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

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Cláudia Neves, August 18, 2005 EVA - p. 1/19

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

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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

EVT: Modeling the tail of a distribution;

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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

EVT: Modeling the tail of a distribution; Characterization of Super-Heavy tails;

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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

EVT: Modeling the tail of a distribution; Characterization of Super-Heavy tails; Estimation of the index of tail heaviness α ≥ 0;

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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

EVT: Modeling the tail of a distribution; Characterization of Super-Heavy tails; Estimation of the index of tail heaviness α ≥ 0; Testing the presence of Super-Heavy tails;

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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

EVT: Modeling the tail of a distribution; Characterization of Super-Heavy tails; Estimation of the index of tail heaviness α ≥ 0; Testing the presence of Super-Heavy tails; Simulation results.

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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

Let

Xi

n

i=1 be i.i.d. random variables with common

distribution function F.

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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

Let

Xi

n

i=1 be i.i.d. random variables with common

distribution function F. Heavy-tailed models suggested by EVT: There exist constants an > 0, n ∈ N such that lim

n→∞ P

max(X1, . . . , Xn) an

≤ x

= exp
−
1 + x

α −α , for all x for which 1 + α−1x > 0, α > 0.

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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

Let

Xi

n

i=1 be i.i.d. random variables with common

distribution function F. Heavy-tailed models suggested by EVT: There exist constants an > 0, n ∈ N such that lim

n→∞ P

max(X1, . . . , Xn) an

≤ x

= exp
−
1 + x

α −α , for all x for which 1 + α−1x > 0, α > 0. Models for the tail pertaining to α = 0 ?

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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

Let

Xi

n

i=1 be i.i.d. random variables with common

distribution function F. Heavy-tailed models suggested by EVT: There exist constants an > 0, n ∈ N such that lim

n→∞ P

max(X1, . . . , Xn) an

≤ x

= exp
−
1 + x

α −α , for all x for which 1 + α−1x > 0, α > 0. Models for the tail pertaining to α = 0 ? Examples: log-Pareto, log-Cauchy, log-Weibull ...

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 4/19

Characterizing Heavy-tail behavior

F is of regular variation at infinity of index α > 0 if lim

t→∞

1 − F(tx) 1 − F(t) = x−α, for all x > 0. 1 − F ∈ RVα

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 4/19

Characterizing Heavy-tail behavior

F is of regular variation at infinity of index α > 0 if lim

t→∞

1 − F(tx) 1 − F(t) = x−α, for all x > 0. 1 − F ∈ RVα F is a slowly varying function if lim

t→∞

1 − F(tx) 1 − F(t) = 1, for all x > 0. 1 − F ∈ RV0

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 4/19

Characterizing Heavy-tail behavior

F is of regular variation at infinity of index α > 0 if lim

t→∞

1 − F(tx) 1 − F(t) = x−α, for all x > 0. 1 − F ∈ RVα F is a slowly varying function if lim

t→∞

1 − F(tx) 1 − F(t) = 1, for all x > 0. 1 − F ∈ RV0 Heavy tails: Suppose there exits a positive function a such that F satisfies lim

t→∞

F(tx) − F(t) a(t)

= 1 − x−α

α , for all x > 0, with α > 0.

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 4/19

Characterizing Heavy-tail behavior

F is of regular variation at infinity of index α > 0 if lim

t→∞

1 − F(tx) 1 − F(t) = x−α, for all x > 0. 1 − F ∈ RVα F is a slowly varying function if lim

t→∞

1 − F(tx) 1 − F(t) = 1, for all x > 0. 1 − F ∈ RV0 Heavy tails: Suppose there exits a positive function a such that F satisfies lim

t→∞

F(tx) − F(t) a(t)

= 1 − x−α

α , for all x > 0, with α > 0. Super-Heavy tails: lim

t→∞

F(tx) − F(t) a(t)

= log x,

x > 0. F ∈ ERVα, α ≥ 0

de Haan (1984)

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 5/19

Π-variation and Γ-variation

Super-Heavy-tailed distributions

U(t) :=

1

1 − F ←

(t) = inf

x : F(x) ≥ 1 − 1

t

The following are equivalent:

F ∈ Π(a) : lim

t→∞

F(tx) − F(t) a(t)

= log x,

x > 0; U ∈ Γ(q) : lim

t→∞

U

t + x q(t)
U(t)

= ex,

x ∈ R, with q(t) = t2a(U(t)), a ∈ RV0

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 6/19

A step towards estimation

lim

t→∞

∞

1

F(tx) − F(t) a(t) dx x2 =

∞

1

1 − x−α α dx x2 = 1 1 + α

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 6/19

A step towards estimation

lim

t→∞

∞

1

F(tx) − F(t) a(t) dx x3 =

∞

1

1 − x−α α dx x3 = 1 2(2 + α)

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 6/19

A step towards estimation

For j = 1, 2 lim

t→∞

∞

1

F(tx) − F(t) a(t) dx xj+1 =

∞

1

1 − x−α α dx xj+1 = 1 j (j + α)

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 6/19

A step towards estimation

For j = 1, 2 lim

t→∞

∞

1

F(tx) − F(t) a(t) dx xj+1 =

∞

1

1 − x−α α dx xj+1 = 1 j (j + α)

∞

1

F(tx) − F(t)

dx xj+1 = t j j

∞

t

dF(u) u j

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 6/19

A step towards estimation

For j = 1, 2 lim

t→∞

∞

1

F(tx) − F(t) a(t) dx xj+1 =

∞

1

1 − x−α α dx xj+1 = 1 j (j + α)

∞

1

F(tx) − F(t)

dx xj+1 = t j j

∞

t

dF(u) u j 2

∞

1

F(tx) − F(t)
dx/x3

∞

1

F(tx) − F(t)
dx/x2 =

∞

t

t/u

2 dF(u)

∞

t (t/u) dF(u) −

→

t→∞

1 + α 2 + α =: ψ(α)

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 6/19

A step towards estimation

For j = 1, 2 lim

t→∞

∞

1

F(tx) − F(t) a(t) dx xj+1 =

∞

1

1 − x−α α dx xj+1 = 1 j (j + α)

∞

1

F(tx) − F(t)

dx xj+1 = t j j

∞

t

dF(u) u j 2

∞

1

F(tx) − F(t)
dx/x3

∞

1

F(tx) − F(t)
dx/x2 =

∞

t

t/u

2 dF(u)

∞

t (t/u) dF(u) −

→

t→∞

1 + α 2 + α =: ψ(α) Let X1,n ≤ X2,n ≤ . . . ≤ Xn,n be the order statistics corresponding to the random sample (X1, . . . , Xn), ˆ ψn(k) :=

k−1

∑

i=0

Xn−k,n/Xn−i,n

2

k−1

∑

i=0

Xn−k,n/Xn−i,n

P

− →

n→∞ ψ(α),

0 ≤ α < ∞ ?

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 7/19

Estimation of the tail parameter

ˆ αn(k) := 2

k−1

∑

i=0

Xn−k,n/Xn−i,n

2 −

k−1

∑

i=0

(Xn−k,n/Xn−i,n)

k−1

∑

i=0

(Xn−k,n/Xn−i,n) −

k−1

∑

i=0

Xn−k,n/Xn−i,n

2 k = kn is a sequence of positive integers such that kn → ∞ and kn = o(n) as n → ∞ ➸ kn is an intermediate sequence

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 7/19

Estimation of the tail parameter

ˆ αn(k) := 2

k−1

∑

i=0

Xn−k,n/Xn−i,n

2 −

k−1

∑

i=0

(Xn−k,n/Xn−i,n)

k−1

∑

i=0

(Xn−k,n/Xn−i,n) −

k−1

∑

i=0

Xn−k,n/Xn−i,n

2 k = kn is a sequence of positive integers such that kn → ∞ and kn = o(n) as n → ∞ ➸ kn is an intermediate sequence Consistency: If (n/

√

k) a

U(n/k)

−

→

n→∞ ∞, then ˆ

αn(k)

P

− →

n→∞ α ≥ 0.

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 7/19

Estimation of the tail parameter

ˆ αn(k) := 2

k−1

∑

i=0

Xn−k,n/Xn−i,n

2 −

k−1

∑

i=0

(Xn−k,n/Xn−i,n)

k−1

∑

i=0

(Xn−k,n/Xn−i,n) −

k−1

∑

i=0

Xn−k,n/Xn−i,n

2 k = kn is a sequence of positive integers such that kn → ∞ and kn = o(n) as n → ∞ ➸ kn is an intermediate sequence Consistency: If (n/

√

k) a

U(n/k)

−

→

n→∞ ∞, then ˆ

αn(k)

P

− →

n→∞ α ≥ 0.

Since (n/k) a

U(n/k)

−

→

n→∞ α, we have that:

for α > 0, consistency holds for any intermediate sequence kn; for α = 0, we need to impose a lower bound to the sequence kn.

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 8/19

Auxiliary results I

Define, for j = 1, 2, M(j)

n (k)

:= n k j

∞

1

Fn(xXn−k,n) − Fn(Xn−k,n)

dx xj+1

=

1 k

k−1

∑

i=0

Xn−k,n Xn−i,n j

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 8/19

Auxiliary results I

Define, for j = 1, 2, M(j)

n (k)

:= n k j

∞

1

Fn(xXn−k,n) − Fn(Xn−k,n)

dx xj+1

=

1 k

k−1

∑

i=0

Xn−k,n Xn−i,n j Let

Un
n≥1 be a sequence of i.i.d. uniformly distributed r.v.’s.

Define: En(t) := 1 n

n

∑

i=1

I{Ui,n≤t}, t ∈ [0, 1] and

{en(t); 0 ≤ t ≤ 1} := {√

n (En(t) − t); t ∈ [0, 1]} n-th uniform empirical process

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 8/19

Auxiliary results I

Define, for j = 1, 2, M(j)

n (k)

:= n k j

∞

1

Fn(xXn−k,n) − Fn(Xn−k,n)

dx xj+1

=

1 k

k−1

∑

i=0

Xn−k,n Xn−i,n j Define Qn(k) := (k/n)/

1 − F(Xn−k,n)
and t∗ = t∗

n(x) := 1−F(xXn−k,n) 1−F(Xn−k,n) , x ≥ 1.

For all ν ∈ (0, 1/2) as, n → ∞, Fn(xXn−k,n) − Fn(Xn−k,n)

= − √

k n B(t∗) +

F(xXn−k,n) − F(Xn−k,n)
Qn(k) + Op(1) kν

n

t∗(1 − t∗)

ν, where Qn(k) is a r.v. independent of the Brownian bridge B and the Op-term is uniform for x ≥ 1.

[Proposition 4.3.1 of Csörg˝

and Horváth (1993)]

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 8/19

Auxiliary results I

Define, for j = 1, 2, M(j)

n (k)

:= n k j

∞

1

Fn(xXn−k,n) − Fn(Xn−k,n)

dx xj+1

=

1 k

k−1

∑

i=0

Xn−k,n Xn−i,n j k n M(j)

n (k)

a(Xn−k,n) − 1 j + α

= −j

k n 1 a(Xn−k,n) 1

√

k

∞

1

B(t∗) dx xj+1 + 1 j + α (Qn(k) − 1)

+ j Qn(k)

∞

1

F(xXn−k,n) − F(Xn−k,n) a(Xn−k,n)

− 1 − x−α

α dx xj+1

+ Op(1) kν

n

∞

1

t∗(1 − t∗)

ν a(Xn−k,n) dx xj+1 .

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 9/19

Auxiliary results II

2nd order Extended Regular Variation:

Suppose there exists a function A(t) → 0, as t → ∞, of constant sign near infinity and a second order parameter ρ ≤ 0 such that lim

t→∞ F(tx)−F(t) a(t)

− 1−x−α

α

A(t)

= 1

ρ x−α+ρ − 1

−α + ρ + x−α − 1

α

=: Hα,ρ(x),

According to de Haan and Stadtmüller (1996), |A(t)| ∈ RVρ. F ∈ 2ERV(α, ρ), α ≥ 0, ρ ≤ 0

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» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 9/19

Auxiliary results II

2nd order Extended Regular Variation:

Suppose there exists a function A(t) → 0, as t → ∞, of constant sign near infinity and a second order parameter ρ ≤ 0 such that lim

t→∞ F(tx)−F(t) a(t)

− 1−x−α

α

A(t)

= 1

ρ x−α+ρ − 1

−α + ρ + x−α − 1

α

=: Hα,ρ(x),

According to de Haan and Stadtmüller (1996), |A(t)| ∈ RVρ. F ∈ 2ERV(α, ρ), α ≥ 0, ρ ≤ 0

Continuity-modulus property for Brownian bridge:

For any δ ∈ (0, 1/2), we have lim

h↓0 sup x>0

|B(x + h) − B(x)|

h1/2−δ

= 0

a.s.

SLIDE 31

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 10/19

Asymptotic normality

Given an intermediate sequence k = kn such that (n/

√

k) a

U(n/k)

−

→

n→∞ ∞,

if F ∈ 2ERV(α, ρ) with α ≥ 0 and

n a
U(n/k)

1/2 A

U(n/k)

−

→

n→∞ λ ∈ R,

SLIDE 32

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 10/19

Asymptotic normality

Given an intermediate sequence k = kn such that (n/

√

k) a

U(n/k)

−

→

n→∞ ∞,

if F ∈ 2ERV(α, ρ) with α ≥ 0 and

n a
U(n/k)

1/2 A

U(n/k)

−

→

n→∞ λ ∈ R,

cj = cj(α, ρ) :=

∞

1

j Hα,ρ(x) dx xj+1 = 1

(j + α)(j + α − ρ) ,

j = 1, 2 ◆ α > 0,

n a(Xn−k,n)

1/2 k n M(j)

n (k)

a(Xn−k,n) − 1 j + α

d

− →

n→∞ −

j

√α

∞

1

B(x−α) dx xj+1 +

√α

j + α N

+ λ cj,

where {B(t); 0 ≤ t ≤ 1} is a Brownian bridge and N denotes a normal r.v. independent of B;

SLIDE 33

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 10/19

Asymptotic normality

Given an intermediate sequence k = kn such that (n/

√

k) a

U(n/k)

−

→

n→∞ ∞,

if F ∈ 2ERV(α, ρ) with α ≥ 0 and

n a
U(n/k)

1/2 A

U(n/k)

−

→

n→∞ λ ∈ R,

cj = cj(α, ρ) :=

∞

1

j Hα,ρ(x) dx xj+1 = 1

(j + α)(j + α − ρ) ,

j = 1, 2 ◆ α > 0,

n a(Xn−k,n)

1/2 k n M(j)

n (k)

a(Xn−k,n) − 1 j + α

d

− →

n→∞ −

j

√α

∞

1

B(x−α) dx xj+1 +

√α

j + α N

+ λ cj,

where {B(t); 0 ≤ t ≤ 1} is a Brownian bridge and N denotes a normal r.v. independent of B; ◆ α = 0,

n a(Xn−k,n)

1/2 k n M(j)

n (k)

a(Xn−k,n) − 1 j + α

d

− →

n→∞ − j

∞

1

W(log x) dx xj+1 + λ cj, where {W(t); 0 ≤ t < ∞} is a standard Brownian motion.

SLIDE 34

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 11/19

Main result

Given an intermediate sequence k = kn such that (n/

√

k) a

U(n/k)

−

→

n→∞ ∞,

if F ∈ 2ERV(α, ρ) with α ≥ 0 and

n a
U(n/k)

1/2 A

U(n/k)

−

→

n→∞ λ ∈ R,

SLIDE 35

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 11/19

Main result

Given an intermediate sequence k = kn such that (n/

√

k) a

U(n/k)

−

→

n→∞ ∞,

if F ∈ 2ERV(α, ρ) with α ≥ 0 and

n a
U(n/k)

1/2 A

U(n/k)

−

→

n→∞ λ ∈ R,

k−1

∑

i=0

Xn−k,n Xn−i,n 1/2 ˆ αn(k) − α

d

− →

n→∞ N

b1, σ2

1

,

with b1 = b1(α, ρ) :=

−λ √

1 + α (2 + α)

(1 + α − ρ)(2 + α − ρ) ,

σ2

1 = σ2 1(α)

:=

(1 + α)(2 + α) (3 + α)(4 + α) (4 + 3α + α2).

SLIDE 36

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 11/19

Main result

Given an intermediate sequence k = kn such that (n/

√

k) a

U(n/k)

−

→

n→∞ ∞,

if F ∈ 2ERV(α, ρ) with α ≥ 0 and

n a
U(n/k)

1/2 A

U(n/k)

−

→

n→∞ λ ∈ R,

k−1

∑

i=0

Xn−k,n Xn−i,n 1/2 ˆ αn(k) − α

d

− →

n→∞ N

b1, σ2

1

,

with b1 = b1(α, ρ) :=

−λ √

1 + α (2 + α)

(1 + α − ρ)(2 + α − ρ) ,

σ2

1 = σ2 1(α)

:=

(1 + α)(2 + α) (3 + α)(4 + α) (4 + 3α + α2).

An alternative formulation is

n a
U
n/k

1/2ˆ αn(k) − α

d

− →

n→∞ N

b∗

1, σ∗2 1

,

b∗

1 = b∗ 1(α, ρ) = (1 + α)1/2b1(α, ρ)

and σ∗2

1 = σ∗2 1 (α) = (1 + α) σ2 1(α)

SLIDE 37

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 12/19

Finite sample behavior I

50 100 150 200 250 300 k

0.6
0.4
0.2

0.0 0.2 0.4 0.6

Beta=0.2 Beta=0.5 Beta=0.8 Beta=1.0 Beta=1.5

LogPareto

Mean

Patterns of the estimated mean of ˆ αn(k) Let Y be a random variable with standard Pareto distribution. A random variable W follows a log-Pareto distribution with parameter β > 0 if and

nly if W = (eβY − 1)/β.

50 100 150 200 250 300 k

0.1

0.0 0.1 0.2 0.3 0.4

Beta=0.2 Beta=0.5 Beta=0.8 Beta=1.0 Beta=1.5

LogPareto

Mean

Patterns of the estimated mean of the Hill estimator

SLIDE 38

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 13/19

Finite sample behavior II

50 100 150 200 250 300 k

0.6
0.4
0.2

0.0 0.2 0.4 0.6 0.8

Beta=0.2 Beta=0.5 Std.LogPareto

LogWeibull and standard LogPareto

Mean

Patterns of the estimated mean of ˆ αn(k) log-Weibull distribution: F(x) = 1 − exp{−(log x)β}, x ≥ 1, 0 < β < 1 Standard log-Pareto distribution: FX(x) = 1 − (log x)−1, x ≥ e

50 100 150 200 250 300 k 0.0 0.2 0.4 0.6 0.8

Beta=0.2 Beta=0.5 Std.LogPareto

LogWeibull and standard LogPareto

Mean

Patterns of the estimated mean of the Hill estimator

SLIDE 39

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 14/19

Finite sample behavior III

50 100 150 200 250 300 k 0.0 0.5 1.0 1.5 2.0

Alpha=0.2 Alpha=0.5 Alpha=0.8 Alpha=1.0 Alpha=1.5

Generalized Pareto

Mean

Patterns of the estimated mean of ˆ αn(k) Generalized Pareto distribution: F(x) = 1 −

1 + x

α −α , x > 0, α ≥ 0

50 100 150 200 250 300 k 0.0 0.5 1.0 1.5 2.0

Alpha=0.2 Alpha=0.5 Alpha=0.8 Alpha=1.0 Alpha=1.5

Generalized Pareto

Mean

Patterns of the estimated mean of the Hill estimator

SLIDE 40

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 15/19

Drawback...

... of the Hill estimator...

Consider a random sample taken from a standard log-Pareto population X = eY, with Y denoting a standard Pareto random variable. Then, the Hill estimator acquires the simple form Hn(k) = 1 k

k−1

∑

i=0

Yn−i,n − Yn−k,n Hence, Hn(k) d

=Yn−k,n

1 k

k

∑

i=1

Y∗

i − 1

= n

k

Sk + bk
1 + op(1)
,

with {Y∗

i }k i=1 denoting i.i.d. standard Pareto random variables independent of

the random threshold Yn−k,n, bk = O(log k) and where Sk denotes a random variable with limiting sum-stable distribution.

SLIDE 41

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 15/19

Drawback...

... of the Hill estimator...

Undertaking the log-Weibull distribution, the Hill estimator may be written as Hn(k) = 1 k

k−1

∑

i=0

log Yn−i,n

1/β −

log Yn−k,n

1/β Hence, Hn(k) d

=

log Yn−k,n

1/β−1 1 β 1 k

k

∑

i=1

log Y∗

i + op(1)

=
log n

k 1/β−1 1 β

S∗

k +

√

k

1 + op(1)
,

where S∗

k converges to a normal random variable.

SLIDE 42

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 16/19

Test of hypothesis

H0 : α = 0 versus H1 : α > 0

SLIDE 43

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 16/19

Test of hypothesis

H0 : α = 0 versus H1 : α > 0 Test Statistic: Tn(k) :=

√

24 k−1

∑

i=0

Xn−k,n Xn−i,n 1/2 ˆ ψn(k) − 1 2

Reject H0 in favor of the unilateral alternative if Tn(k) > z1−¯

α, where zε denotes

the ε-quantile of the standard normal distribution.

SLIDE 44

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 16/19

Test of hypothesis

H0 : α = 0 versus H1 : α > 0 Test Statistic: Tn(k) :=

√

24 k−1

∑

i=0

Xn−k,n Xn−i,n 1/2 ˆ ψn(k) − 1 2

Reject H0 in favor of the unilateral alternative if Tn(k) > z1−¯

α, where zε denotes

the ε-quantile of the standard normal distribution.

The presented test is asymptotically of size ¯

α;

the test discriminates between distributions lying in the class of ERVα for

which the second order condition holds with any intermediate sequence kn such that

n a
U(n/kn)

1/2A

U(n/kn)

−

→

n→∞ 0.

SLIDE 45

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 17/19

Empirical power

50 100 150 200 k 0.0 0.2 0.4 0.6 0.8 1.0

Power

LogGamma and Burr

LogGamma(0.5,2) Burr(1,0.5,1) Burr(1,1,0.5)

Empirical power at a nominal level ¯ α = 0.05

Underlying distribution function F ∈ ERV0.5

SLIDE 46

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 17/19

Empirical power

50 100 150 200 k 0.0 0.2 0.4 0.6 0.8 1.0

Cauchy, LogGamma and Burr

Cauchy LogGamma(1,2) Burr(1,0.5,2) Power

Empirical power at a nominal level ¯ α = 0.05

Underlying distribution function F ∈ ERV1

SLIDE 47

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 18/19

Estimated type I error

50 100 150 200 k 0.0 0.2 0.4 0.6 0.8 1.0

Type I error

LogPareto

Beta=0.5 Beta=0.8 Beta=1.0 Beta=1.5

Estimated type I error probability at a nominal level ¯ α = 0.05

SLIDE 48

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 18/19

Estimated type I error

50 100 150 200 k 0.0 0.2 0.4 0.6 0.8 1.0

LogCauchy, LogWeibull and standard LogPareto

LogCauchy Beta=0.2 Beta=0.5 Std.LogPareto Type I error

Estimated type I error probability at a nominal level ¯ α = 0.05

SLIDE 49

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 19/19

References

Csörg˝

and Horváth (1993). Weighted Approximations in Probability and Statistics.

Wiley.

de Haan, L. (1984). Slow variation and characterization of domains of attraction.

Statistical Extremes and Applications.Reidel Publishing, Dordrecht.

de Haan, L. and Stadtmüller, U. (1996). Generalized regular variation of second order.

J. Austral. Math. Soc..

SLIDE 50

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 19/19

References

Csörg˝

and Horváth (1993). Weighted Approximations in Probability and Statistics.

Wiley.

de Haan, L. (1984). Slow variation and characterization of domains of attraction.

Statistical Extremes and Applications.Reidel Publishing, Dordrecht.

de Haan, L. and Stadtmüller, U. (1996). Generalized regular variation of second order.

J. Austral. Math. Soc..

Heavy Tails

Adler, R. and Feldman, R. and Taqqu, M. S. (1998). Practical Guide to Heavy Tails.

Birkhäuser Verlag.

Heffernan, J. and Resnick, S. (2004). Limit laws for random vectores with an extreme

component. www.orie.cornell.edu/˜sid/

Diebolt, J. and El-Aroui, M. and Garrido, M. and Girard, S. (2003). Quasi-conjugate

bayes estimates for GPD parameters and application to heavy tails modeling. INRIA. Toulouse, France.

SLIDE 51

» Summary » Heavy-tailed distributions » Characterizing Heavy-tail behavior » Π-variation and Γ-variation » 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. 19/19

References

Csörg˝

and Horváth (1993). Weighted Approximations in Probability and Statistics.

Wiley.

de Haan, L. (1984). Slow variation and characterization of domains of attraction.

Statistical Extremes and Applications.Reidel Publishing, Dordrecht.

de Haan, L. and Stadtmüller, U. (1996). Generalized regular variation of second order.

J. Austral. Math. Soc..

Heavy Tails

Adler, R. and Feldman, R. and Taqqu, M. S. (1998). Practical Guide to Heavy Tails.

Birkhäuser Verlag.

Heffernan, J. and Resnick, S. (2004). Limit laws for random vectores with an extreme

component. www.orie.cornell.edu/˜sid/

Diebolt, J. and El-Aroui, M. and Garrido, M. and Girard, S. (2003). Quasi-conjugate

bayes estimates for GPD parameters and application to heavy tails modeling. INRIA. Toulouse, France.

Super-Heavy tails

Desgagné, A. and Angers, J.-F. (2004). Importance sampling with the generalized

exponential power density. Stat. Comp. (accepted).

Mikosch, T. and Resnick, S. (2004). Activity rates with very heavy tails.

www.orie.cornell.edu/˜sid/

Zeevi, A. and Glynn, P

.W. (2004). Recurrence properties of autoregressive processes with super-heavy tailed innovations. J. Appl. Probab. (to appear).