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CURVED EXPONENTIAL FAMILIES OF STOCHASTIC

PROCESSES AND THEIR ENVELOPE FAMILIES

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UWE KÜCHLER^{ 1} AND
MICHAEL
SØRENSEN^{ 2}

^{1} *Institut für Stochastik, Fachbereich Mathematik,
Humboldt-Universität zu Berlin,*

Unter den Linden 6, Berlin, Germany

^{2} *Department of Theoretical Statistics, Institute of
Mathematics,*

University of Aarhus, 8000 Aarhus C, Denmark
(Received January 17, 1994; revised February 27, 1995)

**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.

*Key words and phrases*:
Censored data, diffusion
processes, Gaussian autoregression, goodness-of-fit test,
Markov processes, Ornstein-Uhlenbeck process, pure birth
process.

**Source**
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