On APF Test for Poisson Process with Shift and Scale Parameters

We propose the goodness of fit test for inhomogeneous Poisson processes with unknown scale and shift parameters. A test statistic of Cramer-von Mises type is proposed and its asymptotic behavior is studied. We show that under null hypothesis the limit distribution of this statistic does not depend on unknown parameters.Comment: 15 page

Nonconsistent estimation by diffusion type observations

We consider the properties of the maximum likelihood, Bayes and minimum distance estimates of the finite-dimensional parameter constructed by the observations of the process of diffusion type with the small coefficient of diffusion in the situation when the trend coefficients corresponding to the different values of the parameter are the same and the consistent estimation is impossible. It is proved that these estimates converge to the different random variables.Diffusion process Parameter estimation Nonconsistency

On unbiased density estimation for ergodic diffusion

Two classes of unbiased estimators of the density function of ergodic distribution for the diffusion process of observations are proposed. The estimators are square-root consistent and asymptotically normal. This curious situation is entirely different from the case of discrete-time models (Davis 1977) where the unbiased estimator rarely exists and usually the estimators are not square-root consistent.Diffusion process Nonparametric estimation Density function estimation Unbiased estimator Asymptotic normality

Some problems of nonparametric estimation by observations of ergodic diffusion process

We consider the problems of the density and distribution function estimation by the observations of diffusion process with ergodic properties. In every problem we first propose a minimax bound on the risk of any estimator and then study the asymptotic behavior of several estimators. It is shown that the empiric distribution function is asymptotically normal and asymptotically efficient (in the minimax sense) estimator of the distribution function. In the density estimation problem, we describe the asymptotic behavior of a kernel-type estimator and one another (unbiased) estimator. Both of them are [radical sign]T-consistent, asymptotically normal and asymptotically efficient.Diffusion process Nonparametric estimation Density estimation Distribution function estimation Minimax bound