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