Summary
This paper deals with asymptotic optimal inference in a time-continuous ergodic Markov chain with countable state space, based on observation of the process up to timet. Let the infinitesimal generator depend on an unknown parameter. Under weak assumptions on the parametrization, we show local asymptotic normality for the statistical model ast→∞. As a consequence, limit distributions of sequences of competing estimators for the unknown parameter are more spread out than a specified normal distribution.
References
Aalen, O.O.: Weak convergence of stochastic integrals related to counting processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.38, 261–277 (1977)
Akritas, M.G., Roussas, G.: Asymptotic inference in continuous-time Semi-Markov processes. Scand. J. Stat., Theory Appl.7, 73–79 (1980)
Andersen, P.K., Gill, R.D.: Cox regression model for counting processes: a large sample study. Ann. Stat.10, 1100–1120 (1982)
Anderson, T.W.: The integral of a symmetric unimodel function over a symmetric convex set and some probability inequalities. Proc. Am. Math. Soc.6, 170–176 (1955)
Basawa, I.V., Prakasa Rao, B.L.S.: Statistical inference for stochastic processes. New York London: Academic Press 1980
Basawa, I.V., Scott, D.J.: Asymptotic optimal inference for non-ergodic models. Lect. Notes Stat. vol. 17, Berlin Heidelberg New York: Springer 1983
Billingsley, P.: Statistical inference for Markov processes. Chicago: University of Chicago Press 1961
Brémaud, P.: Point processes and queues. Berlin Heidelberg New York:Springer 1981
Cinlar, E.: Introduction to stochastic processes. Englewood Cliffs NJ: Prentice-Hall 1975
Droste, W., Wefelmeyer, W.: On Hajek's convolution theorem. Statistics & Decisions2, 131–144 (1984)
Grenander, U.: Abstract inference. New York: Wiley 1981
Hajek, J.: A characterization of limiting distributions of regular estimates. Z. Wahrscheinlichkeitstheor. Verw. Geb.14, 323–330 (1970)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. North-Holland Kodansha 1981
Jacod, J.: Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 235–253 (1975)
Jacod, J.: Theorèmes limite pour les processus. In: par Hennequin, P.-L. (ed.), Ecole d'Eté de Probabilités de Saint-Flour XIII-1983. (Lect. Notes Math., vol. 1117 pp. 299–409) Berlin Heidelberg New York: Springer 1985
Jacod, J., Shiryayev, A.N.: Limit theorems for stochastic processes. Berlin Heidelberg New York: Springer 1987
Kabanov, Yu.M., Liptser, R.S., Shiryayev, A.N.: Criteria for absolute continuity of measures corresponding to multivariate point processes. In: Marumaya, G., Prokhorov, J.V. (eds.) Proceedings of the Third Japan-USSR Symposium on Probability Theory (Lect. Notes Math., vol. 550, pp. 232–252) Berlin Heidelberg New York: Springer 1976
Keiding, N.: Maximum likelihood estimation in the birth-and-death process. Ann. Stat.3, 363–372 (1975)
Lenglart, E.: Relation de domination entre deux processus. Ann. Inst. Henri Poincaré Nouv. Ser., Sect. B13, 171–179 (1977)
Liptser, R.S., Shiryayev, A.N.: A functional central limit theorem for semimartingales. Theory Probab. Appl.25, 667–688 (1980)
Métivier, M.: Semimartingales: a course on stochastic processes. Berlin New York: de Gruyter 1982
Pakes, A.: Some conditions for ergodicity and recurrence of Markov chains. Oper. Res.17, 1058–1061 (1969)
Pruscha, H.: Parametric inference in Markov branching processes with time-dependent random immigration rate. J. Appl. Probab.22, 503–517 (1985)
Rebolledo, R.: Central limit, theorems for local martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb.51, 269–286 (1980)
Tweedie, R.L.: Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes. J. Appl. Probab.18, 122–130 (1981)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Höpfner, R. Asymptotic inference for continuous-time Markov chains. Probab. Th. Rel. Fields 77, 537–550 (1988). https://doi.org/10.1007/BF00959616
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00959616
Keywords
- Normal Distribution
- Statistical Model
- Markov Chain
- State Space
- Stochastic Process