2016; Gloter and Jacod, 2001a, 2001b; Jacod et al., 2009; Ogihara, 2018;
Podolskij and Vetter, 2009). In this paper, we focus on the long-term and high-
frequency observation scheme such that ℎ → 0 and : = ℎ → ∞ as → ∞

as same as Favetto (2014, 2016) which enables us to estimate both and .
Favetto (2014) proposes a quasi-likelihood function simultaneously optimised
with respect to both and , and shows consistency of the corresponding
maximum-likelihood-type estimator; and Favetto (2016) proves asymptotic
normality of the estimator when the variance of the noise term Λ is known.
Our contributions upon these researches consist of two parts as follows: (i) our
adaptive quasi-likelihood functions and estimators require less computational
burden compared to the simultaneous ones because we can optimise the
functions with respect to and separately; (ii) the theoretical results for
convergence such that asymptotic normality holds even if the variance of the
noise term Λ is unknown and convergence of moments as Yoshida (2011),
which are not shown in Favetto (2016).
The contents of this paper are as follows: in the Methodology section, we

show how to extract the information of the latent process { } from noisy
≥0
observation and propose adaptive quasi-likelihood functions based on noisy
observation; in the Result section, we show theoretical properties and
simulation of the adaptive maximum-likelihood-type estimators
corresponding to the adaptive quasi-likelihood functions; in the final section,
we summarise these discussions of advantages of our proposal in comparison
to the existent literatures.

2. Methodology
Firstly we discuss the construction of local means which are used as like
the observation of the latent process { } in quasi-likelihood functions.
≥0
Taking sample means of noisy observation is the core idea to remove the

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