Tail dependence coefficient of generalized hyperbolic distribution Mohalilou Aleiyouka Laboratoire de mathématiques appliquées du Havre Université du Havre Normandie Le Havre France mouhaliloune@gmail.com Alexandre Berred Laboratoire de mathématiques appliquées du Havre Université du Havre Normandie Le Havre France alexandre.berred@univ-lehavre.fr Mohammad Ahsanullah Information Systems and Supply Chain Management Rider University Lawrenceville USA ahsan@rider.edu Received 7 June 6 Accepted July 7 The tail dependence describes the limiting proportion of exceeding one margin over a certain threshold given that the other margin has already exceeded that threshold. In this paper we obtain the limit tail dependence coefficient for the generalized hyperbolic distribution. Keywords: Tail dependence; hyperbolic generalized; extreme value. Mathematics Subject Classification: 6E 6E 6H5 6H 6H.. Introduction The generalized hyperbolic distribution was introduced in [3] and has been developed by many authors see among others [7] [] [6] [] especially in relation with several applications in the Finance see [5] [6]. This family contains and generalizes many familiar distributions such as the Student t Gaussian variance gamma Cauchy and others. The aim of this paper is to investigate the upper tail dependence coefficient in the case of the generalized hyperbolic distribution. More precisely a bivariate random variable X = X X follows a generalized hyperbolic distribution GHλ α β δ γ if X = µ + W + W Z X = µ + W + W Z. Copyright 7 the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license http://creativecommons.org/licenses/by-nc/4./. 375
where Z = Z Z N Σ is the bivariate normal distribution with mean and a correlation matrix Σ and W = W W GIGλδγ is the generalized inverse gaussian with probability density function γ δ λ k λ δγ xλ exp γ x + δ x if x > f W x = otherwise. where γ = α λ Rδ > and γ. The probability density function of GHλαβδµ is given by f X x = α λ k λ α δ + x µ expx µ πα λ k λ δ α β δ λ δ + x µ λ+ where K λ is the modified Bessel function of the third kind given by K λ x = exp x y + y yλ du for x > λ R. see for instance [7] []. [] have given general results for upper tail dependence of skewed grouped t-distributions in the case where its degrees of freedom are different i.e. for a random variable X = X X we can write X = G µ U/µ + Z G µ U/µ X = G ν U/ν + Z G U/ν ν.3 where Z = Z Z is a bivariate normal distribution with mean and correlation matrix Σ = ρ with correlation coefficient ρ and Z is independently distributed of the standard uniform ρ random variance U. The parameter = R controls the asymmetry of the distribution and τν are positive constants. The function G η. is the inverse distribution function of Γ η η with η >. [9] have particularly studied a special case of the skewed grouped t-distributions see [] when η = ν = τ in.3 which gives the case of the skew t-distribution introduced in [5]. This model can be defined for X = X X by X = G τ U + Z Gτ U X = G ν U + Z G ν U where Z Z Στ and ν are defined previously. The coefficient of the lower tail dependence for a bivariate random vector X = X X with marginal distributed functions F and F is defined as λ L = lim u P X F u X F u if this limit exists where F and F are the generalized inverse functions of F and F see []. The coefficient of the upper tail dependence of a random vector X can be defined similarly as λ U = lim u P X F u X F u or equivalently λ U = lim X F x + F x X x..4 376
. Main result The aim of this paper is to provide an analytic result of the tail dependence coefficient of the generalised hyperbolic model. We will consider the upper tail dependence for X in the model. using a method deriving from an equivalent form of.4 for λ U : see [3]. λ U = lim P X F x + F x X = x + lim P X F x + F x X = x Theorem.. The upper tail coefficient of the vector X X defined in. is given by: If = = then λ U = g λγδ tdt If > then 3 If = = < then λ U = λ U =. Proof. Let = =. For x a we use the notation f x gx if f x gx and f x gx if there exists c > such as gx = c f x. Let hx = F F x gx = Q λ δ γ Q λ δ γ x and S = Q λ δ γ U. Equation. can be written as X = µ + gs + Z gs X = µ + S + Z S where µ µ Z and Z are defined previously. The upper tail dependence coefficient can be rewritten as PX hxx x λ U = lim x PX x hx x f X X stdsdt = lim x PX x We use the l Hopital s rule to obtain hx λ U = lim f X X sxds h x x f X X hxtdt x f x = lim PX hx/x = x + h x f x x f x PX x/x = hx.. The distribution function Q W of the generalized inverse gaussian is given by x γ δ Q W x = λ k λ δλ tλ exp γ t δ t dt. 377
It follows that the corresponding distribution function Q W satisfies as x Q W x c λγδ x λ λ exp γ x where c λγδ = γ δ λ k λ δλ. Then Q W x = c γδ x λ λ.exp γ x. We use Lemma 3. see [] to invert the last equation and obtain as u Hence for x QW log u u = γ. / gx = m γ δ x λ exp γ x where m λ γ δ = c λ γ δ λ γ. The density f of the generalized hyperbolic distribution with parameters λαδγ is given by f x = l λ αδk λ α δ + x µ δ + x µ λ expx µ where l λ αδ = F satisfies as x F x π α λ πα λ δ λ k λ δ α β. It follows that the corresponding distribution function x l λ αδt λ / k λ α δ + t µ expt µdt d λ x λ exp α x where d λ = λα l λ αδ. Using the same lemma as above for u we can invert F and obtain F u = log u α. It follows that the strictly increasing function hx = F F x satisfies as x hx α x. The derivative of h is equal to h = f f h and hence as x f x f x h x = f x f hx α λ exp α x α. Note that this quantity tends to zero for α < +. Hence the second term in. goes to zero. It remains to calculate the first term of the equation.. The conditional density of S given X satisfies by Bayes rule p S X =x t p X S =tx p S t t λ / exp γ t + δ + x. t Then S X = x GIGλ /γ δ + x. Now we compute the second term of.. PX x X = x = = PX x X = x s = t f S X =x tdt P g t ρ tx + ρ W x g λ γδ tdt 378
where λ = λ / δ = δ + x and W N. We can re-express the right side of. as lim x P ρ W f xgt ρ txg λ γδ tdt. Using the dominated convergence theorem we obtain λ U = g λ γδ tdt. Let > or < For our quantitative developement we need the modified Bessel function of the third kind see [4] with index λ R: π K λ x = x exp x + o x. According to the equation. we have K λ+ x K λ x = + 8λ + 4 8x + 3λ 8 8x + o x 3.3 K λ+ xk λ x K λ x = + 64 56λ + x 8x 3 + o x 4..4 The marginal density of X is defined by f X x = k λ δ + x µ α πkλ δ α e x µ.5 where K λ δ α β = xλ e is equal to α β x+ δ x dx. The conditional density of X. = X X = x f X X x x = f X X x x f X x k λ δ + ax x α = πkλ δ + x µ α e x µ where ax x = ρ x µ + x µ x µ + x µ. Note that X. has a generalised hyperbolic distribution. It is known that the conditional distribution of X. W = w is normal Nµ + ww where W GIG λ δ + x + µ α +. The proof we propose here requires the use of the Chebyshev s inequality so we compute the first two moments. We have EX. = µ + EW = µ + δ + x + µ α + K λ+ α + δ + x + µ K λ α + δ + x + µ You can easily compute the variance of X. from the variance decomposition formula and obtain δ +x +µ [ Kλ+ α + δ +x +µ VarX. = α + Kλ+ 3 α + δ +x +µ K λ α + δ +x +µ + δ +x +µ α + K λ α + δ +x +µ K λ+ α + δ +x +µ K λ α + δ +x +µ ] 379
By the Chebyshev s Inequality for all positive ε P X. EX. ε x VarX. ε x [ = ε x α +λ +3 4 α + 4 δ + x µ α + ] + o x = o x. α + 3 We will combine the variance of X. with.3 and.4 as x. This gives PEX. ε x X. EX. + ε x o as x x Hence lim x P X F F y X = x = if F F x < EX. ε x = µ α + + ε α + + ε x + λ α + + o x We use the expectation of the random variable X. and.3 as x to show F x < F µ x + λ x Hence for i = F i x = PX i x = x λ 3 e i +α + i x α + + o K λ δ α i + α λ // i + α + i By combining.7 and.6 we have + α + > + α + µ + ε. + b as ε > is arbitrary we obtain +α + +α + µ..6 + o..7 x >. If α + = = R then µ + α + >. Then lim x + P X F F y X = x =. Similarly one can show that lim x + P X F F y X = x = and results in λu =. References [] K. Banachewicz and A. van der Vaart Tail dependence of skewed grouped t-distributions. Statistics and Probability Letters. 78 8 388-399. [] O. E. Barndorff-Nielsen and H. Christian Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Probability Theory and Related Fields Springer. 384 977 39-3. [3] O. E. Barndorff-Nielsen Exponentially decreasing distributions for the logarithm of particle size Royal Society of London. Proceedings A. Mathematical Physical and Engineering Sciences. 353 977 4-49. [4] J. Beirlant Y. Goegebeur J. Teugels and J. Segers Statistics of Extremes: Theory and ApplicationsWiley Chichester. 6. 38
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