On the estimation of the second order parameter in extreme-value theory

Transcription

1 On the estimation of the second order parameter in extreme-value theory by El Hadji DEME (1,3), Laurent Gardes (2) and Stéphane Girard (3) (1) LERSTAD, Université Gaston Berger, Saint-Louis, Sénégal. (2) Université de Strasbourg & CNRS, IRMA UMR rue René Descartes, Strasbourg Cedex, France. (3) Team Mistis, INRIA Rh^one-Alpes & Laboratoire Jean Kuntzmann. 655 avenue de l Europe, Montbonnot, Saint-Ismier Cedex, France. January 16th th 2012.

2 1 Extreme-Value Theory 2 New family of estimators for the second order parameter 3 Asymptotic properties 4 Link with existing estimators 5 Numerical results 2 / 30

3 Exemples of Applications Historically : meteorological events (rains, winds...). Finance : stock market crash and financial crises. Reinsurance. Aviation. Biology : Epidemiology, security food. 3 / 30

4 Main results on extreme value theory Let X 1,..., X n be a sequence of independent copies of a real random variable (r.v.) X with cumulative distribution function F. The order statistics associated to this sample are denoted by : X 1,n X n,n. Fisher-Tippett-Gnedenko theorem Under some conditions of regularity on F, there exists a real parameter γ and two sequences (a n ) n 1 > 0 and (b n ) n 1 R such that for all x R, lim P ( a 1 n n (X n,n b n ) x ) = EV γ (x), ( with EV γ(x) = where y + = max(y, 0). 4 / 30 exp (1 + γx) 1/γ exp ` e x + if γ 0, if γ = 0,

5 Definition The parameter γ is the tail index, the primary parameter of extreme events. EV γ is called the extreme value distribution and F is then said to belong to the domain of attraction of EV γ (F DA(EV γ)). Heavy-tailed model In statistics of Extremes, a model F is said to be heavy-tailed whenever, for some γ > 0, its survival function is of the forme : 1 F (x) = x 1/γ l F (x) U(x) = x γ l U (x) where U(x) = inf{y : F (y) 1 1/x} and l ( ) is a slowly varying function i.e. l (λx)/l (x) 1 as x for all λ 1. The present model is now often restated as the asumption of F regulary variation at infinity with index 1/γ (denoted 1 F (x) RV 1/γ. 5 / 30

6 Inference statistic of γ for heavy-tailed model In Statistics of extremes, inference is often based W i,k = (log X n i+1,n log X n k,n ) Z i,k = i (log X n i+1,n log X n i,n ) the log-excesses the rescaled log-spacings Exemple of γ estimator 6 / 30 H n,k = 1 k M α n,k = 1 k kx W i,k = 1 k i=1 kx i=1 kx Z i,k, Hill Estimator Hill (1975) i=1 W α i,k, Moment Estmator, Deker et al (1989) P W n,k = 1 k k i=1 W ` i Zi,k, Kernel estimator of γ, k+1 W is a positive Kernel function. P S n,k = 1 k ` i k j=1 Gα k+1 W α i,k, G α is a positive weight function and α a positive real number.

7 Second Order Condition The asymptotic distribution of estimators of γ is obtained under a second order condition. Second Order Condition (S.O.C) There exist a function A(x) 0 and a second order parameter ρ 0 such that, for every λ > 0, log U(λx) log U(x) γ log λ lim = K x ρ(λ), A(x) where K ρ(λ) = R λ 1 uρ 1 du. Remarks 7 / 30 A is regularly varying with index ρ. If ρ decreases, the rate of convergence of log U(tx) log U(t) to γ log increases. ρ controls the bias of the estimators of γ. A third order condition is needed to deal with the asymptotic distribution of ρ estimators.

8 Third Order Condition Third Order Condition (T.O.C) There exist functions A(x) 0 and B(x) 0, a second order parameter ρ 0 and a third order parameter β 0 such that, for every λ > 0, where (log U(λx) log U(x) γ log λ)/a(x) K ρ(λ) lim = L ρ,β (λ), x B(x) L (ρ,β) (λ) = Z λ 1 Z s s ρ 1 u β 1 duds, and the functions A and B are regularly varying with index ρ and β respectively. Objective Propose a new class of estimators for the second order parameter and to study their asymptotic properties. 1 8 / 30

9 New family of estimators for the second order parameter The model T n = T n(x 1,..., X n) : a random vector in R d the sample X 1,..., X n. drawn from The statistics can always be expanded as : where I = t (1,..., 1) R d, ωn 1 P (T n χ ni) f (ρ) χ n and ω n : random variables, ξ n R d : a random vector, f : R R d : a function continuously differentiable in a neighborhood of ρ (independent of γ). 9 / 30

10 General approach Notations ψ : R d R such that Invariance properties (Inv-prop) ψ(x + λi) = ψ(x) for all x R d and λ R, ψ(λx) = ψ(x) for all λ R \ {0}, Regularity properties (Reg-prop) ψ is continuously differentiable in a neighborhood of f (ρ), ϕ := ψ f is continuous in a neighborhood of ρ Bijection property (Bij-prop) there exist J 0 R and an open interval J R such that ϕ is a bijection from J 0 to J. 10 / 30

11 The estimator Clearly by the invariance and the regularity properties ψ(ωn 1 P (T n χ ni)) = ψ(t n) ψ(f (ρ)). Z n = ψ(t n) ϕ(ρ). Under the bijection property, our family of estimators of the second order parameter is thus defined by : ˆρ n = ϕ 1 (Z n)1l{z n J}. 1l A is the indicator function of the set A 11 / 30

12 Asymptotic properties Theorems Suppose that Inv-prop, Reg-prop and Bij-prop hold then P ˆρ n ρ as n if there exists two a sequence v n, a function, m : R R d and a d d matrix Σ such that then v n(ωn 1 d (T n χ ni) f (ρ)) N d (m(ρ), Σ). v n(ˆρ n ρ) d N m (ρ)! ψ ϕ (ρ), σψ(ρ) 2 with, (ϕ (ρ)) 2 ϕ (ρ) = t f (ρ) ψ(f (ρ)) m ψ (ρ) = t m(ρ) ψ(f (ρ)) σ 2 ψ(ρ) = t ψ(f (ρ)) Σ ψ(f (ρ)). 12 / 30

13 Link with existing estimators Existing estimators In the literature, at least two ways of estimating the second order parameter can be found : Estimators based on rescaled log-spacings, j(log X n j+1 log X n j ) : Goegebeur et al., (JSPI, 2010). Estimators based on log-spacings, log X n j+1 log X n k : Gomeset al.,(extremes, 2002), Fraga-Alves et al., (Portugaliae Mathematica, 2003), Ciuperca and Mercadier, (Extremes, 2010). 13 / 30

14 Link with existing estimators 1. Estimators based on rescaled log-spacings : j(log X n j+1 log X n j ) R k (τ) = 1 k kx «j H τ j log X n j+1,n, k + 1 X n j,n j=1 H τ is a kernel function integrating to one. This statistic is used for instance by Beirlant et al., (Extremes, 1999) to estimate the extreme value index γ and by Goegebeur et al. (JSPI, 2010) to estimate the second order parameter ρ. They proved asymptotic normality of these estimators under a technical condition on the kernel, denoted by (C1) hereafter. 14 / 30

15 Links with existing estimators Link with our framework Suppose the third order condition and (C1) hold. If the sequence k satisfies k, n/k, k 1/2 A(n/k), k 1/2 A 2 (n/k) λ A and k 1/2 A(n/k)B(n/k) λ B, then the random vector T (R) n := satisfies the model i.e. ωn 1 χ n = 1, ω n = A(n/k)/γ(1 + o P (1)), Z 1 f (R) (ρ) = θ i (R k (τ i )/γ) θ i, i = 1,..., d 0 (T n (R) χ ni), P f (R) (ρ) with «H τi (u)u ρ du, i = 1,..., d 15 / 30

16 Link with existing estimators Choice of the kernel and of the function ψ 16 / 30 d = 8, ψ δ : D R \ {0} ψ δ (x 1,..., x 8) = e ψ δ (x 1 x 2, x 3 x 4, x 5 x 6, x 7 x 8), where δ 0 D = {(x 1,..., x 8) R 8 ; x 1 x 2, x 3 x 4, and (x 5 x 6)(x 7 x 8) > 0}. «δ eψ δ : R 4 R textttis given by : ψ e δ (y 1,..., y 4) = y1 y4. y 2 y 3 H τ (u) = (τ + 1)u τ, the statistic T (R) n depends on 16 parameters {(θ i, τ i ) (0, ) 2, i = 1,..., 8}. Let θ = ( θ 1,..., θ 4) (0, ) 4 with θ 3 θ 4 and consider {θ i = θ i/2, i = 1,..., 8} with δ = ( θ 1 θ 2)/( θ 3 θ 4) with x = inf{n N x n}. τ 1 < τ 2 τ 3 < τ 4, τ 5 < τ 6 τ 7 < τ 8 involves 12 free parameters. Z n (R) T (R) n ϕ (R) δ = ψ δ f (R). = ψ δ (T n (R) ) and

17 Link with existing estimators Our contributions We can thus define the following family of estimators : ˆρ (R) n = ϕ 1 (Z n (R) )1l{Z n (R) J}. New estimators of ρ (explicit or not), with Consistency and Asymptotic normality (consequence of ours theorems) Examples of explicit estimators δ = 1 i.e. θ 1 θ 2 = θ 3 θ 4. The rv Z n (R) is denoted by Z (R) n,1. ˆρ (R) n,1 = τ 5ω(1, θ) τ 1 Z (R) n,1 ω(1, θ) Z (R) n,1 δ = 0 i.e. θ 1 = θ 2. The rv Z (R) n ˆρ (R) n,2 = τ 4ω(0, θ) τ 1 Z (R) n,2 (R) ω(0, θ) Z n,2 1l{Z (R) n,1 ω(1, θ) (1, e ψ 1(τ 4, τ 1, τ 4, τ 5))}. is thus denoted by Z (R) n,2 : 1l{Z (R) n,2 ω(0, θ) (1, e ψ 0(τ 4, τ 1, τ 4, τ 5))}. with ω(δ, θ) = e ψ δ ( θ 1(τ 1 τ 2), θ 2(τ 2 τ 4), θ 2(τ 2 τ 4), θ 4(τ 6 τ 4)) 17 / 30

18 Link with existing estimators 2. Estimators based on log-spacings : log X n j+1 log X n k S k (τ, α) = 1 k kx «j G τ,α log X «α n j+1,n, α > 0, k + 1 X n k,n j=1 G τ,α is a positive function. Dekkers et al. (Annals of statistics, 1989) proposed an estimator of γ based on this statistic in the particular case where G τ,α is constant. Ciuperca and Mercadier (Extremes, 2010) used the general statistic to estimate the parameters γ and ρ. They proved the asymptotic normality under a technical condition on the function G τ,α, denoted by (C2) hereafter. 18 / 30

19 Link with existing estimators Link with our framework Suppose the third order condition, (C2) hold. If the sequence k satisfies k, n/k, k 1/2 A(n/k), then the random vector k 1/2 A 2 (n/k) λ A and k 1/2 A(n/k)B(n/k) λ B, T (S) n = «θi Sk (τ i, α i ), i = 1,..., d! γ α i satisfies the model i.e. ωn 1 (T n (S) P χ ni) f (S) (ρ) with χ n = 1, ω n = A(n/k)/γ(1 + o P (1)), and Z 1 «f (S) (ρ) = θ i α i G τi,α i (u)(log(1/u)) α i 1 K ρ(u)du; i = 1,..., d, 0 19 / 30

20 Link with existing estimators Choice of the weighted function and of the function ψ d = 8, the weighted function G τ,α is given by : g τ (u) G τ,α(u) = R 1 0 gτ (x)( log x)α dx, τ 0, α > 0 where g 0(x) = 1 and g τ (x) = (τ + 1)(1 x τ )/τ if τ > free parameters : {(θ i, τ i, α i ) (0, ) 3, i = 1,..., 8} Let (ζ 1,..., ζ 4) (0, ) 4 with ζ 3 ζ 4, such that {θ i α i = ζ i/2, i = 1,..., 8} with δ = (ζ 1 ζ 2)/(ζ 3 ζ 4). x = inf{n N x n}. (τ 2i 1, α 2i 1 ) (τ 2i, α 2i ), for i = 1,..., 4 and, for i = 3, 4, (τ 2i 1, α 2i 1 ) (τ 2i, α 2i ) 20 / 30

21 Link with existing estimators Our contributions We can thus define the following family of estimators : ˆρ (S) n = ϕ 1 (Z n (S) )1l{Z n (S) J}. New estimators of ρ (not necessarily explicit) with Consistency and Asymptotic normality. Exemples of explicit estimators δ = 0 (i.e. ζ 1 = ζ 2), α 1 = α 2 = α 3 = α 4 = 1, τ 1 = α 5 = α 8 = 2 τ 4 = α 6 = 3. Denoting by Z (S) n,4 the rv Z n (S), the estimator of ρ is given by : ˆρ (S) n,4 = 6(Z (S) n,4 + 2) (S) 1l{Z 3Z (S) n,4 n,4 + 4 ( 2, 4/3)}. Consider ω, a function depends only on δ and ζ 21 / 30

22 Link with existing estimators Exemples of explicit estimators δ = 0, α 1 = α 3 = α 4 = 1, τ 1 = τ 4 = α 2 = α 5 = α 8 = 2 and α 6 = 3. Denoting by Z (S) (S) n,5 the rv Z n, ˆρ (S) n,4 ˆρ (S) n,5 = 2(Z (S) n,5 2) (S) 1l{Z 2Z (S) n,5 n,5 1 (1/2, 2)}. and ˆρ(S) n,5 are estimators Ciuperca and Mercadier, (Extremes, 2010). δ = 1, α 1 = α 3 = α 4 = 1, τ 1 = τ 4 = α 2 = α 5 = α 8 = 2 and α 6 = 3 Z (S) n,6 (S) the rv Z n, a new estimator of ρ is given by ρ is given by ˆρ (S) n,6 (S) 3Z n,6 = 4ω (1, ζ) Z (S) n,8 ω (1, ζ) 1l{Z (S) n,6 ω (1, ζ) (1/2, 2/3)}. δ = 1 (i.e. ζ 1 ζ 2 = ζ 3 ζ 4), α 1 = α 2 = α 3 = α 4 = 1, denoting by Z (S) n,7 the rv Z n (S), a nother new estimator of ρ is given by : ˆρ (S) n,7 = Z (S) n,7 + 4/3ω (1, ζ) (S) 1l{Z 2Z (S) + 4/3ω n,7 (1, ζ) ω (1, ζ) ( 4/3, 2/3)}. 22 / 30

23 Asymptotic comparaison Choice of the parameters The estimator based on rescaled log-spacings. H τi (u) = (τ i + 1)u τ i, i = 1,..., 8. τ 1,..., τ 8 and θ 1, θ 3, θ 4, taken as in Goegebeur et al.. θ 2 = θ 1 + δ( θ 4 θ 3), δ 0. Choice of δ, 1 minimization of the AMSE is impossible (depends on unknown parameters). 2 We use an upper bound on the AMSE : AMSE c(γ, λ A, λ B )π(δ, ρ, β). 3 ρ = β, we minimize the function π in delta and the optimal δ as a function of ρ. 23 / 30

24 δ Asymptotic comparaison Choice of δ δ =1.8 δ= ρ Fig.: Optimal δ as a function of ρ δ = 0, 1, 1.5, 1.8, + 24 / 30

25 Illustration on a Burr distribution Burr distribution, Burr(ζ,λ,η) : 1 F (x) = (ζ/(ζ + x η )) λ, x > 0, ζ, λ, η > 0, γ = 1/λη and ρ = 1/λ. The third order condition is hold with β = ρ, A(x) = γx ρ /(1 x ρ ) and B(x) = ρx ρ /(1 x ρ ). n = 5000, γ = ζ = 1, η = 1/λ, ρ = η and k = 1,..., 4995, 25 / 30

26 ρ = 0.25 ρ = 1 AMSE δ = 0 δ = 1 δ = 1.5 δ = 1.8 δ = + AMSE δ = 0 δ = 1 δ = 1.5 δ = 1.8 δ = k k Fig.: Asymptotic mean squared error of ˆρ (R) n, ρ = 0.25; 1 26 / 30

27 ρ = 2.5 ρ = 3 AMSE δ = 0 δ = 1 δ = 1.5 δ = 1.8 δ = + AMSE δ = 0 δ = 1 δ = 1.5 δ = 1.8 δ = k k Fig.: Asymptotic mean squared error of ˆρ (R) n, ρ = 2.5; 3 27 / 30

28 ρ = 4 ρ = 5 AMSE δ = 0 δ = 1 δ = 1.5 δ = 1.8 δ = + AMSE δ = 0 δ = 1 δ = 1.5 δ = 1.8 δ = k k Fig.: Asymptotic mean squared error of ˆρ (R) n, ρ = 4; 5 28 / 30

29 Concluding Remarks If ρ 4, the smallest AMSE is obtained with δ = 1.8. If 3 ρ 2.5, the best AMSE is given by δ = +. If ρ 1, the smallest AMSE is given by δ = 1.5. The values {1.5, 1.8, + } obtained by minimizing the function π are also of interest to minimize the asymptotic mean-squared error. More generally, the minimization of π should permit to determine optimal values for the parameters of any estimator of ρ. 29 / 30

30 AND THAT S ALL / 30