Statistical Inference for Heavy and Super-Heavy-tailed distributions - PowerPoint PPT Presentation

Statistical Inference for Heavy and Super-Heavy-tailed distributions M. Isabel Fraga Alves DEIO, Faculty of Sciences, University of Lisbon, Portugal Laurens de Haan Econometric Institute, Erasmus University of Rotterdam, The Netherlands Cláudia Neves UIMA, Department of Mathematics, University of Aveiro, Portugal Cláudia Neves, August 18, 2005 EVA - p. 1/19

Summary » Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π -variation and Γ -variation � EVT: Modeling the tail of a distribution; » A step towards estimation » Estimation of the tail parameter » Auxiliary results I » Auxiliary results II » Asymptotic normality » Main result » Finite sample behavior I » Finite sample behavior II » Finite sample behavior III » Drawback... » Test of hypothesis » Empirical power » Estimated type I error » References Cláudia Neves, August 18, 2005 EVA - p. 2/19

Summary » Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π -variation and Γ -variation � EVT: Modeling the tail of a distribution; » A step towards estimation » Estimation of the tail parameter � Characterization of Super-Heavy tails; » Auxiliary results I » Auxiliary results II » Asymptotic normality » Main result » Finite sample behavior I » Finite sample behavior II » Finite sample behavior III » Drawback... » Test of hypothesis » Empirical power » Estimated type I error » References Cláudia Neves, August 18, 2005 EVA - p. 2/19

Summary » Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π -variation and Γ -variation � EVT: Modeling the tail of a distribution; » A step towards estimation » Estimation of the tail parameter � Characterization of Super-Heavy tails; » Auxiliary results I » Auxiliary results II » Asymptotic normality � Estimation of the index of tail heaviness α ≥ 0 ; » Main result » Finite sample behavior I » Finite sample behavior II » Finite sample behavior III » Drawback... » Test of hypothesis » Empirical power » Estimated type I error » References Cláudia Neves, August 18, 2005 EVA - p. 2/19

Summary » Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π -variation and Γ -variation � EVT: Modeling the tail of a distribution; » A step towards estimation » Estimation of the tail parameter � Characterization of Super-Heavy tails; » Auxiliary results I » Auxiliary results II » Asymptotic normality � Estimation of the index of tail heaviness α ≥ 0 ; » Main result » Finite sample behavior I » Finite sample behavior II � Testing the presence of Super-Heavy tails; » Finite sample behavior III » Drawback... » Test of hypothesis » Empirical power » Estimated type I error » References Cláudia Neves, August 18, 2005 EVA - p. 2/19

Summary » Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π -variation and Γ -variation � EVT: Modeling the tail of a distribution; » A step towards estimation » Estimation of the tail parameter � Characterization of Super-Heavy tails; » Auxiliary results I » Auxiliary results II » Asymptotic normality � Estimation of the index of tail heaviness α ≥ 0 ; » Main result » Finite sample behavior I » Finite sample behavior II � Testing the presence of Super-Heavy tails; » Finite sample behavior III » Drawback... » Test of hypothesis � Simulation results. » Empirical power » Estimated type I error » References Cláudia Neves, August 18, 2005 EVA - p. 2/19

Heavy-tailed distributions » Summary » Heavy-tailed distributions � n � » Characterizing Heavy-tail Let X i i = 1 be i.i.d. random variables with common behavior » Π -variation and Γ -variation distribution function F . » A step towards estimation » Estimation of the tail parameter » Auxiliary results I » Auxiliary results II » Asymptotic normality » Main result » Finite sample behavior I » Finite sample behavior II » Finite sample behavior III » Drawback... » Test of hypothesis » Empirical power » Estimated type I error » References Cláudia Neves, August 18, 2005 EVA - p. 3/19

Heavy-tailed distributions » Summary » Heavy-tailed distributions � n � » Characterizing Heavy-tail Let X i i = 1 be i.i.d. random variables with common behavior » Π -variation and Γ -variation distribution function F . » A step towards estimation » Estimation of the tail parameter » Auxiliary results I Heavy-tailed models suggested by EVT: » Auxiliary results II » Asymptotic normality » Main result There exist constants a n > 0, n ∈ N such that » Finite sample behavior I » Finite sample behavior II » Finite sample behavior III � max ( X 1 , . . . , X n ) 1 + x � − α � � � » Drawback... � ≤ x = exp − n → ∞ P lim , » Test of hypothesis a n α » Empirical power » Estimated type I error » References for all x for which 1 + α − 1 x > 0 , α > 0 . Cláudia Neves, August 18, 2005 EVA - p. 3/19

Heavy-tailed distributions » Summary » Heavy-tailed distributions � n � » Characterizing Heavy-tail Let X i i = 1 be i.i.d. random variables with common behavior » Π -variation and Γ -variation distribution function F . » A step towards estimation » Estimation of the tail parameter » Auxiliary results I Heavy-tailed models suggested by EVT: » Auxiliary results II » Asymptotic normality » Main result There exist constants a n > 0, n ∈ N such that » Finite sample behavior I » Finite sample behavior II » Finite sample behavior III � max ( X 1 , . . . , X n ) 1 + x � − α � � � » Drawback... � ≤ x = exp − n → ∞ P lim , » Test of hypothesis a n α » Empirical power » Estimated type I error » References for all x for which 1 + α − 1 x > 0 , α > 0 . Models for the tail pertaining to α = 0 ? Cláudia Neves, August 18, 2005 EVA - p. 3/19

Heavy-tailed distributions » Summary » Heavy-tailed distributions � n � » Characterizing Heavy-tail Let X i i = 1 be i.i.d. random variables with common behavior » Π -variation and Γ -variation distribution function F . » A step towards estimation » Estimation of the tail parameter » Auxiliary results I Heavy-tailed models suggested by EVT: » Auxiliary results II » Asymptotic normality » Main result There exist constants a n > 0, n ∈ N such that » Finite sample behavior I » Finite sample behavior II » Finite sample behavior III � max ( X 1 , . . . , X n ) 1 + x � − α � � � » Drawback... � ≤ x = exp − n → ∞ P lim , » Test of hypothesis a n α » Empirical power » Estimated type I error » References for all x for which 1 + α − 1 x > 0 , α > 0 . Models for the tail pertaining to α = 0 ? Examples: log-Pareto, log-Cauchy, log-Weibull ... Cláudia Neves, August 18, 2005 EVA - p. 3/19

Characterizing Heavy-tail behavior » Summary » Heavy-tailed distributions F is of regular variation at infin- » Characterizing Heavy-tail ity of index α > 0 if behavior » Π -variation and Γ -variation » A step towards estimation 1 − F ( tx ) » Estimation of the tail 1 − F ( t ) = x − α , lim parameter t → ∞ » Auxiliary results I » Auxiliary results II for all x > 0 . » Asymptotic normality » Main result » Finite sample behavior I » Finite sample behavior II 1 − F ∈ RV α » Finite sample behavior III » Drawback... » Test of hypothesis » Empirical power » Estimated type I error » References Cláudia Neves, August 18, 2005 EVA - p. 4/19

Characterizing Heavy-tail behavior » Summary » Heavy-tailed distributions F is of regular variation at infin- » Characterizing Heavy-tail ity of index α > 0 if behavior » Π -variation and Γ -variation » A step towards estimation 1 − F ( tx ) » Estimation of the tail 1 − F ( t ) = x − α , lim parameter t → ∞ » Auxiliary results I » Auxiliary results II for all x > 0 . » Asymptotic normality » Main result » Finite sample behavior I » Finite sample behavior II 1 − F ∈ RV α » Finite sample behavior III » Drawback... » Test of hypothesis » Empirical power F is a slowly varying function if » Estimated type I error » References 1 − F ( tx ) 1 − F ( t ) = 1, lim t → ∞ for all x > 0 . 1 − F ∈ RV 0 Cláudia Neves, August 18, 2005 EVA - p. 4/19

Characterizing Heavy-tail behavior » Summary » Heavy-tailed distributions F is of regular variation at infin- Heavy tails: » Characterizing Heavy-tail ity of index α > 0 if behavior » Π -variation and Γ -variation Suppose there exits a positive function a » A step towards estimation 1 − F ( tx ) » Estimation of the tail such that F satisfies 1 − F ( t ) = x − α , lim parameter t → ∞ » Auxiliary results I = 1 − x − α F ( tx ) − F ( t ) » Auxiliary results II lim , for all x > 0 . » Asymptotic normality a ( t ) α t → ∞ » Main result » Finite sample behavior I for all x > 0 , with α > 0 . » Finite sample behavior II 1 − F ∈ RV α » Finite sample behavior III » Drawback... » Test of hypothesis » Empirical power F is a slowly varying function if » Estimated type I error » References 1 − F ( tx ) 1 − F ( t ) = 1, lim t → ∞ for all x > 0 . 1 − F ∈ RV 0 Cláudia Neves, August 18, 2005 EVA - p. 4/19

Statistical Inference for Heavy and Super-Heavy-tailed distributions - PowerPoint PPT Presentation

Statistical Inference for Heavy and Super-Heavy-tailed distributions M. Isabel Fraga Alves DEIO, Faculty of Sciences, University of Lisbon, Portugal Laurens de Haan Econometric Institute, Erasmus University of Rotterdam, The Netherlands

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Importance Sampling Methodology for Multidimensional Heavy-tailed Random Walks Jose Blanchet

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Approximating the covariance matrix with heavy tailed columns and RIP. Alexander Litvak

Almost Optimal Algorithms for Linear Stochastic Bandits with Heavy-Tailed Payofgs Department of

Nonparametric estimation of extreme risks from heavy-tailed distributions St ephane GIRARD

Localization for Brownian motion in a heavy tailed Poissonian potential Ryoki Fukushima Tokyo

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Degree-degree correlations in directed networks with heavy-tailed degrees Pim van der Hoorn

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Escaping Large Deceptive Basins of Attraction with Heavy-Tailed Mutation Operators Tobias

Smoothing of Variable Bandwidth Kernel Estimate of Heavy-Tailed Density Function Natalia M.

Mixture of Heavy-Tailed distributions for Bivariate Precipitation Data

Precise large deviation probabilities for a heavy-tailed random walk 1 Thomas Mikosch

Functional Central Limit Theorem for Heavy Tailed Stationary Infinitely Divisible Processes

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Heavy-tailed random matrices and the Poisson Weighted In fi nite Tree Charles Bordenave CNRS

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Bayesian analysis for heavy-tailed nonlinear mixed effects models Cibele M. Russo in

Statistical Inference for Heavy and Super-Heavy-tailed distributions - PowerPoint PPT Presentation

Statistical Inference for Heavy and Super-Heavy-tailed distributions M. Isabel Fraga Alves DEIO, Faculty of Sciences, University of Lisbon, Portugal Laurens de Haan Econometric Institute, Erasmus University of Rotterdam, The Netherlands

Heavy tails: right skew ! Right skew ! normal distribution (not heavy tailed) ! e.g. heights of

Processing Quantities with Result for Addition . . . Heavy-Tailed Distribution of Case of a

Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S.

Concentration bounds for CVaR estimation: The cases of light-tailed and heavy-tailed

On the Statistical Rate of Nonlinear Recovery in Generative Models with Heavy-tailed Data Xiaohan

Importance Sampling Methodology for Multidimensional Heavy-tailed Random Walks Jose Blanchet

Heavy-tailed Distribu1on of Parallel I/O System Response Time

Approximating the covariance matrix with heavy tailed columns and RIP. Alexander Litvak

Almost Optimal Algorithms for Linear Stochastic Bandits with Heavy-Tailed Payofgs Department of

Nonparametric estimation of extreme risks from heavy-tailed distributions St ephane GIRARD

Localization for Brownian motion in a heavy tailed Poissonian potential Ryoki Fukushima Tokyo

LARGE DEVIATION ESTIMATES FOR CERTAIN HEAVY-TAILED DEPENDENT SEQUENCES ARISING IN RISK

Degree-degree correlations in directed networks with heavy-tailed degrees Pim van der Hoorn

Bayesian Optimization under Heavy-tailed Payoffs Sayak Ray Chowdhury Joint work with Aditya

Long range dependence for heavy Joint work with R. Kulik (U Ottawa), tailed random functions V.

Escaping Large Deceptive Basins of Attraction with Heavy-Tailed Mutation Operators Tobias

Smoothing of Variable Bandwidth Kernel Estimate of Heavy-Tailed Density Function Natalia M.

Mixture of Heavy-Tailed distributions for Bivariate Precipitation Data

Precise large deviation probabilities for a heavy-tailed random walk 1 Thomas Mikosch

Functional Central Limit Theorem for Heavy Tailed Stationary Infinitely Divisible Processes

How to Detect How to Actually . . . Linear Dependence Resulting Algorithm Heavy-Tailed . . .

Heavy-tailed random matrices and the Poisson Weighted In fi nite Tree Charles Bordenave CNRS

Traditional and Heavy-Tailed Self Regularization in Neural Network Models Charles H. Martin &amp;

Bayesian analysis for heavy-tailed nonlinear mixed effects models Cibele M. Russo in

Traditional and Heavy-Tailed Self Regularization in Neural Network Models Charles H. Martin &