Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-24T03:08:03.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum likelihood estimation for continuous-time stochastic processes

Published online by Cambridge University Press:  01 July 2016

Paul David Feigin
Paul David Feigin*
Affiliation:
Australian National University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is mainly concerned with the asymptotic theory of maximum likelihood estimation for continuous-time stochastic processes. The role of martingale limit theory in this theory is developed. Some analogues of classical statistical concepts and quantities are also suggested. Various examples that illustrate parts of the theory are worked through, producing new results in some cases. The role of diffusion approximations in estimation is also explored.

Keywords

MAXIMUM LIKELIHOOD ESTIMATIONCONTINUOUS-TIME STOCHASTIC PROCESSESASYMPTOTIC THEORYMARTINGALE LIMIT THEORYDIFFUSION APPROXIMATIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 4 , December 1976 , pp. 712 - 736
Copyright
Copyright © Applied Probability Trust 1976 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Athreya, K. B. and Keiding, N. (1975) Estimation theory for continuous-time branching processes. Preprint No. 6, Institute of Mathematical Statistics, University of Copenhagen.Google Scholar
Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Bartlett, M. S. (1966) Introduction to Stochastic Processes, 2nd edn. Cambridge University Press.Google Scholar
Basawa, I. V. (1974) Maximum likelihood estimation of parameters in renewal and Markov-renewal processes. Austral. J. Statist. 16, 33–43.Google Scholar
Basawa, I. V., Feigin, P. D. and Heyde, C. C. (1975) Asymptotic properties of maximum likelihood estimators for stochastic processes. Sankhyā, to appear.Google Scholar
Basawa, I. V. and Scott, D. J. (1975a) Efficient tests for stochastic processes. Research Report 170/IVB + DJS 1, Department of Probability and Statistics, University of Sheffield.Google Scholar
Basawa, I. V. and Scott, D. J. (1975b) Efficient tests for branching processes. Unpublished.Google Scholar
Billingsley, P. (1961) Statistical Inference for Markov Processes. University of Chicago Press.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass.Google Scholar
Brown, B. M. and Hewitt, J. I. (1975a) Asymptotic likelihood theory for diffusion processes. J. Appl. Prob. 12, 228–238.Google Scholar
Brown, B. M. and Hewitt, J. I. (1975b) Inference for the diffusion branching process. J. Appl. Prob. 12, 588–594.Google Scholar
Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.CrossRefGoogle Scholar
Dion, J.-P. (1973) Theorèmes limites fonctionnels pour une somme d'un nombre aléatoire de variables aléatoire dependantes. Préprint, Université de Québec à Montréal.Google Scholar
Dion, J.-P. (1974) Estimation of the mean and the initial probabilities of a branching process. J. Appl. Prob. 11, 687–694.CrossRefGoogle Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Feigin, P. D. (1975) Maximum Likelihood Estimation for Stochastic Processes — A Martingale Approach. Ph. D. Thesis, Australian National University.Google Scholar
Godambe, V. P. (1960) An optimum property of regular maximum likelihood estimation. Ann. Math. Statist. 31, 1208–1211.CrossRefGoogle Scholar
Grenander, U. (1951) Stochastic processes and statistical inference. Ark. Mat. 1, 195–277.CrossRefGoogle Scholar
Heyde, C. C. and Feigin, P. D. (1975) On efficiency and exponential families in stochastic process estimation. In Statistical Distributions in Scientific Work Vol. 1, ed. Patil, G. P., Kotz, S. and Ord, J. K., D. Riedel, Utrecht and Boston, 227–240.Google Scholar
Jagers, P. (1971) Diffusion approximations of branching processes. Ann. Math. Statist. 42, 2074–2078.CrossRefGoogle Scholar
Kailath, T. and Zakai, M. (1971) Absolute continuity and Radon–Nikodym derivatives for certain measures relative to Wiener measure. Ann. Math. Statist. 42, 130–140.CrossRefGoogle Scholar
Keiding, N. (1975) Maximum likelihood estimation in the birth-death process. Ann. Statist. 3, 363–372.CrossRefGoogle Scholar
Kunita, H. and Watanabe, S. (1960) On square integrable martingales. Nagoya Math. J. 30, 209–245.Google Scholar
Lamperti, J. (1967) Continuous state branching processes. Bull. Amer. Math. Soc. 73, 382–386.CrossRefGoogle Scholar
McKean, H. P. jr. (1969) Stochastic Integrals. Academic Press, New York.Google Scholar
McNeil, D. R. and Schach, S. (1973) Central limit analogues for Markov population processes. J. R. Statist. Soc. B 35, 1–23.Google Scholar
Meyer, P. A. (1962) A decomposition theorem for supermartingales. Illinois J. Math. 6, 193–205.Google Scholar
Rao, C. R. (1973) Linear Statistical Inference and its Applications, 2nd edn. Wiley, New York.Google Scholar
Striebel, C. T. (1959) Densities for stochastic processes. Ann. Math. Statist. 30, 559–567.CrossRefGoogle Scholar
Watanabe, S. (1969) On two-dimensional Markov processes with the branching property. Trans. Amer. Math. Soc. 136, 447–466.Google Scholar