A large deviations approach to the statistics of extreme events

Cees de Valk

Research output: ThesisDoctoral ThesisScientific

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Abstract

A large deviations approach to the statistics of extreme events addresses the statistical analysis of extreme events with very low probabilities: given a random sample of data of size n, the probability is much smaller than 1/n. In particular, it takes a close look at the regularity assumptions on the tail of the (univariate or multivariate) distribution function. The classical assumptions, cast in the form of limits of ratios of probabilities of extreme events, are not directly applicable in this setting. Therefore, additional assumptions are commonly imposed. Because these may be very restrictive, this thesis proposes an alternative regularity assumption, taking the form of asymptotic bounds on ratios of logarithms of probabilities of extreme events, i.e., a large deviation principle (LDP). In the univariate case, this tail LDP is equivalent to the log-Generalised Weibull (log-GW) tail limit, which generalises the Weibull tail limit and the classical Pareto tail limit, amongst others. Its application to the estimation of high quantiles is discussed. In the multivariate case, the tail LDP implies marginal log-GW tail limits together with a standardised tail LDP describing tail dependence. Its application to the estimation of very low probabilities of multivariate extreme events is discussed, and a connection is established to hidden regular variation (residual tail dependence) and similar models.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Tilburg University
Supervisors/Advisors
  • Einmahl, John, Promotor
Award date7 Dec 2016
Place of PublicationTilburg
Publisher
Print ISBNs9789056684952
Publication statusPublished - 2016

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Statistics of Extremes
Extreme Events
Large Deviations
Tail
Large Deviation Principle
Weibull
Tail Dependence
Univariate
Regularity
Multivariate Extremes
Regular Variation
Multivariate Functions
Multivariate Distribution
Quantile
Pareto
Logarithm
Statistical Analysis
Distribution Function
Imply

Cite this

de Valk, C. (2016). A large deviations approach to the statistics of extreme events. Tilburg: CentER, Center for Economic Research.
de Valk, Cees. / A large deviations approach to the statistics of extreme events. Tilburg : CentER, Center for Economic Research, 2016. 133 p.
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de Valk, C 2016, 'A large deviations approach to the statistics of extreme events', Doctor of Philosophy, Tilburg University, Tilburg.

A large deviations approach to the statistics of extreme events. / de Valk, Cees.

Tilburg : CentER, Center for Economic Research, 2016. 133 p.

Research output: ThesisDoctoral ThesisScientific

TY - THES

T1 - A large deviations approach to the statistics of extreme events

AU - de Valk, Cees

PY - 2016

Y1 - 2016

N2 - A large deviations approach to the statistics of extreme events addresses the statistical analysis of extreme events with very low probabilities: given a random sample of data of size n, the probability is much smaller than 1/n. In particular, it takes a close look at the regularity assumptions on the tail of the (univariate or multivariate) distribution function. The classical assumptions, cast in the form of limits of ratios of probabilities of extreme events, are not directly applicable in this setting. Therefore, additional assumptions are commonly imposed. Because these may be very restrictive, this thesis proposes an alternative regularity assumption, taking the form of asymptotic bounds on ratios of logarithms of probabilities of extreme events, i.e., a large deviation principle (LDP). In the univariate case, this tail LDP is equivalent to the log-Generalised Weibull (log-GW) tail limit, which generalises the Weibull tail limit and the classical Pareto tail limit, amongst others. Its application to the estimation of high quantiles is discussed. In the multivariate case, the tail LDP implies marginal log-GW tail limits together with a standardised tail LDP describing tail dependence. Its application to the estimation of very low probabilities of multivariate extreme events is discussed, and a connection is established to hidden regular variation (residual tail dependence) and similar models.

AB - A large deviations approach to the statistics of extreme events addresses the statistical analysis of extreme events with very low probabilities: given a random sample of data of size n, the probability is much smaller than 1/n. In particular, it takes a close look at the regularity assumptions on the tail of the (univariate or multivariate) distribution function. The classical assumptions, cast in the form of limits of ratios of probabilities of extreme events, are not directly applicable in this setting. Therefore, additional assumptions are commonly imposed. Because these may be very restrictive, this thesis proposes an alternative regularity assumption, taking the form of asymptotic bounds on ratios of logarithms of probabilities of extreme events, i.e., a large deviation principle (LDP). In the univariate case, this tail LDP is equivalent to the log-Generalised Weibull (log-GW) tail limit, which generalises the Weibull tail limit and the classical Pareto tail limit, amongst others. Its application to the estimation of high quantiles is discussed. In the multivariate case, the tail LDP implies marginal log-GW tail limits together with a standardised tail LDP describing tail dependence. Its application to the estimation of very low probabilities of multivariate extreme events is discussed, and a connection is established to hidden regular variation (residual tail dependence) and similar models.

M3 - Doctoral Thesis

SN - 9789056684952

T3 - CentER Dissertation Series

PB - CentER, Center for Economic Research

CY - Tilburg

ER -

de Valk C. A large deviations approach to the statistics of extreme events. Tilburg: CentER, Center for Economic Research, 2016. 133 p. (CentER Dissertation Series).