Data driven time scale in Gaussian quasi-likelihood inference

Abstract

We study parametric estimation of ergodic diffusions observed at high frequency. Different from the previous studies, we suppose that sampling stepsize is unknown, thereby making the conventional Gaussian quasi-likelihood not directly applicable. In this situation, we construct estimators of both model parameters and sampling stepsize in a fully explicit way, and prove that they are jointly asymptotically normally distributed. High order uniform integrability of the obtained estimator is also derived. Further, we propose the Schwarz (BIC) type statistics for model selection and show its model-selection consistency. We conducted some numerical experiments and found that the observed finite-sample performance well supports our theoretical findings.

Stochastic expansion of the quasi-marginal log likelihood

We here step away from the main context and present a set of conditions under which a quasi-marginal log likelihood admits a Schwarz type stochastic expansion, by making use of Jasra et al. (2018, Proof of Theorem 2.1).

Let ${\mathbb {H}}_{n}: \varTheta \times \varOmega \rightarrow {\mathbb {R}}$ be a ${\mathcal {C}}^{3}(\varTheta )$-random function where $\varTheta \subset {\mathbb {R}}^{p}$ is a bounded convex domain. Set $\theta =(\alpha ,\beta )\in {\mathbb {R}}^{p_{\alpha }}\times {\mathbb {R}}^{p_{\beta }}$, and let $\theta _{0}=(\alpha _{0},\beta _{0})\in \varTheta$ be a constant, and $D_{n}=D_{n}(\theta _{0})={\mathrm{diag}}\big (\sqrt{r_{1,n}}I_{p_{\alpha }},\, \sqrt{r_{2,n}}I_{p_{\beta }}\big )$, where $(r_{1,n})$ and $(r_{2,n})$ are positive sequences possibly depending on $\theta _{0}$ and satisfying that $r_{1,n}\wedge r_{2,n}\rightarrow \infty$ and that $r_{2,n}/r_{1,n}\rightarrow 0$ as $n\rightarrow \infty$. We then introduce the random field on ${\mathbb {R}}^{p}$ associated with ${\mathbb {H}}_{n}$:

$$\begin{aligned} {\mathbb {Z}}_{n}(u):=\exp \left\{ {\mathbb {H}}_{n}(\theta _{0}+D_{n}^{-1}u) - {\mathbb {H}}_{n}(\theta _{0}) \right\} . \end{aligned}$$

Here we set ${\mathbb {Z}}_{n}\equiv 0$ outside the set ${\mathbb {U}}_{n}={\mathbb {U}}_{n}(\theta _{0}):=D_{n}(\varTheta -\theta _{0})\subset {\mathbb {R}}^{p}$. Let ${\mathfrak {p}}(\theta )$ be a bounded prior probability density on $\varTheta$, which is assumed to be continuous and positive at $\theta _{0}$. Let $\varDelta _{n}(\theta _{0}):=D_{n}^{-1}\partial _{\theta }{\mathbb {H}}_{n}(\theta _{0})$ and

$$\begin{aligned} \varGamma _{0}:={\mathrm{diag}}(\varGamma _{1,0},\,\varGamma _{2,0}), \end{aligned}$$

where $\varGamma _{1,0}\in {\mathbb {R}}^{p_{\alpha }}\otimes {\mathbb {R}}^{p_{\alpha }}$ and $\varGamma _{2,0}\in {\mathbb {R}}^{p_{\beta }}\otimes {\mathbb {R}}^{p_{\beta }}$ are a.s. positive definite random matrices. Further, let

$$\begin{aligned} {\mathbb {Y}}_{1,n}(\theta )&:=\frac{1}{r_{1,n}}\{{\mathbb {H}}_{n}(\alpha ,\beta ) - {\mathbb {H}}_{n}(\alpha _{0},\beta )\}, \\ {\mathbb {Y}}_{2,n}(\beta )&:= \frac{1}{r_{2,n}} \{{\mathbb {H}}_{n}(\alpha _{0},\beta ) - {\mathbb {H}}_{n}(\alpha _{0},\beta _{0})\}, \end{aligned}$$

and ${\mathbb {Y}}_{1}(\alpha )$ and ${\mathbb {Y}}_{2}(\beta )$ be ${\mathbb {R}}$-valued random functions. Finally, we introduce the quadratic random field

$$\begin{aligned} {\mathbb {Z}}^{0}_{n}(u) =\exp \bigg ( \varDelta _{n}(\theta _{0})[u] - \frac{1}{2}\varGamma _{0}[u,u] \bigg ). \end{aligned}$$

Theorem A.1

In addition to the aforementioned setting, suppose the following conditions.

• There exists an a.s. positive definite random matrix $\varSigma _{0}\in {\mathbb {R}}^{p}\otimes {\mathbb {R}}^{p}$ such that

  $$\begin{aligned} \left( \varDelta _{n}(\theta _{0}),\, -D_{n}^{-1}\partial _{\theta }^{2}{\mathbb {H}}_{n}(\theta _{0})D_{n}^{-1}\right) \overset{{\mathcal {L}}}{\rightarrow }\Big ( \varSigma _{0}^{1/2}\eta ,\, \varGamma _{0}\Big ), \end{aligned}$$

  (A.1)

  where $\eta \sim N_{p}(0,I_{p})$ is a random variable defined on an extension of the original probability space.

• We have

  $$\begin{aligned}&\sup _{\beta }\bigg \Vert \frac{1}{\sqrt{r_{1,n}}} \partial _{\alpha }{\mathbb {H}}_{n}(\alpha _{0},\beta ) \bigg \Vert = O_{p}(1), \end{aligned}$$

  (A.2)
  $$\begin{aligned}&\sup _{\beta } \bigg \Vert -\frac{1}{r_{1,n}}\partial _{\alpha }^{2}{\mathbb {H}}_{n}(\alpha _{0},\beta ) - \varGamma _{1,0}\bigg \Vert =o_{p}(1), \end{aligned}$$

  (A.3)
  $$\begin{aligned}&\sup _{\theta }\left\| D_{n}^{-1}\partial _{\theta }^{3}{\mathbb {H}}_{n}(\theta )D_{n}^{-1} \right\| = O_{p}(1). \end{aligned}$$

  (A.4)

• There exists a constant $q\in (0,1)$ for which

  $$\begin{aligned} r_{1,n}^{q/2}\sup _{\theta }\left| {\mathbb {Y}}_{1,n}(\theta )-{\mathbb {Y}}_{1}(\alpha )\right| \vee r_{2,n}^{q/2}\sup _{\beta }\left| {\mathbb {Y}}_{2,n}(\beta )-{\mathbb {Y}}_{2}(\beta )\right| \xrightarrow {{\mathbb {P}}} 0. \end{aligned}$$

  (A.5)

• There exists an a.s. positive random variable $\chi _{0}$ such that for each $\kappa >0$,

  $$\begin{aligned} \sup _{\alpha ;\, |\alpha -\alpha _{0}|\ge \kappa }{\mathbb {Y}}_{1}(\alpha ) \vee \sup _{\beta ;\, |\beta -\beta _{0}|\ge \kappa }{\mathbb {Y}}_{2}(\beta ) \le -\chi _{0}\kappa ^{2}\qquad \text {a.s.} \end{aligned}$$

  (A.6)

Then, any ${\hat{\theta }}_{n} \in {\mathrm{argmax}}{\mathbb {H}}_{n}$ satisfies that $D_{n}({\hat{\theta }}_{n}-\theta _{0})\xrightarrow {{\mathcal {L}}} \varGamma _{0}^{-1}\varSigma _{0}^{1/2}\eta$, and we have

$$\begin{aligned} \int \bigg |{\mathbb {Z}}_{n}(u)\pi (\theta _{0}+D_{n}^{-1}u) - {\mathbb {Z}}_{n}^{0}(u)\pi (\theta _{0}) \bigg |du \xrightarrow {{\mathbb {P}}} 0 \end{aligned}$$

and

$$\begin{aligned} {\mathfrak {L}}_{n}:=\log \bigg (\int _{\varTheta }\exp \{{\mathbb {H}}_{n}(\theta )\}{\mathfrak {p}}(\theta )d\theta \bigg )&={\mathbb {H}}_{n}(\theta _{0}) + \log |D_{n}^{-1}| + \log {\mathfrak {p}}(\theta _{0}) + \frac{p}{2}\log (2\pi ) \\&\quad -\frac{1}{2}\log |\varGamma _{0}| + \frac{1}{2}\varGamma _{0}^{-1}\left[ \varDelta _{n}(\theta _{0})^{\otimes 2}\right] + o_{p}(1). \end{aligned}$$

Further, if $\log |D_{n}^{-1}| = \log |D_{n}^{-1}({\hat{\theta }}_{n})| + o_{p}(1)$ and $\log {\mathfrak {p}}(\theta _{0})=\log {\mathfrak {p}}({\hat{\theta }}_{n})+o_{p}(1)$, then

$$\begin{aligned} {\mathfrak {L}}_{n}&={\mathbb {H}}_{n}({\hat{\theta }}_{n}) + \log |D_{n}^{-1}({\hat{\theta }}_{n})| + \log {\mathfrak {p}}({\hat{\theta }}_{n}) + \frac{p}{2}\log (2\pi ) \\&\quad -\frac{1}{2}\log |-D_{n}^{-1}\partial _{\theta }^{2} {\mathbb {H}}_{n}({\hat{\theta }}_{n})D_{n}^{-1}| + o_{p}(1). \end{aligned}$$

Theorem A.1 can apply to general locally asymptotically quadratic models under weaker conditions compared with Eguchi and Masuda (2018, Theorem 3.7). A formal extension of Theorem A.1 to cases of more than two rates is straightforward.