Characterization through Hazard Rate of heavy tailed distributions and some Convolution Closure Properties

Transcription

1 Characterization through Hazard Rate of heavy tailed distributions and some Convolution Closure Properties Department of Statistics and Actuarial - Financial Mathematics, University of the Aegean October 13, 2010

2 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions INTRODUCTION We will assume for the whole paper that the distribution function F is supported on [0, ) and that F has positive Lebesque density function f. We use the Matuszweska indices and its properties to show asymptotics inequalities for the hazard rates. We discuss about the relation membership in dominatedly or subversively varying tail distribution and a hazard rate condition. Convolution closure is establish for the class of distributions with subexponential and subversiverly varying tails.

3 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions The class L A distribution function F belongs to the class L if F (x y) lim = 1 x F (x) for all constants y R. This distribution function F is said to have a long tail.

4 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions The class D A distribution function F belongs to the class D if lim sup x F (ux) F (x) < (1) for any 0 < u < 1 (or equivalently for u = 1/2). Such a distribution function F is said to have a dominatedly varying tail. An equivelant way to right (1) is for any u > 1. F (u) := lim inf x F (ux) F (x) > 0 (2)

5 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions The class E A distribution function F belongs to the class E, if for some u > 1 F (u) := lim sup x F (ux) F (x) < 1. Such a distribution function F is said to have subversively varying tail.

6 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions The class S A distribution function F belongs to the class S if F lim n (x) = n x F (x) for any n 2 (or equivalently for n = 2), where F n denotes the nth convolution of F. Such a distribution function F is said to have a subexponetial tail.

7 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions The class A A distribution function F belongs to the class A if F S and F (u) < 1, for some u > 1. In others words A = E S.

8 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions It is well known the following inclusions D A D L S L. D A D E

9 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions Definition of Matuszewska indices Upper Matuszewska index Let g be positive. Then The upper Matuszewska index α(g) is the infimum of those α for which there exists a constant C = C(α) > 0 such that for each Λ > 1 g(λx) g(x) C(1 + o(1))λα, as x and uniformly in λ [1, Λ].

10 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions Lower Matuszewska index Let g be positive. Then The lower Matuszewska index β(g) is the supremum of those β for which there exists a constant D = D(β) > 0 such that for each Λ > 1 g(λx) g(x) D(1 + o(1))λβ, as x, and uniformly in λ [1, Λ].

11 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions Let consider the following notation. The positive function g has 1 bounded increase, (g BI ), if α(g) < 2 bounded decrease, (g BD), if β(g) > 3 positive increase, (g PI ), if β(g) > 0 4 positive decrease, (g PD), if α(g) < 0 An equation that holds between the Matusweska indices is ( ) 1 β(g) = α, for g positive (3) g For more details of the Matuszewska indices, see Chapter 2.1 of Bingham et al. (1987).

12 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions Potter Inequalities Let g(.) be positive. If g BI then for every α > α(g) there exist positive constants C, x 0 such that g(y) ( y ) α g(x) C, (y x x0 ) (4) x If g BD then for every β < β(g) there exist positive constants C, x 0 such that g(y) ( g(x) C y ) β, (y x x0 ) (5) x

13 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions Matuszewska indices for the distributions tails From D. B. H. Cline and G. Samorodnisky(1994) The upper Matuszewska index γ F for a distribution function F, was introduced as follows. γ F := inf { log F (u) : u > 1 log u } = lim u log F (u). (6) log u The lower Matuszewska index δ F for a distribution function F, was introduced as follows. { δ F := sup log F } (u) log F (u) : u > 1 = lim. (7) log u u log u

14 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions Inequalities for the distributions functions If the Upper Matuszeska index γ F < then there exist constants C, x 0 such that ( ) F (x) x γ F (y) C (8) y for every x y x 0 and γ F < γ <. If the Lower Matuszeska index δ F > 0 is finite then there exist constants C, x 0 such that F (x) F (y) C for every x y x 0 and 0 < δ < δ F. ( ) x δ (9) y

15 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions Matuszewska indices for the density functions We introduced the upper Matuszewska index γ f for the density function, as follows { γ f := inf log f } (u) log f (u) : u > 1 = lim log u u log u. (10) f (ux) where f (u) = lim inf x f (x). We introduced the lower Matuszewska index δ f for the density function, as follows δ f := sup { log f (u) : u > 1 log u where f (u) = lim sup x f (ux) f (x). } log f (u) = lim u log u. (11)

16 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions Definition We will say that the density function has 1 bounded decrease, (f BD), if γ f = α(f 1 ) < 2 positive decrease, ( f PD), if δ f = β(f 1 ) > 0

17 Matuszewska Indices Matuszewska indices for the distributions tails Matuszewska indices for the density functions In analogy to inequalities (8) and (9) we introduce inequalities for density function Corollary If f BD then there exist constants C, x 0 such that f (y) ( f (x) C y ) γ (12) x for every y x x 0. If f PD then there exist constants C, x 0 such that for every y x x 0. f (y) ( y ) δ f (x) C (13) x

18 Previous results Main Results The Subversively Class is a large class that extends out of the class of Heavy Tail. We can see the following Example Example The Exponential distribution function is F (x) = exp{ λx}. As we can see F (ux) lim x F (x) = 0 < 1 for all u > 1.

19 Previous results Main Results Lemma (Konstantinides et.al 2002) Let F be a d.f. with a density function f which is eventually non-increasing. Then the following statements are equivelant: 1 F (u) < 1 holds for some u > 1; 2 F (u) < 1 holds for any u > 1; 3 the hazard rate function of F,h(x) = f (x) F (x) satisfies lim inf x xh(x) > 0.

20 Previous results Main Results Eventually non increasing? If a function g is eventually non increasing then y x x 0 g(y) g(x) (14)

21 Previous results Main Results Eventually non increasing? If a function g is eventually non increasing then y x x 0 g(y) g(x) (14) How easy we check that condition (14) holds?

22 Previous results Main Results Eventually non increasing? If a function g is eventually non increasing then y x x 0 g(y) g(x) (14) How easy we check that condition (14) holds? Does the condition (14) hold for every function g?

23 Previous results Main Results Eventually non increasing? If a function g is eventually non increasing then y x x 0 g(y) g(x) (14) How easy we check that condition (14) holds? Does the condition (14) hold for every function g? We can obtain that condition (14) is the Potter inequality (4) for C 1 and α(g) < 0. Obviously

24 Previous results Main Results Eventually non increasing? If a function g is eventually non increasing then y x x 0 g(y) g(x) (14) How easy we check that condition (14) holds? Does the condition (14) hold for every function g? We can obtain that condition (14) is the Potter inequality (4) for C 1 and α(g) < 0. Obviously g(y) g(x) C ( y x ) α C, (x y x0 ). (15) for all α(g) < α < 0. But to obtain, if the Potter inequality holds we only need to calculate the Upper Matuswzeska index

25 Previous results Main Results The main idea is to replace the condition of eventually non-increasing(or decreasing) with the Potter Inequalities. When we have density functions we can obtain an analogous inequality of the previous one, if the δ f > 0.

26 Previous results Main Results Theorem Let F be an absolute continuous distribution function supported on [0, ) with a density function f (x) with δ f > 0 (f PD).Then F E if and only if lim inf x xh(x) > 0. (16)

27 Previous results Main Results A sufficient condition for F E is given from the following theorem Theorem If lim inf x xh(x) > 0 then F E A similar theorem is found in Klüppelberg C.(1988) for the class D L.

28 Previous results Main Results Lemma If f PD, δ f > 1 for any δ (1, δ f ) then : for all x x 0 and C > 0. xh(x) = x f (x) (δ 1) > 0 F (x) C From Lemma and previous theorem we obtain

29 Previous results Main Results Lemma If f PD, δ f > 1 for any δ (1, δ f ) then : for all x x 0 and C > 0. xh(x) = x f (x) (δ 1) > 0 F (x) C From Lemma and previous theorem we obtain Corollary If f PD with δ f > 1 then F E.

30 Previous results Main Results Theorem If F 1, F 2 E and the following statements hold 1 f 1 PD ( ) 1 2 x δ = O F 1 (x)) where δ = min (δ F 1, δ F 2 ), then F 1 F 2 E.

31 Main Results Main Results A sufficient condition for F D L is given by the following theorem. Theorem (Klüppelberg C. 1988). If lim sup x xh(x) < then F D L Corollary (Klüppelberg C. 1988) Let F have an eventually decreasing density f. Then the following statements are equivelant: 1 F D 2 F D L 3 lim sup x xh(x) <

32 Main Results Main Results Theorem Let F be an absolute continuous distribution function supported on [0, ) with a density function f (x) with δ f > 0 (f PD).Then 1 F D 2 F D L 3 lim sup x xh(x) <

33 Main Results Main Results Lemma If f BD then there is positive x 0, such that for all x x 0 and all λ > 1: xh(x) = x f (x) F (x) γ + 1 C (λ γ+1 1) Furthermore if f BD then F D L.

34 Theorem Let F be an absolute continuous distribution function with a density function f (x) with δ f > 0 then F A D if and only if one of the following statements holds 1 0 < lim inf x xh(x) lim sup x xh(x) < 2 0 < F (u) F (u) < 1

35 (A Characterization Theorem for S ) Theorem ((1980)) Suppose F is absolutely continuous with density function f and hazard rate h(x) eventually decreasing to 0. Then F S if and only if x lim exp {yh(x)} f (y)dy = 1 (17) x 0

36 Theorem Suppose F is absolutely continuous with density function f and hazard rate h(x) with α(h) < 0. Then F S if and only if for all k > 0. x lim x 0 exp {kyh(x)} f (y)dy = 1 (18)

37 BIBLIOGRAPHY Bingham, N.H., Goldie, C.M., Teugels, J.L Regular Variation. Cambridge University Press, New York. Embrechts,P.,Goldie, C.M. (1980) On closure and factorization properties of subexponential and related distributions. J Austral.Math.Soc. 29, no.1, Cai Jun (2004) On max-sum equivelance and convolution closure of heavy-tailed distributions and their applications. J Appl. Prob. 41, D. B. H. Cline and G. Samorodnisky, Subexponentiality of the product of independent random variables, Stochastic Processes. Appl. 49 (1994), no. 1,

38 Embrechts,P. Klüppelberg, C. and Mikosch, T. (1997) Modeling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin, 1997 Klüppelberg C. (1988) Subexponential Distributions and Integrated Tails J Appl. Prob. 25, no.1, Konstantinides D.G. (2008) A Class of Heavy Tailed Distributions Journal of Numerical and Applied Mathematics 96, No.1, Konstantinides D.G., Tang,Q.H, Tsitsiashvili,G.SH (2002) Estimates for the Ruin Probability in the Classical Risk Model with Constant Interest Forse in the Presence of Heavy Tails. Insurance: Mathemematics & Economics 31, no.3,

39 E. J. G. (1980) Subexponential Distribution Function J.Austral.Math.Soc. 29, Pratt W. John (1960) On Interchanging Limits and integrals Ann. Math. Statist.31, Tang, Q. and Tsitsiashvili,G.(2003a). Randomly weighted sums of subexponential random variables with applications to ruin theory, Extremes 6, Tang Qihe (2006) The Subexponentiality of Products Revisited. Extremes 9,

40 Thank you for your attention