### Slepian models and regression approximations in crossing and extreme value theory

(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:
https://lup.lub.lu.se/record/1210411

- author
- Lindgren, Georg
^{LU}and Rychlik, Igor^{LU} - organization
- publishing date
- 1991
- 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}, 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}, }