Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Slepian models and regression approximations in crossing and extreme value theory

Lindgren, Georg LU orcid and Rychlik, Igor LU (1991) In International Statistical Review 59(2). p.195-225
Abstract
In crossing theory for stochastic processes the distribution of quantities such as distances between level crossings, maximum height of an excursion between level crossings, amplitude and wavelength, etc., can only be written in the form of infinite-dimensional integrals, which are difficult to evaluate numerically. A Slepian model is an explicit random function representation of the process after a level crossing and it consists of one regression term and one residual process. The regression approximation of a crossing variable is defined as the corresponding variable in the regression term of the Slepian model, and its distribution can be evaluated numerically as a finite-dimensional integral. This paper reviews the use and structure of... (More)
In crossing theory for stochastic processes the distribution of quantities such as distances between level crossings, maximum height of an excursion between level crossings, amplitude and wavelength, etc., can only be written in the form of infinite-dimensional integrals, which are difficult to evaluate numerically. A Slepian model is an explicit random function representation of the process after a level crossing and it consists of one regression term and one residual process. The regression approximation of a crossing variable is defined as the corresponding variable in the regression term of the Slepian model, and its distribution can be evaluated numerically as a finite-dimensional integral. This paper reviews the use and structure of the Slepian model and the regression method and shows how they can be used to obtain good numerical approximations to various crossing variables. It gives a detailed account of the regression method for Gaussian processes with auxiliary variables chosen in a recursive way. It also presents a package of computer programs for the numerical calculations, and gives numerical examples on excursion lengths as well as wavelength and amplitude distributions. Further examples deal with an engineering 'jump-and-bump' problem, and excursions for a chi-2-process. (Less)
Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
AMPLITUDE AND WAVELENGTH DISTRIBUTION, RELIABILITY, GAUSSIAN PROCESSES, CHI-SQUARED PROCESSES, 1ST-PASSAGE DENSITY, JOINT DISTRIBUTION, WEAK-CONVERGENCE, CHI-2 PROCESSES, WAVELENGTH, AMPLITUDE, BEHAVIOR, DURATION, CLICKS, FIELDS
in
International Statistical Review
volume
59
issue
2
pages
195 - 225
publisher
International Statistical Institute
ISSN
1751-5823
language
English
LU publication?
yes
id
44158560-6681-4a9a-b47c-f31dcddb4e78 (old id 1210411)
date added to LUP
2016-04-01 15:22:38
date last changed
2019-03-08 03:04:23
@article{44158560-6681-4a9a-b47c-f31dcddb4e78,
  abstract     = {{In crossing theory for stochastic processes the distribution of quantities such as distances between level crossings, maximum height of an excursion between level crossings, amplitude and wavelength, etc., can only be written in the form of infinite-dimensional integrals, which are difficult to evaluate numerically. A Slepian model is an explicit random function representation of the process after a level crossing and it consists of one regression term and one residual process. The regression approximation of a crossing variable is defined as the corresponding variable in the regression term of the Slepian model, and its distribution can be evaluated numerically as a finite-dimensional integral. This paper reviews the use and structure of the Slepian model and the regression method and shows how they can be used to obtain good numerical approximations to various crossing variables. It gives a detailed account of the regression method for Gaussian processes with auxiliary variables chosen in a recursive way. It also presents a package of computer programs for the numerical calculations, and gives numerical examples on excursion lengths as well as wavelength and amplitude distributions. Further examples deal with an engineering 'jump-and-bump' problem, and excursions for a chi-2-process.}},
  author       = {{Lindgren, Georg and Rychlik, Igor}},
  issn         = {{1751-5823}},
  keywords     = {{AMPLITUDE AND WAVELENGTH DISTRIBUTION; RELIABILITY; GAUSSIAN PROCESSES; CHI-SQUARED PROCESSES; 1ST-PASSAGE DENSITY; JOINT DISTRIBUTION; WEAK-CONVERGENCE; CHI-2 PROCESSES; WAVELENGTH; AMPLITUDE; BEHAVIOR; DURATION; CLICKS; FIELDS}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{195--225}},
  publisher    = {{International Statistical Institute}},
  series       = {{International Statistical Review}},
  title        = {{Slepian models and regression approximations in crossing and extreme value theory}},
  volume       = {{59}},
  year         = {{1991}},
}