Hybrid estimation for ergodic diffusion processes based on noisy discrete observations

1. Introduction

We consider the $d$ -dimensional ergodic diffusion process defined by the stochastic differential equation

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 $d$

-dimensional random vector independent of

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

α \in Θ_{1}

and

β \in Θ_{2}

are unknown parameters,

Θ_{i} \subset R^{m_{i}}

is bounded, open and convex sets in

R^{m_{i}}

admitting Sobolev’s inequalities for embedding

W^{1, p} (Θ_{i}) ↪ C (_{i})

(see Adams and Fournier, 2003; Yoshida, 2011) for

i = 1, 2

θ^{⋆} = (α^{⋆}, β^{⋆})

is the true parameter vector, 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.

Our concern in this paper is the estimation of $θ := (α, β) \in Θ := Θ_{1} \times Θ_{2}$ with long-term, discrete and noisy observation defined as the sequence of the $d$ -dimensional random vectors ${Y_{i h_{n}}}_{i = 0, \dots, n}$ such that for all $i = 0, \dots, n$ ,

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

where $h_{n} > 0$ is the discretisation step satisfying $h_{n} \to 0$ and $T_{n} := n h_{n} \to \infty$ as $n \to \infty$ , ${ε_{i h_{n}}}_{i = 0, \dots, n}$ is the i.i.d. sequence of $d$ -dimensional random vectors with $E [ε_{0}] = 0$ and $V a r (ε_{0}) = I_{d}$ , the components of $ε_{0}$ are independent of each other and have symmetric distribution with respect to 0, and $Λ$ is a $d \times d$

real matrix being positive semi-definite, defining the variance of noise term

Λ^{1 / 2} ε_{i h_{n}}

. Let us assume the half-vectorisation

θ_{ε} := v e c h Λ

is in the bounded, convex and open parameter space

Θ_{ε} \subset R^{d (d + 1) / 2}

, and denote

Ξ := Θ_{ε} \times Θ_{1} \times Θ_{2}

Statistical inference for ergodic diffusion processes has been researched for the last few decades, for instance, see Florens-Zmirou (1989); Yoshida (1992); Bibby and Sørensen (1995); Kessler (1995, 1997); Kutoyants (2004); Iacus (2008); De Gregorio and Iacus (2013, 2018); Iacus and Yoshida (2018) and references therein. The parametric inference for ergodic diffusion processes with discrete and noisy observations has been researched in Favetto (2014, 2016) and Nakakita and Uchida (2017, 2018a, 2018b, 2018c). For parametric estimation for non-ergodic diffusion processes in the presence of market microstructure noise, see Ogihara (2018). Favetto (2014) proposes a simultaneous quasi likelihood function $H_{n} (α, β)$ which necessitates optimisation with respect to both $α$ and $β$ and shows maximum likelihood (ML) type estimators $({^α}_{n}, {^β}_{n}) = arg max H_{n} (α, β)$ have consistency even if the variance of noise is unknown; Favetto (2016) discusses asymptotic normality of the estimator proposed in Favetto (2014) when the variance of noise is known; Nakakita and Uchida (2017, 2018b) suggest adaptive quasi likelihood functions $H_{1, n} (α)$ and $H_{2, n} (β)$ which succeed in lessening the computational burden in comparison to Favetto (2014, 2016), and prove consistency and asymptotic normality of the adaptive ML type estimators corresponding to the quasi likelihoods; Nakakita and Uchida (2018a)

use those quasi likelihood functions for likelihood-ratio-type test and show the asymptotic behaviour of test statistics under both null hypotheses and alternative ones;

Nakakita and Uchida (2018c) analyse those quasi likelihood functions with the framework of quasi likelihood analysis (QLA) proposed by Yoshida (2011), and show the polynomial large deviation inequality (PLDI) for the quasi likelihood functions and consequently the convergence of moments of adaptive ML type estimators and adaptive Bayes type ones. For details of adaptive estimation for diffusion processes, see Yoshida (1992, 2011); Uchida and Yoshida (2012, 2014).

In general, however, the optimisation of quasi likelihood functions for diffusion processes, regardless of noise existence, is strongly dependent on initial values, especially in the case where the volatility function $a$ or drift function $b$ are nonlinear with respect to parameters. Hence, Kaino et al. (2017) and Kaino and Uchida (2018a, b) propose hybrid multi-step estimation procedure for diffusion processes where initial values in optimisation are derived from Bayes type estimation with reduced sample sizes and the sequential optimisation with these initial values is implemented, which inherits the idea of hybrid multi-step estimation for diffusion processes with full sample sizes in Kamatani and Uchida (2015) (see also Kutoyants, 2017). In this research, we also consider hybrid multi-step estimation and apply the idea into inference problem, in particular, PLDI for the quasi likelihood functions and the convergence of moments of estimators for discretely and noisily observed ergodic diffusion processes since PLDI and convergence of moment of estimators are key tools to show the mathematical validity of information criteria for model selection problems (see Uchida, 2010; Fujii and Uchida, 2014; Eguchi and Masuda, 2018).

This paper consists of the following parts: Section 2 deals with the notation; we define the initial and multi-step estimators and set the main theorem for the polynomial-type large deviation inequalities, moment estimates of the Bayes type estimators and convergences of moments in Section 3; a concrete example and simulation results are given in Section 4, the conclusions of this work are summarised in Section 5, and finally we give the proofs of the results in Section 6.

2. Notation and Assumption

First of all we give the notation used throughout this paper.

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 $B$ of the same size, $A [B] := t r (A B^{T})$ . 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 $i = 1, 2, 3$ and $τ_{i} \in (1, 2]$ , $p_{τ_{i}, n} := h_{n}^{- 1 / τ_{i}}$ , $Δ_{τ_{i}, n} := p_{τ_{i}, n} h_{n}$ , $k_{τ_{i}, n} := n / p_{τ_{i}, n} = n h_{n}^{1 / τ_{i}}$ .
With respect to filtration, for all $i = 1, 2, 3$ , $G_{t} := σ (x_{0}, w_{s} : s \leq t)$ , $G_{j, i, n}^{τ_{i}} := G_{j Δ_{τ_{i}, n} + i h_{n}}$ , $G_{j, n}^{τ_{i}} := G_{j, 0, n}^{τ_{i}}$ , $A_{j, i, n}^{τ_{i}} := σ (ε_{ℓ h_{n}} : ℓ \leq j p_{τ_{i}, n} + i - 1)$ , $A_{j, n}^{τ_{i}} := A_{j, 0, n}^{τ_{i}}$ , $H_{j, i, n}^{τ_{i}} := G_{j, i, n}^{τ_{i}} \lor A_{j, i, n}^{τ_{i}}$ and $H_{j, n}^{τ_{i}} := H_{j, 0, n}^{τ_{i}}$ .

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

- ${inf}_{x, α} det A (x, α) > 0$ .
- For a constant $C$ , for all $x_{1}, x_{2} \in R^{d}$ ,
  
  $sup α \in Θ_{1} ∥ a (x_{1}, α) - a (x_{2}, α) ∥ + sup β \in Θ_{2} | b (x_{1}, β) - a (x_{2}, β) | \leq C | x_{1} - x_{2} | .$
- For all $p \geq 0$ , ${sup}_{t \geq 0} E_{θ^{⋆}} [{| X_{t} |}_{t}^{p}] < \infty$ .
- There exists a unique invariant measure $ν = ν_{θ^{⋆}}$ 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 1.

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

	$Y_{1}^{τ_{3}} (α; ϑ^{⋆})$	$:= - \frac{1}{2} \int {t r (A^{τ_{3}} {(x, α, Λ^{⋆})}^{- 1} A^{τ_{3}} (x, α^{⋆}, Λ^{⋆}) - I_{d}) + log \frac{det A^{τ_{3}} (x, α, Λ^{⋆})}{det A^{τ_{3}} (x, α^{⋆}, Λ^{⋆})}} ν (d x),$
	$Y_{2} (β; ϑ^{⋆})$
	$V_{1} (α; ϑ^{*})$	$:= - \frac{2}{9} \int_{R^{d}} {∥ A^{τ_{1}} (x, α, Λ_{⋆}) - A^{τ_{1}} (x, α^{⋆}, Λ_{⋆}) ∥}^{2} ν (d x) = - \frac{2}{9} \int_{R^{d}} {∥ A (x, α) - A (x, α^{⋆}) ∥}^{2} ν (d x),$
	$V_{2} (β; ϑ^{*})$	$:= - \frac{1}{2} \int_{R^{d}} {\| b (x, β^{⋆}) - b (x, β^{⋆}) \|}^{2} ν (d x),$

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.

There exist $χ_{1} (α^{⋆}) > 0$ and $χ_{1}^{'} (β^{⋆}) > 0$ such that for all $α \in Θ_{1}$ and $β \in Θ_{2}$ , $V_{1} (α; θ^{⋆}) \leq - χ_{1} (θ^{⋆}) {| α - α^{⋆} |}^{2}$ and $V_{2} (β; θ^{⋆}) \leq - χ_{1}^{'} (θ^{⋆}) {| β - β^{⋆} |}^{2}$ .
For all $τ_{3} \in (1, 2]$ , there exist $χ_{2} (α^{⋆}) > 0$ and $χ_{2}^{'} (β^{⋆}) > 0$ such that for all $α \in Θ_{1}$ and $β \in Θ_{2}$ , and $Y_{2} (β; θ^{⋆}) \leq - χ_{2}^{'} (θ^{⋆}) {| β - β^{⋆} |}^{2}$ .

The next assumption is concerned with 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 $E_{θ^{⋆}} [ε_{i h_{n}}^{\otimes 2}] = I_{d}$ .

There exist $γ \in (2 / 3, 1)$ and $γ^{'} \in (0, γ]$ such that $n^{- γ} \leq h_{n} \leq n^{- γ^{'}}$ for sufficiently large $n$ .

Remark 2.

$γ^{'}$ should be smaller than or equal to $γ$ such that $n^{- γ} \leq n^{- γ^{'}}$ . $γ$ should be larger than $2 / 3$ and smaller than $1$ ; otherwise for some $C_{1}, C_{2} > 0$ , $k_{τ, n} Δ_{τ, n}^{2} = n p_{τ, n} h_{n}^{2} = n h_{n}^{2 - 1 / τ} \geq n^{1 - γ (2 - 1 / τ)} \geq n^{1 - 3 γ / 2} > C_{1} > 0$ , which must converge to 0 for $τ = τ_{3}$ in Theorem 2, or $T_{n} = n h_{n} \leq n^{1 - γ^{'}} < C_{2}$ as $γ \geq γ^{'}$ , which must diverge in entire discussion. In addition, note that under [A6], $k_{n} = n h_{n}^{1 / τ_{i}} \geq n^{1 - γ / τ_{i}} \geq n^{1 - γ} \to \infty$ for all $i = 1, 2, 3$ .

3. Multi-step estimator and PLDI

3.1. Setting of the initial and multi-step estimators

We define sequences of local means such that ${{¯ Y}_{τ, j}}_{j = 0, \dots, k_{τ, n} - 1}$ , ${{¯ X}_{τ, j}}_{j = 0, \dots, k_{τ, n} - 1}$ and ${{¯ ε}_{τ, j}}_{j = 0, \dots, k_{τ, n} - 1}$ , where

{¯ Y}_{τ, j}

= \frac{1}{p_{τ, n}} p_{τ, n} - 1 \sum i = 0 Y_{j Δ_{τ, n} + i h_{n}},

{¯ X}_{τ, j}

= \frac{1}{p_{τ, n}} p_{τ, n} - 1 \sum i = 0 X_{j Δ_{τ, n} + i h_{n}},

{¯ ε}_{τ, j}

= \frac{1}{p_{τ, n}} p_{τ, n} - 1 \sum i = 0 ε_{j Δ_{τ, n} + i h_{n}}

for $j = 0, \dots, k_{τ, n} - 1$ and $τ = τ_{1}, τ_{2}, τ_{3}$ . For the detailed properties of local means, see Favetto (2014, 2016); Nakakita and Uchida (2017, 2018b, 2018c).

We set for $i = 1, 2$ , $η_{i} \in (γ, 1]$ and ${n - -}_{η_{i}}$ such that ${n - -}_{η_{i}} = n^{η_{i}} \leq n$ satisfying ${T - -}_{η_{i}, n} := {n - -}_{η_{i}} h_{n} \leq T_{n} := n h_{n}$ (then it holds ${T - -}_{η_{i}, n} \to \infty$ ), and correspondingly ${k - -}_{η_{i}, τ_{i}, n} := {n - -}_{η_{i}} / p_{τ_{i}, n} = n^{η_{i}} h_{n}^{1 / τ_{i}}$ .

Remark 3.

$η_{1}$ and $η_{2}$ should be larger than $γ$ to support the divergence ${T - -}_{η_{i}, n} := n^{η_{i}} h_{n} \to \infty$ . $η_{1}$ and $η_{2}$ actually work to make $(η_{1} - γ / τ_{1}) / (1 - γ^{'} / τ_{3}) > 0$ and $(η_{2} - γ) / (1 - γ^{'}) > 0$ , which are the quantities appearing in Remark 6.

Let us set $q_{1}, q_{2} \in (0, 1 / 2]$ and the following quasi likelihood functions:

	$W_{1, τ_{1}, n} (α \| Λ)$	$:= - \frac{1}{2} {k - -}_{η_{1}, τ_{1}, n} - 2 \sum j = 1 {∥ ∥ ∥ Δ_{τ_{1}, n}^{- 1} {({¯ Y}_{τ_{1}, j + 1} - {¯ Y}_{τ_{1}, j})}_{τ_{1}, j + 1}^{\otimes 2} - \frac{2}{3} A_{τ_{1}, n} ({¯ Y}_{τ_{1}, j - 1}, α, Λ) ∥ ∥ ∥}_{τ_{1}, n}^{2},$
	$W_{2, τ_{2}, n} (β)$	$:= - \frac{1}{2} {k - -}_{η_{2}, τ_{2}, n} - 2 \sum j = 1 Δ_{τ_{2}, n}^{- 1} {∣ ∣ {¯ Y}_{τ_{2}, j + 1} - {¯ Y}_{τ_{2}, j} - Δ_{τ_{2}, n} b ({¯ Y}_{τ_{2}, j - 1}, β) ∣ ∣}_{τ_{2}, j + 1}^{2} .$

Using these quasi likelihood functions, we also define the next two functions such that

H_{1, τ_{1}, n}^{(0)} (α | Λ)

= \frac{1}{{k - -}_{η_{1}, τ_{1}, n}^{1 - 2 q_{1}}} W_{1, τ_{1}, n} (α | Λ),

H_{2, τ_{2}, n}^{(0)} (β)

= \frac{1}{{T - -}_{η_{2}, n}^{1 - 2 q_{2}}} W_{2, τ_{2}, n} (β) .

Then, the initial estimators are defined as follows:

	${^Λ}_{n}$	$:= \frac{1}{2 n} n - 1 \sum i = 0 {(Y_{(i + 1) h_{n}} - Y_{i h_{n}})}_{(i + 1) h_{n}}^{\otimes 2},$
	${~ α}_{q_{1}, τ_{1}, n}^{(0)}$
	${~ β}_{q_{2}, τ_{2}, n}^{(0)}$	$:= \frac{\int_{Θ_{2}} β exp (H_{2, τ_{2}, n}^{(0)} (β)) π_{2} (β) d β}{\int_{Θ_{2}} exp (H_{2, τ_{2}, n}^{(0)} (β)) π_{2} (β) d β},$

where $0 < {inf}_{α \in Θ_{1}} π_{1} (α) \leq {sup}_{α \in Θ_{1}} π_{1} (α) < \infty$ and $0 < {inf}_{β \in Θ_{2}} π_{2} (β) \leq {sup}_{β \in Θ_{2}} π_{2} (β) < \infty$ . Note that ${^Λ}_{n}$ uses the whole data.

In the next place, we define the hybrid multi-step estimators. We introduce the quasi likelihood functions in Nakakita and Uchida (2018c) such that

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

where $A_{n}^{τ_{3}} (x, α, Λ) := A (x, α) + 3 Δ_{n}^{\frac{2 - τ_{3}}{τ_{3} - 1}} Λ$ . As Kamatani and Uchida (2015) and Kamatani et al. (2016), let us denote

$J_{1, n} (α)$	$:= \frac{1}{k_{τ_{3}, n}} \partial_{α}^{2} H_{1, τ_{3}, n} (α \| {^Λ}_{n}),$	$J_{1, n} (β)$	$:= \frac{1}{T_{n}} \partial_{β}^{2} H_{2, τ_{3}, n} (β \| {^α}_{J_{1}, n}),$
$K_{1, n} (α)$	$:= {J_{1, n} (α) is invertible.},$	$K_{2, n} (β)$	$:= {J_{2, n} (β) is invertible.},$
${¯ ¯¯ ¯ J}_{1, n} (α)$		${¯ ¯¯ ¯ J}_{2, n} (β)$	$:= J_{2, n} (β) 1_{K_{2, n} (β)} + I_{m_{2}} 1_{K_{2, n}^{c} (β)},$

such that for all $k = 1, \dots, J_{1}$ , $J_{1} := ⌊ - {log}_{2} (q_{1} (η_{1} - γ / τ_{1}) / (1 - γ^{'} / τ_{3})) ⌋$ , and ${^α}_{0, n} := {~ α}_{q_{1}, τ_{1}, n}^{(0)}$

{^α}_{k, n}

:= {^α}_{k - 1, n} - {¯ ¯¯ ¯ J}_{1, n}^{- 1} ({^α}_{k - 1, n}) \frac{1}{k_{τ_{3}, n}} \partial_{α} H_{1, τ_{3}, n} ({^α}_{k - 1, n} | {^Λ}_{n}),

and for all $k = 1, \dots, J_{2}$ , $J_{2} = ⌊ - {log}_{2} (q_{2} (η_{2} - γ) / (1 - γ^{'})) ⌋$ , and ${^β}_{0, n} := {~ β}_{q_{2}, τ_{2}, n}^{(0)}$ ,

{^β}_{k, n}

:= {^β}_{k - 1, n} - {¯ ¯¯ ¯ J}_{2, n}^{- 1} ({^β}_{k - 1, n}) \frac{1}{T_{n}} \partial_{β} H_{2, τ_{3}, n} (^β_{k - 1, n} | {^α}_{J_{1}, n}) .

3.2. PLDIs for the quasi likelihood functions

To examine the asymptotic behaviours of ${^α}_{J_{1}}$ and ${^β}_{J_{2}, n}$ , firstly we will see that the $L^{p}$ -boundedness such that

sup n \in N E [{∣ ∣ {k - -}_{η_{1}, τ_{1}, n}^{q_{1}} ({~ α}_{q_{1}, τ_{1}, n}^{(0)} - α^{⋆}) ∣ ∣}^{M}] + sup n \in N E [{∣ ∣ {T - -}_{η_{2}, n}^{q_{2}} ({~ β}_{q_{2}, τ_{2}, n}^{(0)} - β^{⋆}) ∣ ∣}^{M}] < \infty .

To show these boundedness, we define some random quantities: random fields such that

	$V_{1, τ_{1}, n} (α; ϑ^{*})$
	$:= \frac{1}{{k - -}_{η_{1}, τ_{1}, n}} (W_{1, τ_{1}, n} (α \| {^Λ}_{n}) - W_{1, τ_{1}, n} (α^{⋆} \| {^Λ}_{n}))$

	$- 2 (Δ_{τ_{1}, n}^{- 1} {({¯ Y}_{τ_{1}, j + 1} - {¯ Y}_{τ_{1}, j})}_{τ_{1}, j + 1}^{\otimes 2}) [\frac{2}{3} A_{τ_{1}, n} ({¯ Y}_{τ_{1}, j - 1}, α, {^Λ}_{n})]$
	$+ 2 (Δ_{τ_{1}, n}^{- 1} {({¯ Y}_{τ_{1}, j + 1} - {¯ Y}_{τ_{1}, j})}_{τ_{1}, j + 1}^{\otimes 2}) [\frac{2}{3} A_{τ_{1}, n} ({¯ Y}_{τ_{1}, j - 1}, α^{⋆}, {^Λ}_{n})]),$
	$V_{2, τ_{2}, n} (β; ϑ^{⋆})$

	$= - \frac{1}{2 {k - -}_{η_{2}, τ_{2}, n}} {k - -}_{η_{2}, τ_{2}, n} - 2 \sum j = 1 ({∣ ∣ b ({¯ Y}_{τ_{2}, j - 1}, β) ∣ ∣}^{2} - {∣ ∣ b ({¯ Y}_{τ_{2}, j - 1}, β^{⋆}) ∣ ∣}^{2}$
	$- 2 Δ_{τ_{2}, n}^{- 1} ({¯ Y}_{τ_{2}, j + 1} - {¯ Y}_{τ_{2}, j}) [b ({¯ Y}_{τ_{2}, j - 1}, β)]$
	$+ 2 Δ_{τ_{2}, n}^{- 1} ({¯ Y}_{τ_{2}, j + 1} - {¯ Y}_{τ_{2}, j}) [b ({¯ Y}_{τ_{2}, j - 1}, β^{⋆})]);$

score functions such that

	$S_{1, τ_{1}, n} (ϑ^{⋆})$	$:= - \frac{2}{3 {k - -}_{η_{1}, τ_{1}, n}^{1 - q_{1}}} {k - -}_{η_{1}, τ_{1}, n} - 2 \sum j = 1 (\partial_{α} A (¯ Y_{j - 1}, α^{⋆}))$
		$[Δ_{τ_{1}, n}^{- 1} {({¯ Y}_{τ_{1}, j + 1} - {¯ Y}_{τ_{1}, j})}_{τ_{1}, j + 1}^{\otimes 2} - \frac{2}{3} A_{τ_{1}, n} ({¯ Y}_{τ_{1}, j - 1}, α^{⋆}, {^Λ}_{n})],$
	$S_{2, τ_{2}, n} (ϑ^{⋆})$	$:= - \frac{1}{{T - -}_{η_{1}, n}^{1 - q_{2}}} {k - -}_{η_{2}, τ_{2}, n} - 2 \sum j = 1 (\partial_{β} b ({¯ Y}_{τ_{2}, j - 1}, β^{⋆})) [({¯ Y}_{τ_{2}, j + 1} - {¯ Y}_{τ_{2}, j}) - Δ_{τ_{2}, n} b ({¯ Y}_{τ_{2}, j - 1}, β^{⋆})];$

the observed information matrices such that

	$Γ_{1, τ_{1}, n} (α; ϑ^{⋆}) [u_{1}^{\otimes 2}]$
	$:= - \frac{2}{3 {k - -}_{η_{1}, τ_{1}, n}} {k - -}_{η_{1}, τ_{1}, n} - 2 \sum j = 1 (\partial_{α}^{2} A ({¯ Y}_{j - 1}, α))$
	$[u_{1}^{\otimes 2}, Δ_{τ_{1}, n}^{- 1} {({¯ Y}_{τ_{1}, j + 1} - {¯ Y}_{τ_{1}, j})}_{τ_{1}, j + 1}^{\otimes 2} - \frac{2}{3} A_{τ_{1}, n} ({¯ Y}_{τ_{1}, j - 1}, α, {^Λ}_{n})]$

	$Γ_{2, τ_{2}, n} (β; ϑ^{⋆}) [u_{2}^{\otimes 2}]$
	$:= - \frac{1}{{k - -}_{η_{2}, τ_{2}, n}} {k - -}_{η_{2}, τ_{2}, n} - 2 \sum j = 1 (\partial_{β}^{2} b ({¯ Y}_{j - 1}, β)) [u_{2}^{\otimes 2}, Δ_{τ_{2}, n}^{- 1} ({¯ Y}_{τ_{2}, j + 1} - {¯ Y}_{τ_{2}, j}) - b ({¯ Y}_{τ_{2}, j - 1}, β)]$
	$+ \frac{1}{{k - -}_{η_{2}, τ_{2}, n}} {k - -}_{η_{2}, τ_{2}, n} - 2 \sum j = 1 (\partial_{β} b ({¯ Y}_{τ_{2}, j - 1}, β)) [u_{2}^{\otimes 2}, \partial_{β} b ({¯ Y}_{τ_{2}, j - 1}, β)];$

and the limiting information matrices such that

Γ_{1, τ_{1}} (ϑ^{⋆}) [u_{1}^{\otimes 2}]

:= \frac{4}{9} \int_{R^{d}} (\partial_{α} A (x, α^{⋆})) [u_{1}^{\otimes 2}, \partial_{α} A (x, α^{⋆})

Hybrid estimation for ergodic diffusion processes based on noisy discrete observations

Adaptive and non-adaptive estimation for degenerate diffusion processes

Quasi-likelihood analysis of an ergodic diffusion plus noise

Adaptive estimation for small diffusion processes based on sampled data

Estimation of Cusp Location of Stochastic Processes: a Survey

Empirical and Full Bayes estimation of the type of a Pitman-Yor process

Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed multiscale diffusions

Poisson Source Localization on the Plane. Cusp Case

1. Introduction

2. Notation and Assumption

Remark 1.

Remark 2.

3. Multi-step estimator and PLDI

3.1. Setting of the initial and multi-step estimators

Remark 3.

3.2. PLDIs for the quasi likelihood functions

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

Hybrid estimation for ergodic diffusion processes based on noisy discrete observations

Related Research

Adaptive and non-adaptive estimation for degenerate diffusion processes

Quasi-likelihood analysis of an ergodic diffusion plus noise

Adaptive estimation for small diffusion processes based on sampled data

Estimation of Cusp Location of Stochastic Processes: a Survey

Empirical and Full Bayes estimation of the type of a Pitman-Yor process

Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed multiscale diffusions

Poisson Source Localization on the Plane. Cusp Case

1. Introduction

2. Notation and Assumption

Remark 1.

Remark 2.

3. Multi-step estimator and PLDI

3.1. Setting of the initial and multi-step estimators

Remark 3.

3.2. PLDIs for the quasi likelihood functions