Modelling Multivariate Peaks-over-Thresholds using Generalized Pareto Distributions

Transcription

1 Modelling Multivariate Peaks-over-Thresholds using Generalized Pareto Distributions Anna Kiriliouk 1 Holger Rootzén 2 Johan Segers 1 Jennifer L. Wadsworth 3 1 Université catholique de Louvain (BE) 2 Chalmers University of Technology (S) 3 Lancaster University (UK) Risk, Extremes and Contagion Université de Nanterre, May 25 26, 2016 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

2 Multivariate Peaks-over-Thresholds ξ exceeds a multivariate threshold u: ξ u j = 1,..., d : ξ j > u j ξ 2 u 2 Distribution of excess ξ u given ξ u? ξ 1 u 1 Other exceedances (Thibaud and Opitz, 2013; Dombry and Ribatet, 2015): j = 1,..., d : ξ j > u j ξ ξ d > u u d... Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

3 Multivariate Generalized Pareto Distributions Maxima and exceedances Parametrization Representations and densities Stability Inference Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

4 Generalized Extreme-Value distributions Random vector ξ in R d, cdf F ξ (x) = P(ξ x). Assumption: Max-domain of attraction There exist scaling a n > 0 and location b n R d sequences such that F n ξ (a nx + b n ) G(x), n F n is cdf of vector of componentwise maxima of iid sample from F. Limit G is necessarily Generalized Extreme-Value = Max-Stable Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

5 Multivariate Generalized Pareto distribution If F n ξ (a nx + b n ) G(x) as n, then, provided 0 < G(0) < 1 (w.l.o.g.), L ( an 1 ) d (ξ b n ) η ξ b n Multivariate Generalized Pareto H where η j [, ) is the lower endpoint of G j. Cumulative distribution function H = GP(G): for x such that G(x) > 0, H(x) = log G(x 0) ( log G(x)) log G(0) Beirlant et al. (2004) Rootzén and Tajvidi (2006) Falk et al. (2010) Ferreira and de Haan (2014) Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

6 Independent maxima: single-component exceedances Generalized Extreme-Value G: ( G(x 1, x 2 ) = exp 1 x ) x x 1 > 1, x 2 > 1 Generalized Pareto H: law of { ( 1, Y) wp 1/2, (X 1, X 2 ) = (Y, 1) wp 1/2 X with P(Y t) = y for y 0 (univariate Generalized Pareto). X 1 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

7 Completely dependent maxima and exceedances Generalized Extreme-Value G: ( ) 1 G(x 1, x 2 ) = exp min(x 1, x 2 ) + 1 x 1 > 1, x 2 > 1 Generalized Pareto H: law of (Y, Y) X with P(Y y) = y for y 0 (univariate Generalized Pareto). X 1 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

8 1 Example: logistic Generalized Extreme-Value G(x 1, x 2 ): exp[ { 2 j=1 (1 + x j) 1/θ } θ ] x 1 > 1, x 2 > 1 Generalized Pareto H: pdf h(x 1, x 2 ): 2 θ 2 x 1 x 2 { 2 j=1 (1+x j) 1/θ } θ Support: x 1 > 1 x 2 > 1 x 1 x 2 > θ = Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

9 Multivariate Generalized Pareto Distributions Maxima and exceedances Parametrization Representations and densities Stability Inference Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

10 Generalized Extreme-Value distributions Margins: shape γ j R, location µ j R, scale α j (0, ) exp[ {1 + γ j (x j µ j )/α j } 1/γ j ] if γ j 0, G j (x j ) = exp[ exp{ (x j µ j )/α j }] if γ j = 0 Stable tail dependence function (stdf) l : [0, ) d [0, ) G(x) = exp[ l{ log G 1 (x 1 ),..., log G d (x d )}] Notation: G = GEV(γ, µ, α, l) Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

11 Identifiability issue Max-stability: G = GEV(γ, µ, α, l) = G t = GEV(γ, µ(t), α(t), l), 0 < t < with µ(t) =... and α(t) =... (exercise). However: GP(G t ) = GP(G). Therefore: not all GEV parameters are identifiable from H = GP(G). Solution? Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

12 Cumulative distribution function Let G = GEV(γ, µ, α, l). Suppose σ = α γµ (0, ) Then 0 < G(0) < 1. Let H = GP(G). For x R such that σ + γx > 0, ( H(x) = l π (1 + γ(x 0)/σ) 1/γ) l (π (1 + γx/σ) 1/γ) where, in case also x 0, H j (0) = π j H j (x j ) H j (0) = (1 + γ jx j /σ j ) 1/γ j H(x) = l ( H 1 (x 1 ),..., H d (x d ) ) Parametrization: H = GP(γ, σ, π, l). Constraint: l(π) = 1. Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

13 Multivariate Generalized Pareto Distributions Maxima and exceedances Parametrization Representations and densities Stability Inference Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

14 Standardization We have X GP(γ, σ, π, l) if and only if we have and Z GP(0, 1, π, l). X = σ eγz 1 γ The support of Z is contained in [, ) \ [, 0] and P[Z z] = l ( πe (z 0)) l ( πe z) How to construct Z? Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

15 Spectral representation A random vector S in [, 0] d is a spectral random vector if: P[max(S 1,..., S d ) = 0] = 1 P[S j > ] > 0, j = 1,..., d Let E be a unit exponential random variable independent of S. Then where Z := S + E GP(0, 1, π, l) π j = E[e S j ] [ l(y) = E max j=1,...,d { e S }] j y j π j H(z) = 1 E[1 e max(s z) ] Conversely: given π and l with l(π) = 1, there exists a unique (in law) spectral random vector S such that the above holds. Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

16 Constructing spectral random vectors (1) Let T be a random vector in [, ) d such that Then 0 < E[e T j ] <, j = 1,..., d P[max(T 1,..., T d ) > ] = 1 S = T max(t) is a spectral random vector. The associated GP(0, 1, π, l) = GP S (0, 1, L(S)) distribution is determined by π j = E[e Tj max(t) ] [ { l(y) = E max j=1,...,d y j }] e T j max(t) E[e Tj max(t) ] H(z) = 1 E[1 e max(t z) max(t) ] Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

17 Density (1) Suppose T has support included in R d and is absolutely continuous with Lebesgue density f T. Then the GP T (0, 1, L(T)) distribution has Lebesgue density h given by h(z) = 1(z 0) 1 e max(z) 0 f T (z + log t) t 1 dt. Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

18 Example: max-normalized log-gaussian generators (1) Pdf of H = GP T (0, 1, N d (0, Σ)): h(z) = (2π)(1 d)/2 Σ 1/2 (1 T Σ 1 1) 1/2 e max(z) exp { 12 (Σ zt 1 Σ 1 11 T Σ 1 ) } 1 T z Σ 1 1 z R d \ (, 0] d Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

19 Example: max-normalized log-gaussian generators (2) pdf of (Z 1, Z 2 ) GP T with γ 1 = γ 2 = 0 σ 1 = σ 2 = 0 (T 1, T 2 ) N 2 (0, ( )) 1 ρ ρ ρ =.5 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

20 Example: max-normalized log-gaussian generators (2) pdf of (Z 1, Z 2 ) GP T with γ 1 = γ 2 = 0 σ 1 = σ 2 = 0 (T 1, T 2 ) N 2 (0, ( )) 1 ρ ρ ρ = 0 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

21 Example: max-normalized log-gaussian generators (2) pdf of (Z 1, Z 2 ) GP T with γ 1 = γ 2 = 0 σ 1 = σ 2 = 0 (T 1, T 2 ) N 2 (0, ( )) 1 ρ ρ ρ =.5 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

22 Constructing spectral random vectors (2) Let U be a random vector in [, ) d such that 0 < E[e U j ] < for all j. Let S be the spectral random vector defined in distribution by P[S ] = E [ 1{U max(u) } e max(u)] E[e max(u) ] The associated GP(0, 1, π, l) = GP S (0, 1, L(S)) distribution is given by π j = E[eU j ] E[e max(u) ] [ l(y) = E max j=1,...,d { y j e U }] j E[e U j] H(z) = 1 E[emax(U) e max(u z) ] E[e max(u) ] Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

23 Density (2) Suppose U has support included in R d and is absolutely continuous with Lebesgue density f T. Then the GP U (0, 1, L(U)) distribution has Lebesgue density h given by h(z) = 1(z 0) 1 E[e max(u) ] 0 f U (z + log t) dt. Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

24 Example: independent Fréchet generators e U j independent Fréchet shape α > 0 scales λ j > 0 GP U (0, 1, L(U)) density: h(z) e α d j=1 z j ( d j=1 ( ez j λ j ) α ) d 1/α z R d \ (, 0] d explicit proportionality constant margins dependence γ 1 = γ 2 = 0 α = 2 σ 1 = σ 2 = 1 λ 1 = 2, λ 2 = 1 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

25 Example: independent Beta generators e U j independent Beta shapes (α j, 1) scales λ j GP U (0, 1, L(U)) density: h(z) z R d \ (, 0] d d j=1 ez j/α j max(λe z ) d j=1 1/α j explicit proportionality constant margins dependence γ 1 = γ 2 = 0 α 1 = 2, α 2 = 3 σ 1 = σ 2 = 1 λ 1 = 2, λ 2 = 1 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

26 Point process representation iid U, U 1, U 2,... in [, ) d such that 0 < E[e U j ] < for all j unit-rate Poisson process 0 < R 1 < R 2 <... (U i ) i and (R i ) i are independent Point process consisting of points ξ i (i = 1, 2,...) where ξ i = U i log R i The law, G, of max i ξ i is GEV (de Haan, 1984): log G(z) = E[e max(u z) ] Associated GP: H = GP(G) = GP U (0, 1, L(U)) Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

27 Multivariate Generalized Pareto Distributions Maxima and exceedances Parametrization Representations and densities Stability Inference Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

28 Lower-dimensional margins Let X be GP and let J {1,..., d}. The law of X J = (X j ) j J is not GP. The law of X J given that max j J X j > 0 is GP: X GP(γ, σ, π, l) = L(X J X J 0) = GP (γ J, σ J, (P[X j > u j X J 0]) j J, l J ) X GP U (γ, σ, U) = L(X J X J 0) = GP U (γ J, σ J, L(U J )) If J = {j}, we find that L(X j X j > 0) = GP(γ j, σ j ). Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

29 Threshold stability Univariate GP distributions are threshold stable. What if multivariate? Let X GP(γ, σ, π, l). Let u [0, ) d be such that P(X j > u j ) > 0 for all j. Then L(X u X u) = GP (γ, σ + γu, (P[X j > u j X u]) j, l) Change of marginal parameters as in univariate case Change from π j = P[X j > 0] to P[X j > u j X u] Same stdf l Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

30 Linear combinations Let X GP S (γ, σ, π, l) be such that γ 1 =... = γ d =: γ. Let a [0, ) d and write a X = d j=1 a jx j. If P[a X > 0] > 0, then L(a X a X > 0) = GP(γ, a σ). Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

31 Linear transformations Let X GP S (γ, σ, L(S)) be such that γ 1 =... = γ d =: γ. Let A = (a i,j ) i,j [0, ) m d be such that P[A ix > 0] > 0 for all i. For x R m such that A iσ + γx i > 0 for all i = 1,..., m, [ ] P[AX x] = E 1 max {(1 + γx i/a iσ) 1/γ e U i } i=1,...,m where U = (U 1,..., U m ) is given by γ 1 log ( d j=1 U i = p i,j e γs ) j if γ 0, d j=1 p i,j S j if γ = 0, where p i,j = a i,j σ j /A iσ j. As a consequence, L(AX AX 0) = GP U (γ, Aσ, L(U)). Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

32 Multivariate Generalized Pareto Distributions Maxima and exceedances Parametrization Representations and densities Stability Inference Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

33 Inference problem Data ξ 1,..., ξ n. Threshold u. Threshold exceedances: {ξ i ξ i u}. Model: L(ξ ξ u) = u + σ eγz 1 γ GP(0, 1, π, l) Z GP T (0, 1, L(T)) GP U (0, 1, L(U)) Parametric model for l or L(T) or L(U), parameter α = Parametric model for f Z = Likelihood inference on θ = (γ, σ, α)? Nonparametric inference: empirical stdf ˆl n? Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

34 Data points with some components far beneath the threshold Exceedance x u (on original scale): excess x u 0. Model by Generalized Pareto? Ill-justified if x j u j for some j. Censor components that are too low. X 1 X 2 u 1 u 2 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

35 Censored likelihood 1. Choose second, lower threshold v < u. 2. If x j v j, replace x j by v j : censoring from below. 3. Likelihood contribution of a point x u but x v: h(w C u C, x D u D ) dw C (,v C ] C = {j = 1,..., d : x j v j } censored variables; D = {j = 1,..., d : x j > v j } uncensored variables. Comparative study of likelihood-based estimators: Huser et al. (2014). Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

36 Censored likelihood: Proof of concept X GP U (γ, σ, log Beta) α 1 α 2 α 3 λ 1 λ 2 σ 1 σ 2 γ 1 γ 2 Boxplots of normalized parameter estimates, 30 repetitions Censoring at v = u = 0 Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

37 Overall summary Modelling excess X = ξ u conditionally on ξ u: = Multivariate Generalized Pareto distribution H Q How does H look like? A X = σ eγz 1 γ with Z GP(0, 1, π, l) Q How to construct parametric models for it? A via spectral representations Q How to fit such models? A Censored likelihood Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

38 Outlook Computational challenges Likelihood: proportionality constant E[e max(u) ]? Simulation: change-of-measure f U (u) e max(u) f U (u)? Likelihood optimisation in case many parameters?... Statistical challenges Graceful blending with models for the bulk of the distribution? Covariates? Threshold choice? Model construction: parsimony vs flexibility?... Thank you! Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37

39 Bibliography Beirlant, J., Y. Goegebeur, J. Segers, and J. Teugels (2004). Statistics of Extremes: Theory and Applications. John Wiley & Sons. de Haan, L. (1984). A spectral representation for max-stable processes. The Annals of Probability 12(4), Dombry, C. and M. Ribatet (2015). Functional regular variations, pareto processes and peaks over threshold. Statistics and Its Interface 8, Falk, M., J. Hüsler, and R.-D. Reiss (2010). Laws of small numbers: extremes and rare events. Springer Science & Business Media. Ferreira, A. and L. de Haan (2014). The generalized Pareto process; with a view towards application and simulation. Bernoulli 20(4), Huser, R., A. C. Davison, and M. G. Genton (2014). A comparative study of likelihood estimators for multivariate extremes. Extremes (arxiv: ). Rootzén, H. and N. Tajvidi (2006). Multivariate generalized Pareto distributions. Bernoulli 12(5), Thibaud, E. and T. Opitz (2013). Efficient inference and simulation for elliptical Pareto processes. arxiv preprint arxiv: Kiriliouk/Rootzén/Segers/Wadsworth Multivariate Generalized Pareto Distributions Nanterre, 26 May / 37