### Prediction from a random time point

(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)

Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1273148

- author
- Lindgren, Georg
^{LU} - organization
- publishing date
- 1975
- 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
- 2016-04-01 15:30:23
- date last changed
- 2019-03-08 03:04:21

@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}}, url = {{http://www.jstor.org/stable/2959464}}, volume = {{3}}, year = {{1975}}, }