Abstract
Exponential families of stochastic processes are usually curved. The full exponential families generated by the finite sample exponential families are called the envelope families to emphasize that their interpretation as stochastic process models is not straightforward. A general result on how to calculate the envelope families is given, and the interpretation of these families as stochastic process models is considered. For Markov processes rather explicit answers are given. Three examples are considered some in detail: Gaussian autoregressions, the pure birth process and the Ornstein-Uhlenbeck process. Finally, a goodness-of-fit test for censored data is discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hoffmann-Jørgensen, J. (1994). Probability with a View to Statistics, Chapman and Hall, New York.
Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam.
Jacobsen, M. (1982). Statistical analysis of counting processes, Lecture Notes in Statist, 12, Springer, New York.
Jacod, J. and Mémin, J. (1976). Caractéristiques locales et conditions de continuité absolue pour les semi-martingales. Z. Wahrscheinlichkeitstheorie verw. Geb. 35, 1–37.
Jensen, J. L. (1987). On asymptotic expansions in non-ergodic models, Scand. J. Statist., 14, 305–318.
Keiding, N. (1974). Estimation in the birth process, Biometrika, 61, 71–80.
Küchler, U. and Sørensen, M. (1981). Exponential families of stochastic processes: A unifying semimartingale approach, Internat. Statist. Rev., 57, 123–144.
Küchler, U. and Sørensen, M. (1991). On exponential families of Markov processes, Research Report, No. 233, Department of Theoretical Statistics, Aarhus University, Denmark.
Küchler, U. and Sørensen, M. (1994a). Exponential families of stochastic processes with time-continuous likelihood functions, Scand. J. Statist., 21, 421–431.
Küchler, U. and Sørensen, M. (1994b). Exponential families of stochastic processes and Lévy processes, J. Statist. Plann. Inference, 39, 211–237.
Puri, P. S. (1966). On the homogeneous birth-and-death process and its integral, Biometrika, 53, 61–71.
Sørensen, M. (1986). On sequential maximum likelihood estimation for exponential families of stochastic processes, Internat. Statist. Rev., 54, 191–210.
Sørensen, M. (1993). The natural exponential family generated by a semimartingale, Research Report, No. 269, Department of Theoretical Statistics, Aarhus University, Proceedings of the Fourth Russian-Finnish Symposium on Probability Theory and Mathematical Statistics, TVP, Moscow (to appear).
Sørensen, M. (1995). On conditional inference for the Ornstein-Uhlenbeck process (in preparation).
Væth, M. (1980). A test for the exponential distribution with censored data, Research Report, No. 60, Department of Theoretical Statistics, Aarhus University, Denmark.
Væth, M. (1982). A sampling experiment leading to the full exponential family generated by the censored exponential distribution, Research Report, No. 78, Department of Theoretical Statistics, Aarhus University, Denmark.
White, J. S. (1958). The limiting distribution of the serial correlation coefficient in the explosive case, Ann. Math. Statist., 29, 1188–1197.
Author information
Authors and Affiliations
About this article
Cite this article
Küchler, U., Sørensen, M. Curved exponential families of stochastic processes and their envelope families. Ann Inst Stat Math 48, 61–74 (1996). https://doi.org/10.1007/BF00049289
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00049289