An M-estimator of multivariate tail dependence

A. Krajina

Research output: ThesisDoctoral ThesisScientific

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Abstract

AN M-ESTIMATOR OF TAIL DEPENDENCE. Extreme value theory is the part of probability and statistics that provides the theoretical background for modeling events that almost never happen. The estimation of the dependence between two or more such unlikely events (tail dependence) is the topic of this thesis. The tail dependence structure is modeled by the stable tail dependence function. In Chapter 2 a semiparametric model is considered in which the stable tail dependence function is parametrically modeled. A method of moments estimator of the unknown parameter is proposed, where an integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. This estimator is applied in Chapter 3 to estimate the tail dependence structure of the family of meta-elliptical distributions. The estimator introduced in Chapter 2 is extended in two respects in Chapter 4: (i) the number of variables is arbitrary; (ii) the number of moment equations can exceed the dimension of the parameter space. This estimator is defined as the value of the parameter vector that minimizes the distance between a vector of weighted integrals of the tail dependence function on the one hand and empirical counterparts of these integrals on the other hand. The method, not being likelihood based, applies to discrete and continuous models alike. Under minimal conditions all estimators introduced are consistent and asymptotically normal. The performance and applicability of the estimators is demonstrated by examples.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Tilburg University
Supervisors/Advisors
  • Segers, J.J.J., Co-promotor
  • Einmahl, John, Promotor
Award date23 Apr 2010
Place of PublicationTilburg
Publisher
Print ISBNs9789056682521
Publication statusPublished - 2010

Fingerprint

Tail Dependence
M-estimator
Dependence Function
Estimator
Dependence Structure
Elliptical Distribution
Moment Equations
Moment Estimator
Extreme Value Theory
Alike
Semiparametric Model
Method of Moments
Unknown Parameters
Parameter Space
Likelihood
Exceed
Statistics
Minimise

Cite this

Krajina, A. (2010). An M-estimator of multivariate tail dependence. Tilburg: CentER, Center for Economic Research.
Krajina, A.. / An M-estimator of multivariate tail dependence. Tilburg : CentER, Center for Economic Research, 2010. 105 p.
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Krajina, A 2010, 'An M-estimator of multivariate tail dependence', Doctor of Philosophy, Tilburg University, Tilburg.

An M-estimator of multivariate tail dependence. / Krajina, A.

Tilburg : CentER, Center for Economic Research, 2010. 105 p.

Research output: ThesisDoctoral ThesisScientific

TY - THES

T1 - An M-estimator of multivariate tail dependence

AU - Krajina, A.

PY - 2010

Y1 - 2010

N2 - AN M-ESTIMATOR OF TAIL DEPENDENCE. Extreme value theory is the part of probability and statistics that provides the theoretical background for modeling events that almost never happen. The estimation of the dependence between two or more such unlikely events (tail dependence) is the topic of this thesis. The tail dependence structure is modeled by the stable tail dependence function. In Chapter 2 a semiparametric model is considered in which the stable tail dependence function is parametrically modeled. A method of moments estimator of the unknown parameter is proposed, where an integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. This estimator is applied in Chapter 3 to estimate the tail dependence structure of the family of meta-elliptical distributions. The estimator introduced in Chapter 2 is extended in two respects in Chapter 4: (i) the number of variables is arbitrary; (ii) the number of moment equations can exceed the dimension of the parameter space. This estimator is defined as the value of the parameter vector that minimizes the distance between a vector of weighted integrals of the tail dependence function on the one hand and empirical counterparts of these integrals on the other hand. The method, not being likelihood based, applies to discrete and continuous models alike. Under minimal conditions all estimators introduced are consistent and asymptotically normal. The performance and applicability of the estimators is demonstrated by examples.

AB - AN M-ESTIMATOR OF TAIL DEPENDENCE. Extreme value theory is the part of probability and statistics that provides the theoretical background for modeling events that almost never happen. The estimation of the dependence between two or more such unlikely events (tail dependence) is the topic of this thesis. The tail dependence structure is modeled by the stable tail dependence function. In Chapter 2 a semiparametric model is considered in which the stable tail dependence function is parametrically modeled. A method of moments estimator of the unknown parameter is proposed, where an integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. This estimator is applied in Chapter 3 to estimate the tail dependence structure of the family of meta-elliptical distributions. The estimator introduced in Chapter 2 is extended in two respects in Chapter 4: (i) the number of variables is arbitrary; (ii) the number of moment equations can exceed the dimension of the parameter space. This estimator is defined as the value of the parameter vector that minimizes the distance between a vector of weighted integrals of the tail dependence function on the one hand and empirical counterparts of these integrals on the other hand. The method, not being likelihood based, applies to discrete and continuous models alike. Under minimal conditions all estimators introduced are consistent and asymptotically normal. The performance and applicability of the estimators is demonstrated by examples.

M3 - Doctoral Thesis

SN - 9789056682521

T3 - CentER Dissertation Series

PB - CentER, Center for Economic Research

CY - Tilburg

ER -

Krajina A. An M-estimator of multivariate tail dependence. Tilburg: CentER, Center for Economic Research, 2010. 105 p. (CentER Dissertation Series).