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Prediction from a random time point

Lindgren, Georg LU (1975) In Annals of Probability 3(3). p.412-423
Abstract
In prediction (Wiener-, Kalman-) of a random normal process $\{X(t), t \in R\}$ it is normally required that the time $t_0$ from which prediction is made does not depend on the values of the process. If prediction is made only from time points at which the process takes a certain value $u,$ given a priori, ("prediction under panic"), the Wiener-prediction is not necessarily optimal; optimal should then mean best in the long run, for each single realization. The main theorem in this paper shows that when predicting only from upcrossing zeros $t_\nu$, the Wiener-prediction gives optimal prediction of $X(t_\nu + t)$ as $t_\nu$ runs through the set of zero upcrossings, if and only if the derivative $X'(t_\nu)$ at the crossing points is... (More)
In prediction (Wiener-, Kalman-) of a random normal process $\{X(t), t \in R\}$ it is normally required that the time $t_0$ from which prediction is made does not depend on the values of the process. If prediction is made only from time points at which the process takes a certain value $u,$ given a priori, ("prediction under panic"), the Wiener-prediction is not necessarily optimal; optimal should then mean best in the long run, for each single realization. The main theorem in this paper shows that when predicting only from upcrossing zeros $t_\nu$, the Wiener-prediction gives optimal prediction of $X(t_\nu + t)$ as $t_\nu$ runs through the set of zero upcrossings, if and only if the derivative $X'(t_\nu)$ at the crossing points is observed. The paper also gives the conditional distribution from which the optimal predictor can be computed. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Annals of Probability
volume
3
issue
3
pages
412 - 423
publisher
Institute of Mathematical Statistics
ISSN
0091-1798
language
English
LU publication?
yes
id
4cded402-8d48-4b98-a5f6-3fcb7a7ce05c (old id 1273148)
alternative location
http://www.jstor.org/stable/2959464
date added to LUP
2009-06-03 16:47:29
date last changed
2017-03-15 13:26:59
@article{4cded402-8d48-4b98-a5f6-3fcb7a7ce05c,
  abstract     = {In prediction (Wiener-, Kalman-) of a random normal process $\{X(t), t \in R\}$ it is normally required that the time $t_0$ from which prediction is made does not depend on the values of the process. If prediction is made only from time points at which the process takes a certain value $u,$ given a priori, ("prediction under panic"), the Wiener-prediction is not necessarily optimal; optimal should then mean best in the long run, for each single realization. The main theorem in this paper shows that when predicting only from upcrossing zeros $t_\nu$, the Wiener-prediction gives optimal prediction of $X(t_\nu + t)$ as $t_\nu$ runs through the set of zero upcrossings, if and only if the derivative $X'(t_\nu)$ at the crossing points is observed. The paper also gives the conditional distribution from which the optimal predictor can be computed.},
  author       = {Lindgren, Georg},
  issn         = {0091-1798},
  language     = {eng},
  number       = {3},
  pages        = {412--423},
  publisher    = {Institute of Mathematical Statistics},
  series       = {Annals of Probability},
  title        = {Prediction from a random time point},
  volume       = {3},
  year         = {1975},
}