This paper deals with some change point problems based on the general theory of weak convergence in Hilbert spaces. In particular, we develop an L-2 space approach to handle Anderson-Darling type statistics, which was taken for goodness of fit test problems, to change point problems for which more delicate arguments are necessary. (C) 2014 Elsevier B.V. All rights reserved.

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 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) < t(1)(n) < ... < t(n)(n) such that t(n)(n) -> infinity and n Delta(1+ p)(n) -> 0, where Delta(n) = max(1 <= i <= 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.

It is well known that the kernel estimator [image omitted] for the probability density f on 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 L2(d) could be possible. In this paper, we deny this conjecture. That is, letting [image omitted], we prove that for any sequence {rn} of positive constants such that rn=o(root n), if the rescaled residual rn(f<SU</SUn-fn) converges weakly to a Borel limit in L2(d), then the limit is necessarily degenerate.

It is often assumed in statistics that the random variables under consideration come from a continuous distribution. However, real data is always given in a rounded (discretized) form. The rounding errors become serious when the sample size is large. In this paper, we consider the situation where the mesh of discretization tends to zero as the sample size tends to infinity, and give some sets of sufficient conditions under which the rounding errors can be asymptotically ignored, in the context of Z-estimation. It is theoretically proved that the mid-point discretization is preferable. (C) 2010 Elsevier B.V. All rights reserved.

This study extends the rate of convergence theorem of M-estimators presented by van der Vaart and Wellner (weak convergence and empirical processes: with applications to statistics, Springer-Verlag, Newyork, 1996) who gave a result of the form r(n)((theta) over cap (n)-theta(0)) = O(P)(1) to a result of the form sup(n)E vertical bar r(n)((theta) over cap (n) - theta(0)) vertical bar (p) < infinity, for any p >= 1. This result is useful for deriving the moment convergence of the rescaled residual. An application to maximum likelihood estimators is discussed.

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.

This paper generalizes a part of the theory of Z-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation theta (sic) Psi(n) (theta, (h) over cap (n)) = 0 with an abstract nuisance parameter h when the compensator of Psi(n) is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter theta and the diffusion coefficient is indexed by a nuisance parameter h from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a Z-estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model.

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.

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.

This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685-712]. A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach is based on the entropy techniques developed in the modem theory of empirical processes.

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 weak convergence of stochastic integrals with respect to multivariate point processes. The results are given in terms of an entropy condition for partitioning of the index set of the integrands, which is a sort of L-2-bracketing. We also consider l(infinity)-valued martingale difference arrays, and present natural generalizations of Jain-Marcus's and Ossiander's central limit theorems. As an application, the asymptotic behavior of log-likelihood ratio random fields in general statistical experiments with abstract parameters is derived.

Some sufficient conditions to establish the rate of convergence of certain M-estimators in a Gaussian white noise model are presented. They are applied to some concrete problems, including jump point estimation and nonparametric maximum Likelihood estimation, for the regression function. The results are shown by means of a maximal inequality for continuous martingales and some techniques developed recently in the context of empirical processes.

This paper is devoted to the generalization of central limit theorems for empirical processes to several types of l(infinity)(Psi)-valued continuous-time stochastic processes t squiggly right arrow X-t(n) = (X-t(n, psi) is an element of Psi), where Psi is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t squiggly right arrow X-t(n, psi) is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived.

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.