Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banach S. (1925) Sur les lignes rectifiables et les surfaces dont l’aire est finie. Fundamenta Mathematicae 7: 225–236
Bosq, D. (1998). Nonparametric statistics for stochastic processes (2nd ed.). Lecture notes in statistics (Vol. 110). New York: Springer.
Bosq D., Davydov Yu. (1999) Local time and density estimation in continuous time. Mathematical Methods of Statistics 8: 22–45
Dachian S., Kutoyants Yu. A. (2009) Hypotheses testing: Poisson versus stress-release. Journal of Statistical Planning and Inference 139: 1668–1684
Feller, W. (1971). An introduction to probability theory and its applications (2nd ed., Vol. II). Wiley: New York.
Fujii, T. (2010) On the invariant density of the stress release process (submitted).
Hayashi T. (1986) Laws of large numbers in self-correcting point processes. Stochastic Processes and Applications 23: 319–326
Inagaki N., Hayashi T. (1990) Parameter estimation for the simple self-correcting point process. Annals of the Institute of Statistical Mathematics 42: 89–98
Koul H.L., Stute W. (1999) Nonparametric model checks for time series. The Annals of Statistics 27: 204–236
Kutoyants, Yu. A. (1996). Efficient density estimation for ergodic diffusion. Research Memorandum No. 607. The Institute of Statistical Mathematics, Tokyo.
Kutoyants Yu. A. (1998) Efficient density estimation for ergodic diffusion processes. Statistical Inference for Stochastic Processes 1: 131–155
Kutoyants Yu. A. (2004) Statistical inference for ergodic diffusion processes. Springer, London
Negri I., Nishiyama Y. (2009) Goodness of fit test for ergodic diffusion processes. Annals of the Institute of Statistical Mathematics 61: 919–928
Nishiyama Y. (2000) Weak convergence of some classes of martingales with jumps. The Annals of Probability 28: 685–712
Ogata, Y., Vere-Jones, D. (1984). Inference for earthquake models: a self-correcting model. Stochastic Processes and Applications, 17, 337–347.
Shorack G.R., Wellner J.A. (1986) Empirical processes with applications to statistics. Wiley, New York
van der Vaart A.W., Wellner J.A. (1996) Weak convergence and empirical processes. Springer, New York
van Zanten J. H. (2000) On the uniform convergence of the empirical density of an ergodic diffusion. Statistical Inference for Stochastic Processes 3: 251–262
Vere-Jones D. (1988) On the variance properties of stress release models. The Australian Journal of Statistics 30A: 123–135
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Fujii, T., Nishiyama, Y. Some problems in nonparametric inference for the stress release process related to the local time. Ann Inst Stat Math 64, 991–1007 (2012). https://doi.org/10.1007/s10463-011-0344-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-011-0344-7