Estimation of the second order parameter for heavy-tailed distributions

Transcription

1 Estimation of the second order parameter for heavy-tailed distributions Stéphane Girard Inria Grenoble Rhône-Alpes, France joint work with El Hadji Deme (Université Gaston-Berger, Sénégal) and Laurent Gardes (Université de Strasbourg, France)

2 1 Extreme value theory 2 Estimation of the second order parameter 3 Asymptotic properties 4 Link with existing estimators 5 Illustration on simulations 2

3 Extreme value theory Let X 1,..., X n be independent copies of a real random variable X with survival function F = 1 F. The order statistics associated to this sample are denoted by : X 1,n X n,n. Fréchet Maximum domain of attraction The cumulative distribution function F belongs to the Fréchet maximum domain of attraction if and only if F (x) = x 1/γ l(x), where γ > 0 is the extreme-value index and l is a slowly varying function i.e. l(λx) l(x) 1 as x for all λ 1. This condition is equivalent to F is regularly varying with index 1/γ (heavy-tailed distribution). The asymptotic distribution of estimators of γ is obtained under a second order condition. 3

4 Extreme value theory Second order condition There exist a function A(x) 0 and a second order parameter ρ 0 such that, for all λ > 0, «Z 1 l(λx) λ lim x A(x) log = K ρ(λ) := u ρ 1 du. l(x) A is regularly varying with index ρ. If ρ is small, the rate of convergence of l(λx)/l(x) to one is high (and conversely). ρ controls the bias of the estimators of γ. ρ is of primordial importance in the adaptative choice of k which is the number of upper order statistics X n k,+1,n X n,n used in the estimation of γ. A third order condition is needed to deal with the asymptotic distribution of ρ estimators. 1 4

5 Extreme value theory Third order condition 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, with (log l(λx) log l(x)) /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 where the functions A and B are regularly varying with index ρ and β respectively. 1 Contributions A new class of estimators for the second order parameter ρ, Asymptotic properties, Links with existing estimators, New estimators. 5

6 Definition of the family of estimators for the second order parameter Model The two main ingredients of our approach are a random vector T n = T n(x 1,..., X n) R d a function ψ : R d R verifying the following assumptions : There exist a random variable ω n such that where I = t (1,..., 1) R d. Invariance properties for all x R d and λ R \ {0}. ωn 1 P (T n I) f (ρ), ψ(x + λi) = ψ(x) and ψ(λx) = ψ(x) 6

7 Definition of the family of estimators for the second order parameter Idea Invariance (and regularity) properties entail ψ(t n) = ψ(ωn 1 P (T n I)) ψ(f (ρ)) Letting Z n := ψ(t n) and ϕ := ψ f, one obtains Z n P ϕ(ρ). Suppose there exist J 0 R and J R such that ϕ is a bijection J 0 J. Definition The family of estimators of the second order parameter is thus defined by : j ϕ 1 (Z ˆρ n = n) if Z n J, 0 otherwise. 7

8 Asymptotic properties Theorem Under the invariance (and regularity) conditions, If ω 1 n (T n I) P f (ρ) then ˆρ n P ρ. If, moreover, v n(ωn 1 d (T n I) f (ρ)) N d (m(ρ), γ 2 Σ) where v n, m R d and Σ is a regular d d matrix then v n(ˆρ n ρ) d N t m ψ(f (ρ)) ϕ (ρ), γ 2 t ψ(f (ρ)) Σ ψ(f (ρ)) (ϕ (ρ)) 2 «. 8

9 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,n log X n j,n ), j = 1,..., k Hall & Welsh (Annals of Statistics, 1985), Goegebeur et al. (JSPI, 2010), De Wet et al. (SPL, 2012),... Estimators based on log-excesses, (log X n j+1,n log X n k,n ), j = 1,..., k Gomes et al. (Extremes, 2002), Fraga-Alves et al. (Portugaliae Mathematica, 2003), Ciuperca & Mercadier (Extremes, 2010),... 9

10 Link with existing estimators based on rescaled log-spacings 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 log X n j,n ), k + 1 j=1 H τ is a kernel function indexed by a parameter τ > 0. This statistics is used for instance by Beirlant et al. (Extremes, 1999) to estimate the extreme-value index γ and by Hall & Welsh (Annals of Statistics, 1985), Goegebeur et al. (JSPI, 2010), De Wet et al. (SPL, 2012) 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. 10

11 Links with existing estimators based on rescaled log-spacings Statistics T n Suppose the third order condition and (C1) hold. If the sequence k satisfies then the random vector 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, T n := (R k (τ i )/γ) θ i, i = 1,..., d, properly normalised in asymptotically Gaussian. More precisely, ω n = A(n/k)/γ(1 + o P (1)), v n = k 1/2 A(n/k) and Z 1 «f (ρ) = θ i H τi (u)u ρ du, i = 1,..., d. 0 11

12 The estimator still depends on 12 parameters. The estimator is not necessarily explicit, the inverse of ϕ has to be computed numerically. 12 Link with existing estimators based on rescaled log-spacings Statistics T n Let d = 8, T n depends on 16 parameters θ 1,..., θ 8, τ 1,..., τ 8. Suppose θ 1 = θ 2, θ 3 = θ 4, θ 5 = θ 6 and θ 7 = θ 8. Function ψ The chosen function ψ is given by : ψ(x 1,..., x 8) = x1 x2 x 3 x 4 x7 x 8 x 5 x 6 «(θ1 θ 3 )/(θ 5 θ 7 ) Z n = Rθ 1 k (τ1) Rθ 1 k (τ2) Rk (τ7) Rθ 7 k (τ8) R θ 3 k (τ3) Rθ 3 k (τ4) R θ 5 k (τ5) Rθ 5 k (τ6) Asymptotic normality θ 7 and thus! (θ1 θ 3 )/(θ 5 θ 7 ) In this situation, the estimator of ρ is asymptotically Gaussian.

13 Examples Let H τ (u) = (τ + 1)u τ, To simplify, we assume that τ 2 = τ 3, τ 4 = τ 8 and τ 6 = τ 7. There are 9 remaining parameters and ϕ is given in this case by : ϕ(ρ) = cste»» (θ1 θ τ4 ρ τ5 ρ 3 )/(θ 5 θ 7 ) τ 1 ρ τ 4 ρ Three explicit estimators can be derived : θ 1 θ 3 = θ 5 θ 7, Goegebeur et al. (JSPI, 2010), 8 free parameters, θ 1 = θ 3, new estimator, 8 free parameters, ˆρ = τ1zn cste τ4 Z n cste τ 1 = τ 5, new estimator, 8 free parameters, ˆρ = τ4z 1/(δ 1) n cste 1/(δ 1) τ 1 Z 1/(δ 1) n cste 1/(δ 1) 13

14 Link with existing estimators based on log-excesses 2. Estimators based on log-excesses : (log X n j+1,n log X n k,n ) S k (τ, α) = 1 k kx «j G τ,α (log X n j+1,n log X n k,n ) α, α > 0, k + 1 j=1 G τ,α is a positive function. This statistics is used for instance by Dekkers et al. (Annals of statistics, 1989), Gomes & Martins (JSPI, 2001), Segers (JSPI, 2001) to estimate the extreme-value index γ and by Hall & Welsh (Annals of Statistics, 1985), Peng (SPL, 1998), Fraga et al. (MMS, 2003), Ciuperca & Mercadier (Extremes, 2010), to estimate the second order parameter ρ. They proved the asymptotic normality under a technical condition on the function G τ,α, denoted by (C2) hereafter. 14

15 Links with existing estimators based on log-excesses Statistics T n Suppose the third order condition and (C2) hold. If the sequence k satisfies then the random vector 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, T n = 0 «θi Sk (τ i, α i ), i = 1,..., d! γ α i properly normalised in asymptotically Gaussian. More precisely, ω n = A(n/k)/γ(1 + o P (1)), v n = k 1/2 A(n/k) and Z 1 «f (ρ) = θ i α i G τi,α i (u)(log(1/u)) α i 1 K ρ(u)du, i = 1,..., d, 15

16 The estimator still depends on 20 parameters. The estimator is not necessarily explicit, the inverse of ϕ has to be computed numerically. 16 Link with existing estimators based on log-excesses Statistics T n Let d = 8, T n depends on 24 parameters θ 1,..., θ 8, τ 1,..., τ 8, α 1,..., α 8. Suppose θ 1α 1 = θ 2α 2, θ 3α 3 = θ 4α 4, θ 5α 5 = θ 6α 6 and θ 7α 7 = θ 8α 8. Function ψ The chosen function ψ is given by : ψ(x 1,..., x 8) = x1 x2 x 3 x 4 x7 x 8 x 5 x 6 «(θ1 α 1 θ 3 α 3 )/(θ 5 α 5 θ 7 α 7 ) Z n = S θ 1 k (τ1, α1) S θ 2 k (τ2, α2) Sk (τ7, α7) S θ 8 k (τ8, α8) S θ 3 k (τ3, α3) S θ 4 k (τ4, α4) S θ 5 k (τ5, α5) S θ 6 k (τ6, α6) Asymptotic normality In this situation, the estimator of ρ is asymptotically Gaussian. θ 7 and thus! (θ1 α 1 θ 3 α 3 )/(θ 5 α 5 θ 7 α 7 )

17 Link with existing estimators based on log-excesses Examples Let G τ,α(u) = (1 u τ )/ R 1 0 (1 x τ )( log x) α dx, To simplify, we assume that τ 2 = τ 3 = τ 5 = τ 6 = τ 7 = τ 8 = α 7 = 1, α 6 = 3 and α 8 = 2. There are 11 remaining parameters. Three explicit estimators can be recovered : θ 1 = θ 3, α 1 = α 2 = α 3 = α 4 = 1, τ 1 = 2 and τ 4 = 3, Ciuperca & Mercadier (Extremes, 2010), no free parameter θ 1 = θ 3, α 1 = α 3 = α 4 = 1 and τ 1 = τ 4 = α 2 = 2, Ciuperca & Mercadier (Extremes, 2010), no free parameter θ 1 = θ 3 = θ 6 = θ 8 = α 2 = α 4 = α 5 = τ 1 = τ 4 = 1, α 3 = θ 4 = θ 7 = 2, θ 5 = 3 and α 1 = 4, Gomes et al. (Extremes, 2002), no free parameter 17

18 Link with existing estimators based on log-excesses Examples Three new estimators can be built : θ 1 θ 2 = 2θ 5 θ 7, α 1 = α 2 = α 3 = α 4 = 1, τ 1 = α 5 = 2 and τ 4 = 3, 3 free parameters 6Zn + 4 cste ˆρ = 3Z n + 4 cste θ 1 θ 2 = 2θ 5 θ 7, α 1 = α 3 = α 4 = 1 and τ 1 = τ 4 = α 2 = α 5 = 2, 3 free parameters 6Zn 4 cste ˆρ = 2Z n 1 cste τ 1 = τ 4 = α 1 = 1, α 2 = α 3 = α 5 = 2 and α 4 = 3, 4 free parameters 1/(δ+1) 3Zn cste 1/(δ+1) ˆρ = cste 1/(δ+1) Z 1/(δ+1) n If δ = 0, we find back the estimator introduced in Fraga-Alves et al. (Portugaliae Mathematica, 2003) 18

19 Illustration on simulations Estimators 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 2, θ 3, θ 5,..., θ 8 chosen as in Goegebeur et al. (JSPI, 2010) and De Wet et al. (SPL, 2012) : τ 1 = 1.25, τ 2 = τ 3 = 1.75, τ 4 = τ 8 = 2, τ 5 = 1.5, τ 6 = τ 7 = 1.75, θ 1 = θ 2 = 0.01, θ 5 = θ 6 = 0.02 and θ 7 = θ 8 = θ 3 = θ 4 = δ for δ 0. A simple expression if obtained for ϕ :»» δ 2 ρ 1.5 ρ ϕ(ρ) = cste 1.25 ρ 2 ρ Its inverse is explicit when δ = 0 (new explicit estimator) or δ = 1 (Goegebeur et al. (JSPI, 2010)). We shall also consider the case δ = 1.5 which can be shown to be in some sense optimal when ρ = 0 (new implicit estimator). 19

20 Illustration on simulations Burr distribution Survival distribution function : with x 0 and ρ < 0. 1 F (x) = (1 + x ρ ) 1/ρ Extreme-value index γ = 1, second order parameter ρ < 0. The third order condition holds with β = ρ, A(x) = γx ρ /(1 x ρ ) and B(x) = ρx ρ /(1 x ρ ). Experimental design Sample size n = 5000, 500 replications. Intermediate sequence k = 1500,..., Second order parameter ρ = 0.25 and ρ = 1. 20

21 Asymptotic mean-squared error & empirical mean-squared error ρ = 0.25 ρ = 0.25 AMSE δ = 0 δ = 1 δ = 1.5 MSE δ = 0 δ = 1 δ = k k Left : Asymptotic mean-squared error, Right : empirical mean-squared error. 21

22 Asymptotic mean-squared error & mean-squared error ρ = 1 ρ = 1 AMSE δ = 0 δ = 1 δ = 1.5 MSE δ = 0 δ = 1 δ = k k Left : Asymptotic mean-squared error, Right : empirical mean-squared error. 22

23 Conclusion + General framework for building estimators of the second order parameter. + Asymptotic normality of the estimators is direclty derived from the asymptotic behavior of rescaled log-spacings or log-excesses. + Efficient tool for studying existing estimators or defining new ones. But... How to compare in practice estimators depending on so many parameters? E. Deme, L. Gardes and S. Girard. On the estimation of the second order parameter for heavy-tailed distributions, REVSTAT - Statistical Journal, 11, ,