Quantile prediction of a random eld extending the gaussian setting

Transcription

1 Quantile prediction of a random eld extending the gaussian setting 1 Joint work with : Véronique Maume-Deschamps 1 and Didier Rullière 2 1 Institut Camille Jordan Université Lyon 1 2 Laboratoire des Sciences Actuarielle et Financière Université Lyon 1 Nantes, March 2018

2 1 Introduction Methodology Estimation 5 L p quantiles Haezendonck-Goovaerts risk measures 6

3 1 Introduction Methodology Estimation 5 L p quantiles Haezendonck-Goovaerts risk measures 6

4 Introduction Conditional quantiles of a random eld may be usefull (Probability of Improvement, condence bounds for kriging prediction). Quite simple in the gaussian case. What about elliptical random elds?

5 1 Introduction Methodology Estimation 5 L p quantiles Haezendonck-Goovaerts risk measures 6

6 Elliptical distributions Denition ([Cambanis et al., 1981]) Let X be a random vector of dimension d. X is said elliptical i there exists a unique µ R d, a positive denite matrix Σ R d d, and a non-negative random variable R such that : X d = µ + RΛU (d) with ΛΛ T = Σ, U (d) is a uniform distribution on the unit sphere of dimension d, independant of R. Furthermore, X is consistent if R = d χ 2 dξ, where ξ is a non-negative random variable independant of χ 2 d and d. [Kano, 1994]

7 Elliptical distributions Theorem (Elliptical density) If X E d (µ, Σ, R), then : f X (x) = c d det(σ) 1 2 g d ((x µ)σ 1 (x µ)) where c d g d (t) = Γ( d 2) (d 1) t fr ( t), and f R (t) is the p.d.f of R. 2π d 2 g d is called the generator of X.

8 Examples Distribution Constant c d Generator g d (t) ξ 1 Gaussian exp( t ) 2 1 ( ) Student, ν > t d+ν 2 ν ν Gaussian Mixture Slash, a > 0 (2π) d 2 Γ( d+ν 2 ) 1 Γ( ν 2 ) (νπ) d 2 1 (2π) d 2 2 a 2 1 aγ( d+a 2 ) π d 2 n k=1 π k θ d k exp χ 2 d+a (t) t d+a 2 ( 1 t 2θk 2 χ 2 ν ) n π k δ θk k=1 Pareto(1, a)

9 Denition A random eld {X (t)} t T is ξ elliptical if for all N N and all t 1,..., t N T, the vector (X (t 1 ),..., X (t N )) is (ξ, N) elliptical.

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11 Problem Let {X (t)} t T be a ξ elliptical random eld. The aim is to provide q α (Y X = x) where Y = X (t), t T and X = (X (t 1 ),..., X (t N )), t 1,..., t N T. Notice that α = 1/2 leads to classical kriging. Theoretical conditional quantiles : q α (Y X = x) = µ Y x + σ Y X Φ 1 R (α) with { µy x = µ Y + Σ YX Σ 1 X (x µ X ) σy 2 X = Σ Y Σ YX Σ 1 X Σ XY Problem : Estimation of Φ 1 R (α).

12 1 Introduction Methodology Estimation 5 L p quantiles Haezendonck-Goovaerts risk measures 6

13 Quantile Regression [Koenker and Bassett, 1978] A rst idea is to use Quantile Regression : q QR α (Y X = x) = β X + β 0, where and (β, β0) = arg min E [ ( )] S α Y β X β 0 β R N,β 0 R S α (s) = (α 1)s + max {s, 0}.

14 Quantile Regression [Koenker and Bassett, 1978] Theorem ([Maume-Deschamps et al., 2017a]) The quantile regression vector (β, β 0) of Y (X = x), satisfying the previous problem, is given by β = Σ 1 X Σ XY, β 0 = µ Y Σ YX Σ 1 X µ X + σ Y X Φ 1 R (α). The quantile regression predictor with level α [0, 1] is given by qα QR (Y X = x) = µ Y x + σ Y X Φ 1 R (α). Furthermore, the distribution of qα QR (Y X ) is { qα QR (Y X ) E 1 µy + σ Y X Φ 1 R (α), Σ YX Σ 1 X Σ XY, ξ }.

15 Comments Inecient linear model. Φ 1 R (α) instead of Φ 1 R (α).

16 Methodology Estimation 1 Introduction Methodology Estimation 5 L p quantiles Haezendonck-Goovaerts risk measures 6

17 Methodology Estimation Some asymptotic relationships Theorem ([Maume-Deschamps et al., 2017a]) Under some assumptions, there exist 0 < l < + and η R such that : [ ( Φ 1 R 1 1 l 1 α +2(1 l) )] 1 η α 1 Φ 1 R (α) We thus dene the following approximation for q α (Y X = x) : [ ( qα E (Y X = x) = µ Y x + σ Y X Φ 1 R 1 1 l 1 α +2(1 l) )] 1 η

18 Methodology Estimation Examples Proposition ([Maume-Deschamps et al., 2017a]) The Gaussian, Student, Gaussian Mixture, and Slash distributions satisfy the previous assumptions, with coecients η and l given in the table below. Distribution η l Gaussian 1 1 N Student, ν > 0 ν + 1 Γ( ν+n+1 2 )Γ( ν 2 ) ( 1 + q 1 Γ( ν+n 2 )Γ( ν+1 2 ) Gaussian Mixture 1 Slash, a > 0 N a + 1 ) N+ν 2 ν N 2 +1 ν ν+n ) min(θ 1 1,...,θ 1 n ) N exp ( min(θ 1 1,...,θ 1 n ) 2 2 q 1 ( ) n π k θ N k exp 1 2θ k 2 q 1 k=1 Γ( N+1+a 2 )q1 N+a 2 Γ( N+a 2 )(N+a)χ 2 N+a (q1)2 a 2 1 Γ( 1+a 2 )

19 Methodology Estimation Estimation ([Usseglio-Carleve, 2017]) Let (X 1, Y 1 ),..., (X n, Y n ) be a random sample independant and identically distributed from an (ξ, N + 1) elliptical vector with (µ, Σ) known. Assumption We assume that there exist a function A such that A(t) 0 as t +, and lim t + where γ > 0 and ρ 0. Φ 1 R (1 1 ωt ) Φ 1 R (1 1 t ) ωγ A(t) = ω γ ωρ 1 ρ

20 Methodology Estimation Parameters estimation Proposition Under the previous assumption, parameters η and l exist, and are expressed : η = l = ( Γ N+γ ( ) Γ γ ) 1 + γn where m(x) = (x µ X ) T Σ 1 X (x µ X ). γ 1 π N 2 (N+γ 1 )c N g N (m(x)).

21 Methodology Estimation Parameters estimation Denition We dene the two following estimators : ˆη kn = Nˆγ kn + 1 ˆl kn,h n = ( Γ ( Γ ) N+ ˆγ 1 kn +1 2 ) ˆγ 1 kn +1 2 (N+ˆγ 1 kn ) m(x) 1 N 2 Γ( N 2 ) N π 2 nhn ˆγ 1 N kn π 2 ( n m(x) (Xi µ K X ) T Σ 1 ) X (X i µ X ) hn i=1, k n where ˆγ kn = 1 k n i=1 ( W[i] ln W ), [kn+1] k n = o(n), h n = o(1), k n +, nh n + as n + and W is the rst (or indierently any) component of the vector Λ 1 X (X µ X ).

22 Methodology Estimation Parameters estimation Proposition ( ) n Under our assumption, and if k n A k n 0 as n +, then the following asymptotic relationships hold : If nh n /k n where V 2 = Γ ( N 2 ) m(x) N 2 1 π N 2 n + 0, then ) (ˆl kn,h nhn n l ˆη kn η c N g N (m(x)) N n + (( ) 0, 0 ( )) V ( ) Γ N+γ 1 +1 ( K(u) 2 2 N + γ 1 ) 2 1 γ 1 π N 2 du ( ) Γ γ 1 +1 c 2 2 N g N(m(x)) 2

23 Methodology Estimation Parameters estimation Proposition ( ) n Under our assumption, and if k n A k n 0 as n +, then the following asymptotic relationships hold : If nh n /k n n + +, then ) (ˆl kn,h kn n l ˆη kn η where V 1 = π N γ 2 Γ cn 2 g N(m(x)) 2 N n + ( ) N+γ Ψ 2 ( ) Γ γ (( ) ( 0, 0 ( ) γ 1 +1 Ψ 2 V 1 Nγ V 1 Nγ V 1 N 2 γ 2 2γ 2 (Nγ + 1) ( ) N+γ )) 2 N (Nγ + 1) 2

24 Methodology Estimation Parameters estimation Proposition Under our assumption, and if ( ) n k n A k n 0 as n +, then the following asymptotic relationships hold : If nh n /k n c R +, then n + ) (ˆl kn,h kn n l ˆη kn η N n + (( ) 0, 0 ( V1 + 1 c V 2 Nγ )) V 1 Nγ V 1 N 2 γ 2

25 Methodology Estimation Quantile estimation We consider (α n ) n such that α n 1 as n +, and distinguish two cases : Intermediate quantiles, i.e we suppose n(1 α n ) +. It entails that the estimation of the α n quantile leads to an interpolation of sample results. High quantiles. We suppose n(1 α n ) 0, i.e we need to extrapolate sample results to areas where no data are observed.

26 Methodology Estimation Quantile estimation Denition We dene the two following estimators, respectively for intermediate and high quantiles : ( ) 1 ˆq α E ˆηkn n (Y X = x) = µ Y x + σ Y X W [ kn+1] ( ( ( ] 1 )))ˆγkn ˆq α E ˆηkn, n (Y X = x) = µ Y x + σ Y X [W knn [kn+1] 2 + ˆl 1 kn,hn 2 1 α n n where k n = ( ) 2+ˆl 1 and W is the rst (or indierently any) component of kn,hn 1 αn 2 the vector Λ 1 X (X µ X ).

27 Methodology Estimation Quantile estimation Theorem Consider that our assumption holds and assume that : k n = o(nh n) and ( ) k n na 0 as n +. kn Then n(1 α n) +, ln(1 α n) = o( k n) and kn ( ln(1 α = o n(1 αn) n) n +. ( ) ( ) kn ˆq E αn (Y X = x) ln (1 α n) qα E n (Y X = x) 1 N N 2 γ 4 0, n + (γn + 1) 4. ) as Therefore : ˆq E α n (Y X = x) q αn (Y X = x) P 1 as n +.

28 Methodology Estimation Quantile estimation Theorem We denote p n = ( 2 + l ( (1 α n) 1 2 )) 1, pn = Consider that our assumption holds and assume that : k n = o(nh n) and ( ) k n na 0 as n +. kn Then ( ln n(1 α n) 0, ln (n(1 α n)) = o( k n) and kn k n n(1 α n) ( 2 + ˆl kn,h n ( (1 αn) 1 2 )) 1. ln(1 α n) ) ln( n θ [0, + [. kn (1 α n) ( ) ( ˆq α E ( ) n (Y X = x) γ qα E n (Y X = x) 1 N 0, n + γn + 1 θ γ 2 ) 2 ) N (γn + 1) 2 Therefore : ˆq E α n (Y X = x) q αn (Y X = x) P 1 as n +.

29 L p quantiles Haezendonck-Goovaerts risk measures 1 Introduction Methodology Estimation 5 L p quantiles Haezendonck-Goovaerts risk measures 6

30 L p quantiles Haezendonck-Goovaerts risk measures L p quantiles Denition ([Chen, 1996]) Let Z be a real random variable. The L p quantiles of Z with level α ]0, 1[ and p > 0, denoted q p,α (Z), is solution of the minimization problem : q p,α (Z) = arg min z R where Z + = Z1 {Z>0}. E [ (1 α) (z Z) p + + α (Z z)p + ]

31 L p quantiles Haezendonck-Goovaerts risk measures L p quantiles According to [Koenker and Bassett, 1978], the case p = 1 leads to the quantile q 1,α (Z) = F 1 Z (α). The case p = 2 is called expectile, which knows a growing interest : [Taylor, 2008], [Cai and Weng, 2016], [Maume-Deschamps et al., 2017b]. Other cases are dicult to deal with : [Bernardi et al., 2017].

32 L p quantiles Haezendonck-Goovaerts risk measures Extreme L p quantiles Theorem ( [Daouia et al., 2017]) If a random variable Z lls our assumption, then q p,α (Z) q α (Z) where β is the beta function. Lemma [ ] γ γ = f α 1 β (p, γ 1 L (γ, p) p + 1) Under our assumption, the conditional distribution Y X = x is attracted to a maximum domain of Pareto-type distribution with tail index (γ 1 + N) 1.

33 L p quantiles Haezendonck-Goovaerts risk measures Conditional extreme L p quantiles estimation Denition Let (α n ) n N be a sequence such that α n 1 as n +. We dene : ˆq p,αn (Y X = x) = µ Y x + σ Y X ( W [n pn+1] ) 1 ˆη kn [ ( ˆq p,αn (Y X = x) = µ Y x + σ Y X W knn [kn+1] (2 + ˆl ( 1 kn,hn ( f L (ˆγ 1 ) ) 1 + N, p kn ( fl (ˆγ 1 ) ) 1 + N, p kn ))) ] 1 ˆγkn 1 αn 2 ˆηkn

34 L p quantiles Haezendonck-Goovaerts risk measures Haezendonck-Goovaerts risk measures Denition ([Haezendonck and Goovaerts, 1982]) Let Z be a real random variable, and ϕ a non negative and convex function with ϕ(0) = 0, ϕ(1) = 1 and ϕ(+ ) = +. The Haezendonck-Goovaerts risk measure of Z with level α ]0, 1[ associated to ϕ, is given by the following : H α (Z) = inf z R {z + H α(z, z)} where H α (Z, z) is the unique solution h to the equation : [ ( )] (Z z)+ E ϕ = 1 α h ϕ is called Young function.

35 L p quantiles Haezendonck-Goovaerts risk measures Extreme Haezendonck-Goovaerts risk measures The case ϕ(t) = t leads to the Tail Value-at-Risk TVaR α (Z). Proposition ([Tang and Yang, 2012]) If Z lls our assumption, and taking a Young function ϕ(t) = t p, p 1, then the following relationship holds : H p,α (Z) q α (Z) α 1 γ 1 (γ 1 p) pγ 1 p γ(p 1) β ( γ 1 p, p ) γ = fh (γ, p)

36 L p quantiles Haezendonck-Goovaerts risk measures Conditional extreme H-G risk measures estimation Denition Let (α n ) n N be a sequence such that α n 1 as n +. We dene : ) ˆη Ĥ p,αn (Y X = x) = µ Y x + Σ Y X (W 1 kn [n pn+1] ( ( Ĥ p,αn (Y X = x) = µ Y x + Σ Y X [W knn [kn+1] ( f H (ˆγ 1 + N kn ( fh (ˆγ 1 ) ) 1 + N, p kn ))) ] 1 ˆγkn 1 αn ˆl ( 1 kn,hn ) ) 1, p ˆηkn

37 1 Introduction Methodology Estimation 5 L p quantiles Haezendonck-Goovaerts risk measures 6

38 We apply our estimators to a sample of a centered Student process with ν = 1.5 degrees of freedom, and compare with theoretical results. σ(t) = e t. We take the sequences k n = n 0.6, h n = n 0.15 and α n = 1 n 1.1. The chosen kernel K is the gaussian p.d.f. Theoretical result is q α (Y X = x) = ν+m(x) ν+n Φ 1 ν+n (α)

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40 References I Bernardi, M., Bignozzi, V., and Petrella, L. (2017). On the Lp-quantiles for the Student t distribution. Statistics & Probability Letters. Cai, J. and Weng, C. (2016). Optimal reinsurance with expectile. Scandinavian Actuarial Journal, 2016(7) : Cambanis, S., Huang, S., and Simons, G. (1981). On the theory of elliptically contoured distributions. Journal of Multivariate Analysis, 11 : Chen, Z. (1996). Conditional Lp-quantiles and their application to the testing of symmetry in non-parametric regression. Statistics & Probability Letters, 29(2) : Daouia, A., Girard, S., and Stuper, G. (2017). Extreme M-quantiles as risk measures : from L1 to Lp optimization. Bernoulli. Haezendonck, J. and Goovaerts, M. (1982). A new premium calculation principle based on Orlicz norms. Insurance : Mathematics and Economics, 1 :4153.

41 References II Kano, Y. (1994). Consistency property of elliptical probability density functions. Journal of Multivariate Analysis, 51 : Koenker, R. and Bassett, G. J. (1978). Regression quantiles. Econometrica, 46(1) :3350. Maume-Deschamps, V., Rullière, D., and Usseglio-Carleve, A. (2017a). Quantile predictions for elliptical random elds. Journal of Multivariate Analysis, 159 :1 17. Maume-Deschamps, V., Rullière, D., and Usseglio-Carleve, A. (2017b). Spatial expectile predictions for elliptical random elds. Methodology and Computing in Applied Probability. Tang, Q. and Yang, F. (2012). On the Haezendonck-Goovaerts risk measure for extreme risks. Insurance : Mathematics and Economics, 50 : Taylor, J. W. (2008). Estimating Value at Risk and Expected Shortfall Using Expectiles. Journal of Financial Econometrics, 6(2) :

42 References III Usseglio-Carleve, A. (2017). Estimation of conditional extreme risk measures from heavy-tailed elliptical random vectors. working paper.

43 Merci!