Abstract
We consider an ℝd dimensional homogeneous diffusion process with a unique invariant density f. We construct a kernel type estimator for the invariant density and study its mean-square convergence. We find that this estimator reaches in a specific minimax sense a rate that is slower than parametric but faster than in classical d-dimensional estimation problems. Finally we examine the almost sure (pointwise and uniform) behavior of the estimator and we give examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bandi, F.M., Moloche, G.: On the functional estimation of multivariate diffusion processes. Preprint (2002), available under http://gsbwww.uchicago.edu/fac/federico.bandi/research/
Blanke, D., Bosq, D.: A family of minimax rates for density estimators in continuous time. Stoch. Anal. Appl., 18, 6, 871–900 (2000)
Blanke, D.: Sample paths adaptive density estimator. Math. Methods Statist., 13, 2, 123–152 (2004)
Bosq, D.: Nonparametric statistics for stochastic processes. Springer-Verlag, New York (1998)
Capasso, V., Bakstein, D.: An Introduction to Continuous–time Stochastic Processes. Birkhäuser, Boston (2005)
Capasso, V.: Mathematical structures of epidemic systems. Springer-Verlag, Berlin (1993)
Castellana, J.V., Leadbetter, M.R.: On smoothed probability density estimation for stationary processes. Stoch. Processes Appl., 21, 179–193 (1986)
Dalalyan, A., Reiß, M.: Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case. Preprint (2005), available under http://www.proba.jussieu.fr/pageperso/dalalyan/Links/AD.html
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, N.J. (1964)
Kutoyants, Yu.A.: Statistical Inference for Ergodic Diffusion Processes. Springer Series in Statistics, New York (2004)
Qian, Z., Russo, F., Zheng, W.: Comparison theorem and estimates for transition probability densities of diffusion processes. Probab. Theory Related Fields 127, 388–406 (2003)
Soize, C.: The Fokker-Planck equation for stochastic dynamical systems and its explicit steady state solutions. World Scientific, Singapore (1994)
Van Zantan, J.H.: On the uniform convergence of the empirical density of an ergodic diffusion. Statist. Inf. Stoch. Proc., 3, 251–262 (2000)
Veretennikov, A.Yu.: On Castellana-Leadbetter's Condition for Diffusion Density Estimation. Statist. Inf. Stoch. Proc., 2, 1–9 (1999)
Veretennikov, A.Yu.: On subexponential mixing rate for Markov processes. Theory Probab. Appl., 49(1), 110–122 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this chapter
Cite this chapter
Bianchi, A. (2007). Invariant Density Estimation for Multidimensional Diffusions. In: Aletti, G., Micheletti, A., Morale, D., Burger, M. (eds) Math Everywhere. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44446-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-44446-6_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44445-9
Online ISBN: 978-3-540-44446-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)