A test procedure to detect a change in the value of the parameter in the drift of a diffusion process is proposed. The test statistic is asymptotically distribution free under the null hypothesis that the true parameter does not change. Also, the test is shown to be consistent under the alternative that there exists a change point.

This paper is concerned with nonparametric statistics for the stress release process. We propose the local time estimator (LTE) for the stationary density and show that it is unbiased and uniformly consistent. The LTE is used in constructing an estimator for the intensity function. A goodness of fit test for the intensity function is also presented. In these studies, the local time of the stress release process plays an important role.

We consider <var>k</var>-sample and change point problems for independent data in a unified way. We propose a test statistic based on the rank statisitcs. The asymptotic distribution under the null hypothesis is shown to be the supremum of the 2-dimensional standard Brownian pillow. Also, the test is shown to be consistent under the alternative that <var>k</var> distribution functions are linearly independent. It is important from practical point of view that our test is not only asymptotically distribution free but also distribution free even for fixed finite sample.

We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional small diffusions. Our test is based on 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 alternative.

Let a one-dimensional ergodic diffusion process X be observed at time points 0 = t(0)(n) &lt; t(1)(n) &lt; ... &lt; t(n)(n) such that t(n)(n) -&gt; infinity and n Delta(1+ p)(n) -&gt; 0, where Delta(n) = max(1 &lt;= i &lt;= n) |t(i)(n) - t(i-1)(n)|, with p is an element of (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.

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.

This paper deals with nonparametric inference problems in the multiplicative intensity model for counting processes. We propose a Nelson-Aalen type estimator based on discrete observation. The functional asymptotic normality of the estimator is proved. The limit process is the same as that in the continuous observation case, thus the proposed estimator based on discrete observation has the same properties as the Nelson-Aalen estimator based on continuous observation. For example, the asymptotic efficiency of proposed estimator is valid based on less information than the continuous observation case. A Kaplan-Meier type estimator is also discussed. Nonparametric goodness of fit test is considered, and an asymptotically distribution free test is proposed.

We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional ergodic diffusions, where the diffusion coefficient is a nuisance function which is estimated in some sense. Using a theory for the continuous observation case, we construct a test based on the data observed discretely in space, that is, the so-called tick time sampled data. It is proved that the asymptotic distribution of our test under the null hypothesis is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. It is also shown that the test is consistent under any fixed alternative. © Springer Science+Business Media B.V. 2010.

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. © 2010 The Authors Economic Notes © 2010 Banca Monte dei Paschi di Siena SpA.

要旨あり研究ノート

要旨あり研究ノート

要旨あり研究ノート

A goodness of fit test for the drift coefficient of an ergodic diffusion process is presented. The test is based on the score marked empirical process. The weak convergence of the proposed test statistic is studied under the null hypothesis and it is proved that the limit process is a continuous Gaussian process. The structure of its covariance function allows to calculate the limit distribution and it turns out that it is a function of a standard Brownian motion and so exact rejection regions can be constructed. The proposed test is asymptotically distribution free and it is consistent under any simple fixed alternative.

Goodness-of-fit test in a general model of nonlinear time series is considered. We present an asymptotically distribution-free test based on a random field of innovation martingales. Its consistency under any fixed alternative is also proved.

要旨あり確率過程の統計解析研究ノート

In the Kolmogorov-Smirnov theorem, the underlying distribution is assumed to be a continuous distribution. On the other hand, real data in practice is always given in a discretized (rounded) form. In this paper we establish a Donsker type theorem in the fashion of the modern empirical process theory to obtain a (right) Kolmogorov-Smirnov test for discretized data.

We consider a nonparametric estimation problem for the Levy measure of time-inhomogeneous process with independent increments. We derive the functional asymptotic normality and efficiency, in an l(infinity)-space, of generalized Nelson-Aalen estimators. Also we propose some asymptotically distribution free tests for time-homogeneity of the Levy measure. Our result is a fruit of the empirical process theory and the martingale theory. (C) 2007 Elsevier B.V. All rights reserved.

In this paper, we consider the problem of testing for a parameter change using the cusum test based on one-step estimators in diffusion processes. It is shown that under regularity conditions the cusum test statistic has the limiting distribution of a functional of Brownian bridge.

This paper deals with statistical inference problems for a special type of marked point processes based on the realization in random time intervals [0, tau(u)]. Sufficient conditions to establish the local asymptotic normality (LAN) of the model are presented, and then, certain class of stopping times tau(u) 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.