# Quasi-likelihood analysis of an ergodic diffusion plus noise

We consider adaptive maximum-likelihood-type estimators and adaptive Bayes-type ones for discretely observed ergodic diffusion processes with observation noise whose variance is constant. The quasi-likelihood functions for the diffusion and drift parameters are introduced and the polynomial-type large deviation inequalities for those quasi-likelihoods are shown to see the convergence of moments for those estimators.

## Authors

• 6 publications
• 16 publications
05/28/2018

### Inference for ergodic diffusions plus noise

We research adaptive maximum likelihood-type estimation for an ergodic d...
11/13/2017

### Adaptive estimation and noise detection for an ergodic diffusion with observation noises

We research adaptive maximum likelihood-type estimation for an ergodic d...
11/27/2018

### Hybrid estimation for ergodic diffusion processes based on noisy discrete observations

We consider parametric estimation for ergodic diffusion processes with n...
05/01/2021

### Local Asymptotic Mixed Normality via Transition Density Approximation and an Application to Ergodic Jump-Diffusion Processes

We study sufficient conditions for local asymptotic mixed normality. We ...
04/29/2020

### Adaptive tests for parameter changes in ergodic diffusion processes from discrete observations

We consider the adaptive test for the parameter change in discretely obs...
04/05/2018

### Adaptive test for ergodic diffusions plus noise

We propose some parametric tests for ergodic diffusion-plus-noise model,...
10/28/2019

### Penalized quasi likelihood estimation for variable selection

Penalized methods are applied to quasi likelihood analysis for stochasti...
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## 1. Introduction

We consider a -dimensional ergodic diffusion process defined by the following stochastic differential equation such that

 dXt=b(Xt,β)dt+a(Xt,α)dwt, X0=x0,

where is an -dimensional Wiener process,

is a random variable independent of

, and are unknown parameters, and are bounded, open and convex sets in admitting Sobolev’s inequalities for embedding for , is the true value of the parameter, and and are known functions.

A matter of interest is to estimate the parameter with partial and indirect observation of : the observation is discretised and contaminated by exogenous noise. The sequence of observation , which our parametric estimation is based on, is defined as

 Yihn=Xihn+Λ1/2εihn, i=0,…,n,

where is the discretisation step such that and , is an i.i.d. sequence of random variables independent of and such that and where

is the identity matrix in

for every , and is a positive semi-definite matrix which is the variance of noise term. We also assume that the half vectorisation of has bounded, open and convex parameter space , and let us denote . We also notate the true parameter of as , its half vectorisation as , and . That is to say, our interest is on parametric inference for an ergodic diffusion with long-term and high-frequency noised observation. One concrete example is the wind velocity data provided by NWTC Information Portal (2018) whose observation is contaminated by exogenous noise with statistical significance according to the test for noise detection (Nakakita and Uchida, 2018b).

As the existent discussion, Nakakita and Uchida (2018b) propose the following estimator , and such that

 ^Λn=12nn−1∑i=0(Y(i+1)hn−Yihn)⊗2, Hτ1,n(^αn;^Λn)=supα∈Θ1Hτ1,n(α;^Λn), H2,n(^βn;^αn)=supβ∈Θ2H2,n(β;^αn),

where for every matrix , is the transpose of and , and are the adaptive quasi-likelihood functions of and respectively defined in Section 3, is a tuning parameter, and Nakakita and Uchida (2018b) show these estimators are asymptotically normal and especially the drift one is asymptotically efficient. To obtain the convergence rates of the estimators, it is necessary to see the composition of the quasi-likelihood functions. Both of them are function of local means of observation defined as

 ¯Yj=1pnpn−1∑i=0YjΔn+ihn, j=0,…,kn−1,

where is the number of partition given for observation, is that of observation in each partition, is the time interval which each partition has, and note that these parameters have the properties , and . Intuitively speaking, and correspond to and in the observation scheme without exogenous noise, and divergence of

works to eliminate the influence of noise by law of large numbers. Hence it should be also easy to understand that we have the asymptotic normality with the convergence rates

and for and ; that is,

 [√kn(^αn−α⋆),√Tn(^βn−β⋆)]→dξ,

where is an

-dimensional Gaussian distribution with zero-mean.

The statistical inference for diffusion processes with discretised observation has been investigated in these decades: see Florens-Zmirou (1989), Yoshida (1992), Bibby and Sørensen (1995), Kessler (1995, 1997). In practice, it is necessary to argue whether exogenous noise exists in observation, and it has been pointed out that the observational noise, known as microstructure noise, certainly exists in high-frequency financial data which is one of the major disciplines where statistics for diffusion processes is applied. Inference for diffusions under such the noisy and discretised observation in fixed time interval is discussed by Jacod et al. (2009), and also Favetto (2014, 2016) examine same problem as our study and shows simultaneous ML-type estimation has consistency under the situation where the variance of noise is unknown and asymptotic normality under the situation where the variance is known. As mentioned above, Nakakita and Uchida (2018b) propose adaptive ML-type estimation which has asymptotic normality even if we do not know the variance of noise, and test for noise detection which succeeds in showing the real data NWTC Information Portal (2018) which is contaminated by observational noise.

Our study aims at polynomial type large deviation inequalities for statistical random fields and construction of the estimators with not only asymptotic normality as shown in Nakakita and Uchida (2018b) but also a certain type of convergence of moments. Asymptotic normality is well-known as one of the hopeful properties that estimators are expected to have; for instance, Nakakita and Uchida (2018a) utilise this result to compose likelihood-ratio-type statistics and related ones for parametric test and proves the convergence in distribution to a

-distribution under null hypothesis and consistency of the test under alternative one. However, it is also known that asymptotic normality is not sufficient to develop some discussion requiring convergence of moments such as information criterion. In concrete terms, it is necessary to shows the convergence of moments such as for every

with at most polynomial growth and adaptive ML-type estimator and ,

 Eϑ⋆[f(√kn(^αn−α⋆),√Tn(^βn−β⋆))]→Eθ⋆[f(ξ)].

This property is stronger than mere asymptotic normality since if we take as a bounded and continuous function, then indeed asymptotic normality follows.

To see the convergence of moments for adaptive ML-type estimator, we can utilise polynomial-type large deviation inequalities (PLDI) and quasi-likelihood analysis (QLA) proposed by Yoshida (2011) which have been widely used to discuss convergence of moments of not only ML-type estimation but also Bayes-type one in statistical inference for continuous-time stochastic processes. This approach is developed from the exponential-type large deviation and likelihood analysis introduced by Ibragimov and Has’minskii (1972, 1973, 1981), and the polynomial-type one discussed by Kutoyants (1984, 1994, 2004). Yoshida (2011) itself discusses convergence of moments in adaptive maximum-likelihood-type estimation, simultaneous Bayes-type one, and adaptive Bayes-type one for ergodic diffusions with and . Uchida and Yoshida (2012, 2014) examine the same problem for adaptive ML-type and adaptive Bayes-type estimation for ergodic diffusions with more relaxed condition: and for some . Ogihara and Yoshida (2011) study convergence of moments for parametric estimators against ergodic jump-diffusion processes in the scheme of and . Other than diffusion processes or jump-diffusions, Clinet and Yoshida (2017) show PLDI for the quasi-likelihood function for ergodic point processes and the convergence of moments for the corresponding ML-type and Bayes-type estimators. As the applications of these discussions, Uchida (2010) composes AIC-type information criterion for ergodic diffusion processes, and Eguchi and Masuda (2018) propose BIC-type one for local-asymptotic quadratic statistical experiments including some schemes for diffusion processes. In this paper, we develop QLA for our ergodic diffusion plus noise model and propose the adaptive Bayes-type estimators of both drift and volatility parameters. Furthermore, we show the convergence of moments of both the adaptive ML-type estimators and the adaptive Bayes-type estimators for the ergodic diffusion plus noise model. Note that Bayes-type estimation itself is important to deal with non-linearity of parameters and multimodality of quasi-likelihood functions which sometimes appear in statistics for diffusion processes. In particular, the hybrid type estimators with initial Bayes-type estimators are considered for diffusion type processes, see Kamatani and Uchida (2015); Kaino and Uchida (2018a, b), and references therein. Moreover, as an application of the Bayes-type estimation proposed in this paper, Kaino et al. (2018) study the hybrid estimators with initial Bayes-type estimators for our ergodic diffusion plus noise model and give an example and simulation results of the hybrid estimator.

## 2. Notation and assumption

We set the following notations.

• For every matrix , is the transpose of , and .

• For every set of matrices and whose dimensions coincide, . Moreover, for any , and , .

• Let us denote the

-th element of any vector

as and -th one of any matrix as .

• For any vector and any matrix , and .

• For every , is the -norm.

• , , and .

• For given , , , and , and we define the sequence of local means such that

 ¯Zj=1pnpn−1∑i=0ZjΔn+ihn, j=0,⋯,kn−1,

where indicates an arbitrary sequence defined on the mesh such as , and .

###### Remark 1.

Since the observation is masked by the exogenous noise, it should be transformed to obtain the undermined process . As illustrated by Nakakita and Uchida (2018b), the sequence can extract the state of the latent process in the sense of the statement of Lemma 2.

• , , , , , and .

• We define the real-valued function as for :

 V((l1,l2),(l3,l4)) +32(Λ(l1,l3)⋆Λ(l2,l4)⋆+Λ(l1,l4)⋆Λ(l2,l3)⋆),

and with the function as for and ,

 σ(i,j):={j if i=1,∑i−1ℓ=1(d−ℓ+1)+j−i+1 if i>1,

we define the matrix as for ,

 W(i1,i2)1:=V(σ−1(i1),σ−1(i2)).
• Let

 {Bκ(x)∣∣κ=1,…,m1, Bκ=(B(j1,j2)κ)j1,j2}, {fλ(x)∣∣λ=1,…,m2, fλ=(f(1)λ,…,f(d)λ)}

be sequences of -valued functions and -valued ones respectively such that the components of themselves and their derivative with respect to are polynomial growth functions for all and . Then we define the following matrix-valued functionals, for ,

 (W(τ)2({Bκ:κ=1,…,m1}))(κ1,κ2) :={ν(tr{(¯Bκ1A¯Bκ2A)(⋅)}) if τ∈(1,2),ν(tr{(¯Bκ1A¯Bκ2A+4¯Bκ1A¯Bκ2Λ⋆+12¯Bκ1Λ⋆¯Bκ2Λ⋆)(⋅)}) if τ=2, (W3({fλ:λ=1,…,m2}))(λ1,λ2) :=ν((fλ1A(fλ2)T)(⋅)),

where is the invariant measure of discussed in the following assumption [A1]-(iv), and for all function on , .

With respect to , we assume the following conditions.

• .

• For some constant , for all ,

 supα∈Θ1∥a(x1,α)−a(x2,α)∥+supβ∈Θ2|b(x1,β)−b(x2,β)|≤C|x1−x2|
• For all , .

• There exists an unique invariant measure on and for all and with polynomial growth,

 1T∫T0f(Xt)dt→P∫Rdf(x)ν(dx).
• For any polynomial growth function satisfying , there exist , with at most polynomial growth for such that for all ,

 Lθ⋆G(x)=−g(x),

where is the infinitesimal generator of .

###### Remark 2.

Paradoux and Veretennikov (2001) show a sufficient condition for [A1]-(v). Uchida and Yoshida (2012) also introduce the sufficient condition for [A1]-(iii)–(v) assuming [A1]-(i)–(ii), and , and such that for all and satisfying ,

 1|x|xTb(x,β) ≤−c0|x|γ.
• There exists such that and have continuous derivatives satisfying

 supα∈Θ1∣∣∂jx∂iαa(x,α)∣∣ ≤C(1+|x|)C, 0≤i≤4, 0≤j≤2, supβ∈Θ2∣∣∂jx∂iβb(x,β)∣∣ ≤C(1+|x|)C, 0≤i≤4, 0≤j≤2.

With the invariant measure , we define

 Yτ1(α;ϑ⋆) :=−12∫{tr(Aτ(x,α,Λ⋆)−1Aτ(x,α⋆,Λ⋆)−Id)+logdetAτ(x,α,Λ⋆)detAτ(x,α⋆,Λ⋆)}ν(dx), Y2(β;ϑ⋆)

where . For these functions, let us assume the following identifiability conditions hold.

• For all , there exists a constant such that for all .

• For all , there exists a constant such that for all .

The next assumption is with respect to the moments of noise.

• For any , has -th moment and the components of are independent of the other components for all , and

. In addition, for all odd integer

, , , and , , and .

The assumption below determines the balance of convergence or divergence of several parameters. Note that is a tuning parameter and hence we can control it arbitrarily in its space .

• , , , , , for . Furthermore, there exists such that for sufficiently large .

###### Remark 3.

Let us denote and where and the components of their derivatives are polynomial growth with respect to uniformly in . Then the discussion in Uchida (2010) verifies under [A1] and [A6], for all ,

 supn∈NEθ⋆⎡⎢⎣supϑ∈Ξ(kϵ1n∣∣ ∣∣1knkn−2∑j=1f(XjΔn,ϑ)−∫Rdf(x,ϑ)ν(dx)∣∣ ∣∣)M⎤⎥⎦<∞.

## 3. Quasi-likelihood analysis

First of all, we introduce and analyse some quasi-likelihood functions and estimators which are defined in Nakakita and Uchida (2018b). The quasi-likelihood functions for the diffusion parameter and the drift one using this sequence are as follows:

 Hτ1,n(α;Λ):=−12kn−2∑j=1((23ΔnAτn(¯Yj−1,α,Λ))−1[(¯Yj+1−¯Yj)⊗2]+logdetAτn(¯Yj−1,α,Λ)), H2,n(β;α):=−12kn−2∑j=1((ΔnA(¯Yj−1,α))−1[(¯Yj+1−¯Yj−Δnb(¯Yj−1,β))⊗2]),

where . We set the adaptive ML-type estimator , and such that

 ^Λn :=12nn−1∑i=0(Y(i+1)hn−Yihn)⊗2, Hτ1,n(^αn;^Λn) =supα∈Θ1Hτ1,n(α;^Λn), H2,n(^βn;^αn) =supβ∈Θ2H2,n(β;^αn).

Assume that , are continuous and , and denote the adaptive Bayes-type estimators

 ~αn :={∫Θ1exp(Hτ1,n(α;^Λn))π1(α)dα}−1∫Θ1αexp(Hτ1,n(α;^Λn))π1(α)dα, ~βn

Our purpose is to show the polynomial-type large deviation inequalities for the quasi-likelihood functions defined above in the framework introduced by Yoshida (2011), and the convergences of moments for these estimators as the application of them. Let us denote the following statistical random fields for and

 Zτ1,n(u1;^Λn,α⋆) :=exp(Hτ1,n(α⋆+k−1/2nu1;^Λn)−Hτ1,n(α⋆;^Λn)), ZML2,n(u2;^αn,β⋆) :=exp(H2,n(β⋆+T−1/2nu2;^αn)−H2,n(β⋆;^αn)), ZBayes2,n(u2;~αn,β⋆) :=exp(H2,n(β⋆+T−1/2nu2;~αn)−H2,n(β⋆;~αn)),

and some sets

 Uτ1,n(α⋆) :={u1∈Rm1;α⋆+k−1/2nu1∈Θ1}, U2,n(β⋆) :={u2∈Rm2;β⋆+T−1/2nu2∈Θ2},

and for ,

 Vτ1,n(r,α⋆) :={u1∈Uτ1,n(α⋆);r≤|u1|}, V2,n(r,β⋆) :={u2∈U2,n(β⋆);r≤|u2|}.

We use the notation as Nakakita and Uchida (2018b) for the information matrices

 Iτ(ϑ⋆) :=diag{W1,I(2,2),τ,I(3,3)}(ϑ⋆), Jτ(ϑ⋆) :=diag{Id(d+1)/2,J(2,2),τ,J(3,3)}(ϑ⋆),

where for ,

 I(2,2),τ(ϑ⋆) :=W(τ)2({34(Aτ)−1(∂α(k1)A)(Aτ)−1(⋅,ϑ⋆):k1=1,…,m1}), J(2,2),τ(ϑ⋆) :=[12ν(tr{(Aτ)−1(∂α(i1)A)(Aτ)−1(∂α(i2)A)}(⋅,ϑ⋆))]i1,i2,

and for ,

 I(3,3)(θ⋆) =J(3,3)(θ⋆):=[ν((A)−1[∂β(j1)b,∂β(j2)b](⋅,θ⋆))]j1,j2.

We also denote and .

###### Theorem 1.

Under [A1]-[A6], we have the following results.

1. The polynomial-type large deviation inequalities hold: for all , there exists a constant such that for all ,

 Pθ⋆⎡⎣supu1∈Vτ1,n(r,α⋆)Zτ1,n(u1;^Λn,α⋆)≥e−r⎤⎦ ≤C(L)rL, Pθ⋆[supu2∈V2,n(r,β⋆)ZML2,n(u2;^αn,β⋆)≥e−r] ≤C(L)rL, Pθ⋆[supu2∈V2,n(r,β⋆)ZBayes2,n(u2;~αn,β⋆)≥e−r] ≤C(L)rL.
2. The convergences of moment hold:

 Eθ⋆[f(√n(^θε,n−θ⋆ε),√kn(^αn−α⋆),√Tn(^βn−β⋆))] →E[f(ζ0,ζ1,ζ2)], Eθ⋆[f(√n(^θε,n−θ⋆ε),√kn(~αn−α⋆),√Tn(~βn−β⋆))] →E[f(ζ0,ζ1,ζ2)],

where

 (ζ0,ζ1,ζ2)∼Nd(d+1)/2+m1+m2(0,(Jτ(ϑ⋆))−1(Iτ(ϑ⋆))(Jτ(ϑ⋆))−1)

and is an arbitrary continuous functions of at most polynomial growth.

### 3.1. Evaluation for local means

In the first place we give some evaluations related to local means. Some of the instruments are inherited from the previous studies by Nakakita and Uchida (2017) and Nakakita and Uchida (2018b). We define the following random variables:

 ζj+1,n:=1pnpn−1∑i=0∫(j+1)ΔnjΔn+ihndws,ζ′j+2,n:=1pnpn−1∑i=0∫(j+1)Δn+ihn(j+1)Δndws.

The next lemma is Lemma 11 in Nakakita and Uchida (2018b).

###### Lemma 1.

and are -measurable, independent of and Gaussian.These variables have the next decompositions:

 ζj+1,n =1pnpn−1∑k=0(k+1)∫jΔn+(k+1)hnjΔn+khndws, ζ′j+1,n =1pnpn−1∑k=0(pn−k−1)∫jΔn+(k+1)hnjΔn+khndws.

The evaluation of the following conditional expectations holds:

 Eθ⋆[ζj,n|Gnj]=Eθ⋆[ζ′j+1,n|Gnj] =0,