Yoichi Nishiyama

Goodness of fit test for ergodic diffusions by discrete time observations: an innovation martingale approach

We consider a nonparametric goodness-of-fit test problem for the drift coefficient of one-dimensional ergodic diffusions. Our test is based on the discrete-time observation of the processes, and the diffusion coefficient is a nuisance function which is estimated in some sense in our testing procedure. We prove that the limit distribution of our test is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. We also show that our test is consistent under any fixed alternatives.

Local asymptotic normality of a sequential model for marked point processes and its applications

This paper deals with statistical inference problems for a special type of marked point processes based on the realization in random time intervals [0,τu]. Sufficient conditions to establish the local asymptotic normality (LAN) of the model are presented, and then, certain class of stopping times τu satisfying them is proposed. Using these stopping rules, one can treat the processes within the framework of LAN, which yields asymptotic optimalities of various inference procedures. Applications for compound Poisson processes and continuous time Markov branching processes (CMBP) are discussed. Especially, asymptotically uniformly most powerful tests for criticality of CMBP can be obtained. Such tests do not exist in the case of the non-sequential approach. Also, asymptotic normality of the sequential maximum likelihood estimators (MLE) of the Malthusian parameter of CMBP can be derived, although the non-sequential MLE is not asymptotically normal in the supercritical case.

Z-process method for change point problems with applications to discretely observed diffusion processes

The aim of this paper is to develop a general, unified approach, based on some partial estimation functions which we call “Z-process”, to some change point problems in mathematical statistics. The method proposed can be applied not only to ergodic models but also to some models where the Fisher information matrix is random. Applications to some concrete models, including a parametric model for volatilities of diffusion processes are presented. Simulations for randomly time-transformed Brownian bridge process appearing as the limit of the proposed test statistics are performed with computer intensive use.

Impossibility of weak convergence of kernel density estimators to a non-degenerate law in L 2(ℝ d )

It is well known that the kernel estimator for the probability density f on ℝ d has pointwise asymptotic normality and that its weak convergence in a function space, especially with the uniform topology, is a difficult problem. One may conjecture that the weak convergence in L 2(ℝ d ) could be possible. In this paper, we deny this conjecture. That is, letting , we prove that for any sequence {r n } of positive constants such that r n =o(√n), if the rescaled residual r n ([fcirc] n −f n ) converges weakly to a Borel limit in L 2(ℝ d ), then the limit is necessarily degenerate.

Estimation for the invariant law of an ergodic diffusion process based on high-frequency data

Let a one-dimensional ergodic diffusion process X be observed at time points such that and , where , with p∈(0, 1) being a constant depending also on some conditions on X. We consider the nonparametric estimation problems for the invariant distribution and the invariant density. In both problems, we propose some estimators which are asymptotically normal and asymptotically efficient in some functional senses.

Review on Goodness of Fit Tests for Ergodic Diffusion Processes by Different Sampling Schemes

We review some recent results on goodness of fit test for the drift coefficient of a one-dimensional ergodic diffusion, where the diffusion coefficient is a nuisance function which however is estimated. Using a theory for the continuous observation case, we first present a test based on deterministic discrete time observations of the process. Then we also propose a test based on the data observed discretely in space, that is, the so-called tick time sample scheme. In both sampling schemes the limit distribution of the test is the supremum of the standard Brownian motion, thus the test is asymptotically distribution free. The tests are also consistent under any fixed alternatives.