Multivariate extreme value statistics for risk assessment

Yi He

Research output: ThesisDoctoral ThesisScientific

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Abstract

This dissertation consists of three essays about statistical estimation and inference methods concerning extremal events and tail risks. The first essay establishes a natural, semi-parametric estimation procedure for the multivariate half-space depth-based extreme quantile region in arbitrary dimensions. In contrast to the failure of fully non-parametric approaches due to the scarcity of extremal observations, the good finite-sample performance of our extreme estimator is clearly demonstrated in simulation studies. The second essay extends this method to various depth functions, and, furthermore, establishes an asymptotic approximation theory of what-we-called (directed) logarithmic distance between our estimated and true quantile region. Confidence sets that asymptotically cover the quantile region or its complement, or both simultaneously, with a pre-specified probability can be therefore constructed under weak regular variation conditions. The finite-sample coverage probabilities of our asymptotic confidence sets are evaluated in a simulation study for the half-space depth and the projection depth. We use the procedures in both chapters for risk management by applying them to stock market returns. The third essay develops a statistical inference theory of a recently proposed tail risk measure. For regulators who are interested in monitoring these what-we-called relative risks of individual banks, we provide a jackknife empirical likelihood inference procedure based on the smoothed nonparametric estimation and a Wilks type of result. A simulation study and real-life data analysis show that the proposed relative risk measure is useful in monitoring systemic risk.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Tilburg University
Supervisors/Advisors
  • Einmahl, John, Promotor
  • Werker, Bas, Promotor
Award date6 Dec 2016
Place of PublicationTilburg
Publisher
Print ISBNs9789056684914
Publication statusPublished - 2016

Fingerprint

Multivariate Extremes
Extreme Value Statistics
Halfspace Depth
Risk Assessment
Confidence Set
Risk Measures
Relative Risk
Simulation Study
Quantile
Statistical Inference
Tail
Extreme Quantiles
Monitoring
Regular Variation
Semiparametric Estimation
Statistical Estimation
Likelihood Inference
Jackknife
Empirical Likelihood
Approximation Theory

Cite this

He, Y. (2016). Multivariate extreme value statistics for risk assessment. Tilburg: CentER, Center for Economic Research.
He, Yi. / Multivariate extreme value statistics for risk assessment. Tilburg : CentER, Center for Economic Research, 2016. 142 p.
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year = "2016",
language = "English",
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series = "CentER Dissertation Series",
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He, Y 2016, 'Multivariate extreme value statistics for risk assessment', Doctor of Philosophy, Tilburg University, Tilburg.

Multivariate extreme value statistics for risk assessment. / He, Yi.

Tilburg : CentER, Center for Economic Research, 2016. 142 p.

Research output: ThesisDoctoral ThesisScientific

TY - THES

T1 - Multivariate extreme value statistics for risk assessment

AU - He, Yi

PY - 2016

Y1 - 2016

N2 - This dissertation consists of three essays about statistical estimation and inference methods concerning extremal events and tail risks. The first essay establishes a natural, semi-parametric estimation procedure for the multivariate half-space depth-based extreme quantile region in arbitrary dimensions. In contrast to the failure of fully non-parametric approaches due to the scarcity of extremal observations, the good finite-sample performance of our extreme estimator is clearly demonstrated in simulation studies. The second essay extends this method to various depth functions, and, furthermore, establishes an asymptotic approximation theory of what-we-called (directed) logarithmic distance between our estimated and true quantile region. Confidence sets that asymptotically cover the quantile region or its complement, or both simultaneously, with a pre-specified probability can be therefore constructed under weak regular variation conditions. The finite-sample coverage probabilities of our asymptotic confidence sets are evaluated in a simulation study for the half-space depth and the projection depth. We use the procedures in both chapters for risk management by applying them to stock market returns. The third essay develops a statistical inference theory of a recently proposed tail risk measure. For regulators who are interested in monitoring these what-we-called relative risks of individual banks, we provide a jackknife empirical likelihood inference procedure based on the smoothed nonparametric estimation and a Wilks type of result. A simulation study and real-life data analysis show that the proposed relative risk measure is useful in monitoring systemic risk.

AB - This dissertation consists of three essays about statistical estimation and inference methods concerning extremal events and tail risks. The first essay establishes a natural, semi-parametric estimation procedure for the multivariate half-space depth-based extreme quantile region in arbitrary dimensions. In contrast to the failure of fully non-parametric approaches due to the scarcity of extremal observations, the good finite-sample performance of our extreme estimator is clearly demonstrated in simulation studies. The second essay extends this method to various depth functions, and, furthermore, establishes an asymptotic approximation theory of what-we-called (directed) logarithmic distance between our estimated and true quantile region. Confidence sets that asymptotically cover the quantile region or its complement, or both simultaneously, with a pre-specified probability can be therefore constructed under weak regular variation conditions. The finite-sample coverage probabilities of our asymptotic confidence sets are evaluated in a simulation study for the half-space depth and the projection depth. We use the procedures in both chapters for risk management by applying them to stock market returns. The third essay develops a statistical inference theory of a recently proposed tail risk measure. For regulators who are interested in monitoring these what-we-called relative risks of individual banks, we provide a jackknife empirical likelihood inference procedure based on the smoothed nonparametric estimation and a Wilks type of result. A simulation study and real-life data analysis show that the proposed relative risk measure is useful in monitoring systemic risk.

M3 - Doctoral Thesis

SN - 9789056684914

T3 - CentER Dissertation Series

PB - CentER, Center for Economic Research

CY - Tilburg

ER -

He Y. Multivariate extreme value statistics for risk assessment. Tilburg: CentER, Center for Economic Research, 2016. 142 p. (CentER Dissertation Series).