Quasi-likelihood analysis of an ergodic diffusion plus noise

1. Introduction

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

d X_{t} = b (X_{t}, β) d t + a (X_{t}, α) d w_{t}, X_{0} = x_{0},

where ${w_{t}}_{t \geq 0}$ is an $r$ -dimensional Wiener process, $x_{0}$

is a random variable independent of

{w_{t}}_{t \geq 0}

α \in Θ_{1}

and

β \in Θ_{2}

are unknown parameters,

Θ_{1} \subset R^{m_{1}}

and

Θ_{2} \subset R^{m_{2}}

are bounded, open and convex sets in

R^{m_{i}}

admitting Sobolev’s inequalities for embedding

W^{1, p} (Θ_{i}) ↪ C ({¯ ¯¯ ¯ Θ}_{i})

for

i = 1, 2

θ^{⋆} = (α^{⋆}, β^{⋆})

is the true value of the parameter, and

a : R^{d} \times Θ_{1} \to R^{d} \otimes R^{r}

and

b : R^{d} \times Θ_{2} \to R^{d}

are known functions.

A matter of interest is to estimate the parameter $θ = (α, β)$ with partial and indirect observation of ${X_{t}}_{t \geq 0}$ : the observation is discretised and contaminated by exogenous noise. The sequence of observation ${Y_{i h_{n}}}_{i = 0, \dots, n}$ , which our parametric estimation is based on, is defined as

Y_{i h_{n}} = X_{i h_{n}} + Λ^{1 / 2} ε_{i h_{n}}, i = 0, \dots, n,

where $h_{n} > 0$ is the discretisation step such that $h_{n} \to 0$ and $T_{n} = n h_{n} \to \infty$ , ${ε_{i h_{n}}}_{i = 0, \dots, n}$ is an i.i.d. sequence of random variables independent of ${w_{t}}_{t \geq 0}$ and $x_{0}$ such that $E_{θ^{⋆}} [ε_{i h_{n}}] = 0$ and ${V a r}_{θ^{⋆}} (ε_{i h_{n}}) = I_{d}$ where $I_{m}$

is the identity matrix in

R^{m} \otimes R^{m}

for every

m \in N

, and

Λ \in R^{d} \otimes R^{d}

is a positive semi-definite matrix which is the variance of noise term. We also assume that the half vectorisation of

Λ

has bounded, open and convex parameter space

Θ_{ε}

, and let us denote

Ξ := Θ_{ε} \times Θ_{1} \times Θ_{2}

. We also notate the true parameter of

Λ

Λ_{⋆}

, its half vectorisation as

θ_{ε}^{⋆} = v e c h Λ_{⋆}

, and

ϑ^{⋆} = (θ_{ε}^{⋆}, α^{⋆}, β^{⋆})

. 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).

Figure 1. plot of wind velocity labelled Sonic x (left) and y (right) (119M) at the M5 tower from 00:00:00 on 1st July, 2017 to 20:00:00 on 5th July, 2017 with 0.05-second resolution (NWTC Information Portal, 2018)

As the existent discussion, Nakakita and Uchida (2018b) propose the following estimator ${^Λ}_{n}$ , ${^α}_{n}$ and ${^β}_{n}$ such that

	${^Λ}_{n} = \frac{1}{2 n} n - 1 \sum i = 0 {(Y_{(i + 1) h_{n}} - Y_{i h_{n}})}_{(i + 1) h_{n}}^{\otimes 2},$
	$H_{1, n}^{τ} ({^α}_{n}; {^Λ}_{n}) = sup α \in Θ_{1} H_{1, n}^{τ} (α; {^Λ}_{n}),$
	$H_{2, n} ({^β}_{n}; {^α}_{n}) = sup β \in Θ_{2} H_{2, n} (β; {^α}_{n}),$

where for every matrix $A$ , $A^{T}$ is the transpose of $A$ and $A^{\otimes 2} = A A^{T}$ , $H_{1, n}^{τ}$ and $H_{2, n}$ are the adaptive quasi-likelihood functions of $α$ and $β$ respectively defined in Section 3, $τ \in (1, 2]$ 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

{¯ Y}_{j} = \frac{1}{p_{n}} p_{n} - 1 \sum i = 0 Y_{j Δ_{n} + i h_{n}}, j = 0, \dots, k_{n} - 1,

where $k_{n}$ is the number of partition given for observation, $p_{n}$ is that of observation in each partition, $Δ_{n} = p_{n} h_{n}$ is the time interval which each partition has, and note that these parameters have the properties $k_{n} \to \infty$ , $p_{n} \to \infty$ and $Δ_{n} \to 0$ . Intuitively speaking, $k_{n}$ and $Δ_{n}$ correspond to $n$ and $h_{n}$ in the observation scheme without exogenous noise, and divergence of $p_{n}$

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

\sqrt{k_{n}}

and

\sqrt{T_{n}}

for

α

and

β

; that is,

[\sqrt{k_{n}} ({^α}_{n} - α^{⋆}), \sqrt{T_{n}} ({^β}_{n} - β^{⋆})] \to^{d} ξ,

where $ξ$ is an $(m_{1} + m_{2})$

-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 $[0, 1]$ 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 $χ^{2}$

-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

f \in C (R^{m_{1}} \times R^{m_{2}})

with at most polynomial growth and adaptive ML-type estimator

{^α}_{n}

and

{^β}_{n}

E_{ϑ^{⋆}} [f (\sqrt{k_{n}} (^α_{n} - α^{⋆}), \sqrt{T_{n}} ({^β}_{n} - β^{⋆}))] \to E_{θ^{⋆}} [f (ξ)] .

This property is stronger than mere asymptotic normality since if we take $f$ 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 $n h_{n} \to \infty$ and $n h_{n}^{2} \to 0$ . 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: $n h_{n} \to \infty$ and $n h_{n}^{p} \to 0$ for some $p \geq 2$ . Ogihara and Yoshida (2011) study convergence of moments for parametric estimators against ergodic jump-diffusion processes in the scheme of $n h_{n} \to \infty$ and $n h_{n}^{2} \to 0$ . 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 $A$ , $A^{T}$ is the transpose of $A$ , and $A^{\otimes 2} := A A^{T}$ .
For every set of matrices $A$ and $B$ whose dimensions coincide, $A [B] := t r (A B^{T})$ . Moreover, for any $m \in N$ , $A \in R^{m} \otimes R^{m}$ and $u, v \in R^{m}$ , $A [u, v] := v^{T} A u$ .
Let us denote the $ℓ$
-th element of any vector
$v$ as $v^{(ℓ)}$ and $(ℓ_{1}, ℓ_{2})$ -th one of any matrix $A$ as $A^{(ℓ_{1}, ℓ_{2})}$ .
For any vector $v$ and any matrix $A$ , $| v | := \sqrt{t r (v^{T} v)}$ and $∥ A ∥ := \sqrt{t r (A^{T} A)}$ .
For every $p > 0$ , ${∥ \cdot ∥}_{p}$ is the $L^{p} (P_{θ^{⋆}})$ -norm.
$A (x, α) := a {(x, α)}^{\otimes 2}$ , $a (x) := a (x, α^{⋆})$ , $A (x) := A (x, α^{⋆})$ and $b (x) := b (x, β^{⋆})$ .
For given $τ \in (1, 2]$ , $p_{n} := h_{n}^{- 1 / τ}$ , $Δ_{n} := p_{n} h_{n}$ , and $k_{n} := n / p_{n}$ , and we define the sequence of local means such that

${¯ Z}_{j} = \frac{1}{p_{n}} p_{n} - 1 \sum i = 0 Z_{j Δ_{n} + i h_{n}}, j = 0, \dots, k_{n} - 1,$

where ${Z_{i h_{n}}}_{i = 0, \dots, n}$ indicates an arbitrary sequence defined on the mesh ${i h_{n}}_{i = 0, \dots, n}$ such as ${Y_{i h_{n}}}_{i = 0, \dots, n}$ , ${X_{i h_{n}}}_{i = 0, \dots, n}$ and ${ε_{i h_{n}}}_{i = 0, \dots, n}$ .

Remark 1.

Since the observation is masked by the exogenous noise, it should be transformed to obtain the undermined process ${X_{t}}_{t \geq 0}$ . As illustrated by Nakakita and Uchida (2018b), the sequence ${{¯ Y}_{j}}_{j = 0, \dots, k_{n} - 1}$ can extract the state of the latent process ${X_{t}}_{t \geq 0}$ in the sense of the statement of Lemma 2.

$G_{t} := σ (x_{0}, w_{s} : s \leq t)$ , $G_{j, i}^{n} := G_{j Δ_{n} + i h_{n}}$ , $G_{j}^{n} := G_{j, 0}^{n}$ , $A_{j, i}^{n} := σ (ε_{ℓ h_{n}} : ℓ \leq j p_{n} + i - 1)$ , $A_{j}^{n} := A_{j, 0}^{n}$ , $H_{j, i}^{n} := G_{j, i}^{n} \lor A_{j, i}^{n}$ and $H_{j}^{n} := H_{j, 0}^{n}$ .
We define the real-valued function as for $l_{1}, l_{2}, l_{3}, l_{4} = 1, \dots, d$ :

$V ((l_{1}, l_{2}), (l_{3}, l_{4}))$

$+ \frac{3}{2} (Λ_{⋆}^{(l_{1}, l_{3})} Λ_{⋆}^{(l_{2}, l_{4})} + Λ_{⋆}^{(l_{1}, l_{4})} Λ_{⋆}^{(l_{2}, l_{3})}),$

and with the function $σ$ as for $i = 1, \dots, d$ and $j = i, \dots, d$ ,

$σ (i, j) := {\begin{matrix} j & if i = 1, \sum_{ℓ = 1}^{i - 1} (d - ℓ + 1) + j - i + 1 & if i > 1, \end{matrix}$

we define the matrix $W_{1}$ as for $i_{1}, i_{2} = 1, \dots, d (d + 1) / 2$ ,

$W_{1}^{(i_{1}, i_{2})} := V (σ^{- 1} (i_{1}), σ^{- 1} (i_{2})) .$

Let

	${B_{κ} (x) ∣ ∣ κ = 1, \dots, m_{1}, B_{κ} = (B_{κ}^{(j_{1}, j_{2})})_{j_{1}, j_{2}}},$
	${f_{λ} (x) ∣ ∣ λ = 1, \dots, m_{2}, f_{λ} = (f_{λ}^{(1)}, \dots, f_{λ}^{(d)})}$

be sequences of $R^{d} \otimes R^{d}$ -valued functions and $R^{d}$ -valued ones respectively such that the components of themselves and their derivative with respect to $x$ are polynomial growth functions for all $κ$ and $λ$ . Then we define the following matrix-valued functionals, for ${¯ B}_{κ} := \frac{1}{2} (B_{κ} + B_{κ}^{T})$ ,

	${(W_{2}^{(τ)} ({B_{κ} : κ = 1, \dots, m_{1}}))}^{(κ_{1}, κ_{2})}$
	$:= {\begin{matrix} ν (t r {({¯ B}_{κ_{1}} A {¯ B}_{κ_{2}} A) (\cdot)}) & if τ \in (1, 2), ν (t r {({¯ B}_{κ_{1}} A {¯ B}_{κ_{2}} A + 4 {¯ B}_{κ_{1}} A {¯ B}_{κ_{2}} Λ_{⋆} + 12 {¯ B}_{κ_{1}} Λ_{⋆} {¯ B}_{κ_{2}} Λ_{⋆}) (\cdot)}) & if τ = 2, \end{matrix}$
	${(W_{3} ({f_{λ} : λ = 1, \dots, m_{2}}))}_{3}^{(λ_{1}, λ_{2})}$
	$:= ν ((f_{λ_{1}} A {(f_{λ_{2}})}_{λ_{2}}^{T}) (\cdot)),$

where $ν = ν_{θ^{⋆}}$ is the invariant measure of $X_{t}$ discussed in the following assumption [A1]-(iv), and for all function $f$ on $R^{d}$ , $ν (f (\cdot)) := \int_{R^{d}} f (x) ν (d x)$ .

With respect to $X_{t}$ , we assume the following conditions.

- ${inf}_{x, α} det A (x, α) > 0$ .
- For some constant $C$ , for all $x_{1}, x_{2} \in R^{d}$ ,
  
  $sup α \in Θ_{1} ∥ a (x_{1}, α) - a (x_{2}, α) ∥ + sup β \in Θ_{2} | b (x_{1}, β) - b (x_{2}, β) | \leq C | x_{1} - x_{2} |$
- For all $p \geq 0$ , ${sup}_{t \geq 0} E_{θ^{⋆}} [{| X_{t} |}_{t}^{p}] < \infty$ .
- There exists an unique invariant measure $ν = ν_{0}$ on $(R^{d}, B (R^{d}))$ and for all $p \geq 1$ and $f \in L^{p} (ν)$ with polynomial growth,
  
  $\frac{1}{T} \int_{0}^{T} f (X_{t}) d t \to^{P} \int_{R^{d}} f (x) ν (d x) .$
- For any polynomial growth function $g : R^{d} \to R$ satisfying $\int_{R^{d}} g (x) ν (d x) = 0$ , there exist $G (x)$ , $\partial_{x^{(i)}} G (x)$ with at most polynomial growth for $i = 1, \dots, d$ such that for all $x \in R^{d}$ ,
  
  $L_{θ^{⋆}} G (x) = - g (x),$
  
  where $L_{θ^{⋆}}$ is the infinitesimal generator of $X_{t}$ .

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), ${sup}_{x, α} A (x, α) < \infty$ and $^{\exists} c_{0} > 0$ , $M_{0} > 0$ and $γ \geq 0$ such that for all $β \in Θ_{2}$ and $x \in R^{d}$ satisfying $| x | \geq M_{0}$ ,

\frac{1}{| x |} x^{T} b (x, β)

\leq - c_{0} {| x |}^{γ} .

There exists $C > 0$ such that $a : R^{d} \times Θ_{1} \to R^{d} \otimes R^{r}$ and $b : R^{d} \times Θ_{2} \to R^{d}$ have continuous derivatives satisfying

$sup α \in Θ_{1} ∣ ∣ \partial_{x}^{j} \partial_{α}^{i} a (x, α) ∣ ∣$ $\leq C {(1 + | x |)}^{C}, 0 \leq i \leq 4, 0 \leq j \leq 2,$

$sup β \in Θ_{2} ∣ ∣ \partial_{x}^{j} \partial_{β}^{i} b (x, β) ∣ ∣$ $\leq C {(1 + | x |)}^{C}, 0 \leq i \leq 4, 0 \leq j \leq 2.$

With the invariant measure $ν$ , we define

	$Y_{1}^{τ} (α; ϑ^{⋆})$	$:= - \frac{1}{2} \int {t r (A^{τ} {(x, α, Λ_{⋆})}_{⋆}^{- 1} A^{τ} (x, α^{⋆}, Λ_{⋆}) - I_{d}) + log \frac{det A^{τ} (x, α, Λ_{⋆})}{det A^{τ} (x, α^{⋆}, Λ_{⋆})}} ν (d x),$
	$Y_{2} (β; ϑ^{⋆})$

where $A^{τ} (x, α, Λ) := A (x, α) + 3 Λ 1_{{2}} (τ)$ . For these functions, let us assume the following identifiability conditions hold.

For all $τ \in (1, 2]$ , there exists a constant $χ (α^{⋆}) > 0$ such that $Y_{1}^{τ} (α; θ^{⋆}) \leq - χ (θ^{⋆}) {| α - α^{⋆} |}^{2}$ for all $α \in Θ_{1}$ .
For all $τ \in (1, 2]$ , there exists a constant $χ^{'} (β^{⋆}) > 0$ such that $Y_{2} (β; θ^{⋆}) \leq - χ^{'} (θ^{⋆}) {| β - β^{⋆} |}^{2}$ for all $β \in Θ_{2}$ .

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

For any $k > 0$ , $ε_{i h_{n}}$ has $k$ -th moment and the components of $ε_{i h_{n}}$ are independent of the other components for all $i$ , ${w_{t}}_{t \geq 0}$ and $x_{0}$
. In addition, for all odd integer
$k$ , $i = 0, \dots, n$ , $n \in N$ , and $ℓ = 1, \dots, d$ , $E_{θ^{⋆}} [{(ε_{i h_{n}}^{(ℓ)})}^{k}] = 0$ , and $E_{θ^{⋆}} [ε_{i h_{n}}^{\otimes 2}] = I_{d}$ .

The assumption below determines the balance of convergence or divergence of several parameters. Note that $τ$ is a tuning parameter and hence we can control it arbitrarily in its space $(1, 2]$ .

$p_{n} = h_{n}^{- 1 / τ}$ , $τ \in (1, 2]$ , $h_{n} \to 0$ , $T_{n} = n h_{n} \to \infty$ , $k_{n} = n / p_{n} \to \infty$ , $k_{n} Δ_{n}^{2} \to 0$ for $Δ_{n} := p_{n} h_{n}$ . Furthermore, there exists $ϵ_{0} > 0$ such that $n h_{n} \geq k_{n}^{ϵ_{0}}$ for sufficiently large $n$ .

Remark 3.

Let us denote $ϵ_{1} = ϵ_{0} / 2$ and $f \in C^{1, 1} (R^{d} \times Ξ)$ where $f$ and the components of their derivatives are polynomial growth with respect to $x$ uniformly in $ϑ \in Ξ$ . Then the discussion in Uchida (2010) verifies under [A1] and [A6], for all $M > 0$ ,

sup n \in N E_{θ^{⋆}} ⎡ ⎢ ⎣ sup ϑ \in Ξ {(k_{n}^{ϵ_{1}} ∣ ∣ ∣ ∣ \frac{1}{k_{n}} k_{n} - 2 \sum j = 1 f (X_{j Δ_{n}}, ϑ) - \int_{R^{d}} f (x, ϑ) ν (d x) ∣ ∣ ∣ ∣)}^{M} ⎤ ⎥ ⎦ < \infty .

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 $α$ and the drift one $β$ using this sequence are as follows:

	$H_{1, n}^{τ} (α; Λ) := - \frac{1}{2} k_{n} - 2 \sum j = 1 ({(\frac{2}{3} Δ_{n} A_{n}^{τ} ({¯ Y}_{j - 1}, α, Λ))}_{n}^{- 1} [{({¯ Y}_{j + 1} - {¯ Y}_{j})}_{j + 1}^{\otimes 2}] + log det A_{n}^{τ} ({¯ Y}_{j - 1}, α, Λ)),$
	$H_{2, n} (β; α) := - \frac{1}{2} k_{n} - 2 \sum j = 1 ({(Δ_{n} A ({¯ Y}_{j - 1}, α))}_{n}^{- 1} [{({¯ Y}_{j + 1} - {¯ Y}_{j} - Δ_{n} b ({¯ Y}_{j - 1}, β))}_{j + 1}^{\otimes 2}]),$

where $A_{n}^{τ} (x, α, Λ) := A (x, α) + 3 Δ_{n}^{\frac{2 - τ}{τ - 1}} Λ$ . We set the adaptive ML-type estimator ${^Λ}_{n}$ , ${^α}_{n}$ and ${^β}_{n}$ such that

	${^Λ}_{n}$	$:= \frac{1}{2 n} n - 1 \sum i = 0 {(Y_{(i + 1) h_{n}} - Y_{i h_{n}})}_{(i + 1) h_{n}}^{\otimes 2},$
	$H_{1, n}^{τ} ({^α}_{n}; {^Λ}_{n})$	$= sup α \in Θ_{1} H_{1, n}^{τ} (α; {^Λ}_{n}),$
	$H_{2, n} ({^β}_{n}; {^α}_{n})$	$= sup β \in Θ_{2} H_{2, n} (β; {^α}_{n}) .$

Assume that $π_{ℓ}$ , $ℓ = 1, 2$ are continuous and $0 < {inf}_{θ_{ℓ} \in Θ_{ℓ}} π_{ℓ} (θ_{ℓ}) < {sup}_{θ_{ℓ} \in Θ_{ℓ}} π_{ℓ} (θ_{ℓ}) < \infty$ , and denote the adaptive Bayes-type estimators

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

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 $u_{1} \in R^{m_{1}}$ and $u_{2} \in R^{m_{2}}$

	$Z_{1, n}^{τ} (u_{1}; {^Λ}_{n}, α^{⋆})$	$:= exp (H_{1, n}^{τ} (α^{⋆} + k_{n}^{- 1 / 2} u_{1}; {^Λ}_{n}) - H_{1, n}^{τ} (α^{⋆}; {^Λ}_{n})),$
	$Z_{2, n}^{M L} (u_{2}; {^α}_{n}, β^{⋆})$	$:= exp (H_{2, n} (β^{⋆} + T_{n}^{- 1 / 2} u_{2}; {^α}_{n}) - H_{2, n} (β^{⋆}; {^α}_{n})),$
	$Z_{2, n}^{B a y e s} (u_{2}; {~ α}_{n}, β^{⋆})$	$:= exp (H_{2, n} (β^{⋆} + T_{n}^{- 1 / 2} u_{2}; {~ α}_{n}) - H_{2, n} (β^{⋆}; {~ α}_{n})),$

and some sets

	$U_{1, n}^{τ} (α^{⋆})$	$:= {u_{1} \in R^{m_{1}}; α^{⋆} + k_{n}^{- 1 / 2} u_{1} \in Θ_{1}},$
	$U_{2, n} (β^{⋆})$	$:= {u_{2} \in R^{m_{2}}; β^{⋆} + T_{n}^{- 1 / 2} u_{2} \in Θ_{2}},$

and for $r \geq 0$ ,

	$V_{1, n}^{τ} (r, α^{⋆})$	$:= {u_{1} \in U_{1, n}^{τ} (α^{⋆}); r \leq \| u_{1} \|},$
	$V_{2, n} (r, β^{⋆})$	$:= {u_{2} \in U_{2, n} (β^{⋆}); r \leq \| u_{2} \|} .$

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

	$I^{τ} (ϑ^{⋆})$	$:= d i a g {W_{1}, I^{(2, 2), τ}, I^{(3, 3)}} (ϑ^{⋆}),$
	$J^{τ} (ϑ^{⋆})$	$:= d i a g {I_{d (d + 1) / 2}, J^{(2, 2), τ}, J^{(3, 3)}} (ϑ^{⋆}),$

where for $i_{1}, i_{2} \in {1, \dots, m_{1}}$ ,

	$I^{(2, 2), τ} (ϑ^{⋆})$	$:= W_{2}^{(τ)} ({\frac{3}{4} {(A^{τ})}^{- 1} (\partial_{α^{(k_{1})}} A) {(A^{τ})}^{- 1} (\cdot, ϑ^{⋆}) : k_{1} = 1, \dots, m_{1}}),$
	$J^{(2, 2), τ} (ϑ^{⋆})$	$:= {[\frac{1}{2} ν (t r {{(A^{τ})}^{- 1} (\partial_{α^{(i_{1})}} A) {(A^{τ})}^{- 1} (\partial_{α^{(i_{2})}} A)} (\cdot, ϑ^{⋆}))]}_{i_{1}, i_{2}},$

and for $j_{1}, j_{2} \in {1, \dots, m_{2}}$ ,

I^{(3, 3)} (θ^{⋆})

= J^{(3, 3)} (θ^{⋆}) := {[ν ({(A)}^{- 1} [\partial_{β^{(j_{1})}} b, \partial_{β^{(j_{2})}} b] (\cdot, θ^{⋆}))]}_{j_{1}, j_{2}} .

We also denote ${^θ}_{ε, n} := v e c h {^Λ}_{n}$ and $θ_{ε}^{⋆} := v e c h Λ_{⋆}$ .

Theorem 1.

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

The polynomial-type large deviation inequalities hold: for all $L > 0$ , there exists a constant $C (L)$ such that for all $r > 0$ ,

	$P_{θ^{⋆}} ⎡ ⎣ sup u_{1} \in V_{1, n}^{τ} (r, α^{⋆}) Z_{1, n}^{τ} (u_{1}; {^Λ}_{n}, α^{⋆}) \geq e^{- r} ⎤ ⎦$	$\leq \frac{C (L)}{r^{L}},$
	$P_{θ^{⋆}} [sup u_{2} \in V_{2, n} (r, β^{⋆}) Z_{2, n}^{M L} (u_{2}; {^α}_{n}, β^{⋆}) \geq e^{- r}]$	$\leq \frac{C (L)}{r^{L}},$
	$P_{θ^{⋆}} [sup u_{2} \in V_{2, n} (r, β^{⋆}) Z_{2, n}^{B a y e s} (u_{2}; {~ α}_{n}, β^{⋆}) \geq e^{- r}]$	$\leq \frac{C (L)}{r^{L}} .$

The convergences of moment hold:

	$E_{θ^{⋆}} [f (\sqrt{n} ({^θ}_{ε, n} - θ_{ε}^{⋆}), \sqrt{k_{n}} ({^α}_{n} - α^{⋆}), \sqrt{T_{n}} ({^β}_{n} - β^{⋆}))]$	$\to E [f (ζ_{0}, ζ_{1}, ζ_{2})],$
	$E_{θ^{⋆}} [f (\sqrt{n} ({^θ}_{ε, n} - θ_{ε}^{⋆}), \sqrt{k_{n}} (~ α_{n} - α^{⋆}), \sqrt{T_{n}} ({~ β}_{n} - β^{⋆}))]$	$\to E [f (ζ_{0}, ζ_{1}, ζ_{2})],$

where

(ζ_{0}, ζ_{1}, ζ_{2}) \sim N_{d (d + 1) / 2 + m_{1} + m_{2}} (0, {(J^{τ} (ϑ^{⋆}))}^{- 1} (I^{τ} (ϑ^{⋆})) {(J^{τ} (ϑ^{⋆}))}^{- 1})

and $f$ 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:

ζ_{j + 1, n} := \frac{1}{p_{n}} p_{n} - 1 \sum i = 0 \int_{j Δ_{n} + i h_{n}}^{(j + 1) Δ_{n}} d w_{s}, ζ_{j + 2, n}^{'} := \frac{1}{p_{n}} p_{n} - 1 \sum i = 0 \int_{(j + 1) Δ_{n}}^{(j + 1) Δ_{n} + i h_{n}} d w_{s} .

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

Lemma 1.

$ζ_{j + 1, n}$ and $ζ_{j + 1, n}^{'}$ are $G_{j + 1}^{n}$ -measurable, independent of $G_{j}^{n}$ and Gaussian.These variables have the next decompositions:

	$ζ_{j + 1, n}$	$= \frac{1}{p_{n}} p_{n} - 1 \sum k = 0 (k + 1) \int_{j Δ_{n} + k h_{n}}^{j Δ_{n} + (k + 1) h_{n}} d w_{s},$
	$ζ_{j + 1, n}^{'}$	$= \frac{1}{p_{n}} p_{n} - 1 \sum k = 0 (p_{n} - k - 1) \int_{j Δ_{n} + k h_{n}}^{j Δ_{n} + (k + 1) h_{n}} d w_{s} .$

The evaluation of the following conditional expectations holds:

	$E_{θ^{⋆}} [ζ_{j, n} \| G_{j}^{n}] = E_{θ^{⋆}} [ζ_{j + 1, n}^{'} \| G_{j}^{n}]$	$= 0,$

Quasi-likelihood analysis of an ergodic diffusion plus noise

Authors

Inference for ergodic diffusions plus noise

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

Hybrid estimation for ergodic diffusion processes based on noisy discrete observations

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

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

Adaptive test for ergodic diffusions plus noise

Penalized quasi likelihood estimation for variable selection

1. Introduction

2. Notation and assumption

Remark 1.

Remark 2.

Remark 3.

3. Quasi-likelihood analysis

Theorem 1.

3.1. Evaluation for local means

Lemma 1.

	$V ((l_{1}, l_{2}), (l_{3}, l_{4}))$

	$+ \frac{3}{2} (Λ_{⋆}^{(l_{1}, l_{3})} Λ_{⋆}^{(l_{2}, l_{4})} + Λ_{⋆}^{(l_{1}, l_{4})} Λ_{⋆}^{(l_{2}, l_{3})}),$

	$sup α \in Θ_{1} ∣ ∣ \partial_{x}^{j} \partial_{α}^{i} a (x, α) ∣ ∣$	$\leq C {(1 + \| x \|)}^{C}, 0 \leq i \leq 4, 0 \leq j \leq 2,$
	$sup β \in Θ_{2} ∣ ∣ \partial_{x}^{j} \partial_{β}^{i} b (x, β) ∣ ∣$	$\leq C {(1 + \| x \|)}^{C}, 0 \leq i \leq 4, 0 \leq j \leq 2.$

Quasi-likelihood analysis of an ergodic diffusion plus noise

Authors

Related Research

Inference for ergodic diffusions plus noise

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

Hybrid estimation for ergodic diffusion processes based on noisy discrete observations

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

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

Adaptive test for ergodic diffusions plus noise

Penalized quasi likelihood estimation for variable selection

This week in AI

1. Introduction

2. Notation and assumption

Remark 1.

Remark 2.

Remark 3.

3. Quasi-likelihood analysis

Theorem 1.

3.1. Evaluation for local means

Lemma 1.