Quantile function expansion using regularly varying functions

Transcription

1 Quantile function expansion using regularly varying functions arxiv: v [math.st] 9 Aug 07 Thomas Fung a, an Eugene Seneta b a Department of Statistics, Macquarie University, NSW 09, Australia b School of Mathematics an Statistics, University of Syney, NSW 006, Australia August 0, 07 Abstract We present a simple result that allows us to evaluate the asymptotic orer of the remainer of a partial asymptotic expansion of the quantile function hu as u 0 + or. This is focusse on important univariate istributions when h has no simple close form, with a view to assessing asymptotic rate of ecay to zero of tail epenence in the context of bivariate copulas. The Introuction motivates the stuy in terms of the stanar Normal. The Normal, Skew-Normal an Gamma are use as initial examples. Finally, we iscuss approximation to the lower quantile of the Variance-Gamma an Skew-Slash istributions. Keywors: Asymptotic expansion; asymptotic tail epenence; Quantile function; Regularly varying functions; Skew-Slash istribution; Variance-Gamma istribution. Corresponing Author. Honorary Associate, University of Syney. aress: thomas.fung@mq.eu.au Thomas Fung.

2 Introuction This paper is motivate by the nee for a generally applicable proceure to stuy the asymptotic behaviour as u 0 + of F i u, i =, an Cu, u = P X F u, X F u where the F u s are the inverse of continuous an strictly increasing cf s i F i u, i =,, an a bivariate copula function respectively. A ranom vector X = X, X with marginal inverse istribution function F i u, i =, has coefficient of lower tail epenence λ L if the limit λ L = lim u 0 + λ L u exists, where λ L u = P X F u X F u = P X F u, X F u P Cu, u X F =. u u If λ L = 0 then X is sai to be asymptotically inepenent in the lower tail. In this situation in particular the asymptotic rate of approach to the limit 0 of λ L u is an inication of the strength of asymptotic inepenence. The classical case is the bivariate Normal with correlation coefficient ρ as iscusse in Embrechts, McNeil an Straumann 0. It was shown in Fung an Seneta 0 that if ρ λu = Φ Φ u u θ Lu + ρ where Φx, < x < is the cf of the stanar Normal, an Lu is a slowly varying function as u 0 +, then λ L u uθ Lu θ +. Fung an Seneta 0 also showe in their Theorem 3 that λu u ρ +ρ Lu where + ρ ρ Lu 4π log u +ρ. ρ

3 The proof within their Theorem epene heavily on the very specific asymptotic relation as x between the cf Φ an the corresponing stanar Normal pf f. We were unable to use in that paper, for this purpose, the expression for the quantile function Φ u of the stanar Normal istribution yu = logu 4π log u, 3 given for example by Lefor an Tawn 997 from a truncate expansion of Φ u, since we i not know the asymptotic orer of the reminer Φ u yu. We were able to prove that Φ u yu as u 0 +, but inasmuch as this asserts only that Φ u = yu + o = log u + o, we can only be sure of the ominant term of a truncate asymptotic expansion, an this was inaequate to procee from. Since However our general result below, when applie to the stanar Normal, gives Φ u = yu + O Φy y π e y log u. as y, from λu kue yu ρ +ρ +O log u where ku = log u π +ρ ρ ; = kue ρ +ρy u +O log u 3

4 = kue ρ +ρy u +O log u = kue ρ +ρy u+o as u 0 +, since y uo = O log u Thus, since y u log u as u 0 +. log u λu kue +ρ ρy u an the right han sie simplifies to the right han sie of. In this note we present a simple result that allows us to evaluate the asymptotic orer of the ifference between yu an hu as u 0 + or, where hu is the quantile function corresponing to a cumulative istribution function g for important istributions where hu has no close form, an yu is an asymptotic close form expression. This is the general result which we iscuss in Section. In Section 3, we will illustrate our results by consiering the quantile function for the Generalise Gamma-type tail which has various commonly use istributions such as Normal, Skew-Normal, Gamma, Variance-Gamma an a Skew-Slash as special cases. Detaile extreme value structure of such istributions is important in a financial mathematics context. These iniviual examples will be iscusse in Section 4. Main Result Our main result is summarise into the following Theorem. Theorem. Suppose that gx is a strictly positive continuous an strictly increasing cumulative istribution function cf on, A], A < 0. Suppose further that some function yu as u 0 + satisfies ygx = x + O ζx 4 4

5 as x, such that ζx 0, x. Suppose finally that ζx = ψ x with ψw = w ρ Lw for some constant ρ 0 an function Lw, w > 0, slowly varying at 0. If hu, u 0, ga] is the inverse function of g, then hu = yu + Oζyu. Proof. We begin with the fact that ghu = u yghu = yu hu + Oζhu = yu, using 4 Thus hu yu as u 0+ an hence ζhu ζyu = ψ /hu ψ /yu 5 by the Uniform Convergence Theorem of slowly varying function of Seneta 976, Theorem.. From 5, ζhu ζyu O ζhu = O ζyu as u 0+. Finally, hu + Oζhu = yu hu = yu + Oζhu hu = yu + Oζhu hu = yu + Oζyu. 5

6 Note that the formulation of the theorem requires, in the case ρ = 0 only, that the function Lw 0, w 0. The conition require in Theorem is simply that the correction term in 4 is relate to a regular varying function which is quite general an shoul apply to a wie range of istributions. The result in Theorem not only ensures hu an yu will be asymptotically equivalent, it will also stipulate how accurate yu will be for hu. We express as a corollary to Theorem the corresponing result for the upper tail quantiles. Corollary. Suppose that gx is a strictly positive continuous an strictly increasing cumulative istribution function cf on [A,, A > 0. Suppose further that some function yu as u satisfies ygx = x + O ζx as x, such that ζx 0, x. If ζx = ψ x with ψw = w ρ Lw for some constant ρ 0 an function Lw, w > 0, slowly varying at 0. If hu = g u, u [ga, is the inverse function of g i.e. the upper tail quantile function then hu = yu + Oζyu. In the next section, we will illustrate our results by consiering the quantile function corresponing to a cf that has a Generalise Gamma-type tail behaviour. 6

7 3 Quantile for Generalise Gamma-type tail behaviour 3. Lower Tail We are intereste in approximation to the quantile function for the Generalise Gammatype tail. We consier a cf g to have a Generalise Gamma-type lower tail behaviour if g can be expresse as gx = a x b e c x + O, x < 0, 6 x e for some constants a, c,, e > 0 an b R. Several istributions of current interest have such tails, an we iscuss them as special cases in the next section. Suppose that an approximation to the quantile function hu = g u to be b yu = c c u b b a log log c b u b, for small u > 0. 7 This can be obtaine via the recursive metho for an inverse function see Chapter.4 of De Bruijn 96 for instance on g. Then ygx = = = { b c b c log c log b log c b [ b c c b x + log a x b e c x +O x e a a x b e c x + O b b x e c b x e c b x + O x e log ca b x b e c b x + O x e c b x + log + O x e log c b x + log ca b b x + log + O x e ]} 7

8 = { [ b c c b x + log log log c b x + O x e ca b b x + O c b x x e ]} { [ b = c ]} log x c b x + O + O x x e log x = x + O + O x e+ x x + O log x, if e x = x + O, otherwise. x +e c log b x As yu c log u, the behaviour of the quantile function h is yu + O hu = yu + O log c log u log u log u e + = yu + O log log u, if e log u, otherwise, 8 by Theorem. We next expan yu to obtain an expansion for hu with an error term of appropriate lower orer. After some algebra, yu = c [ a b log b log u b c log u log u + b 3 log a b ] 3 +O. log u 3 c log u + b 3 + b 3 log u log [ log a b c log u c b ] 8

9 As a result, when e, h becomes log log u hu =yu + O log u = c a b log b log u b c c log u c log c c u b c 3 log + O ; 9 c u log u On the other han, when > e, h becomes hu =yu + O log u e + = c log u b + O log u e + c log u c. a b log b c c log u c 3. Upper Tail Similarly, a cf g is sai to have a Generalise Gamma-type upper tail behaviour if g can be expresse as gx = ax b e cx + O, x, 0 x e 9

10 for some constants a, c,, e > 0 an b R, which woul suggest the following approximation to the upper tail quantile function: b yu = c c u b b a log log c u b b, as u. Then by Corollary, the upper tail quantile function h can be expresse as hu yu + O = yu + O log log u log u c log u + = log u + c +O log u e +, if e;, if < e; b log log u c c log u + b log log u + O c 3 c log u log u e + b log log u c c log u + log log u log u b log a b c c c log u, if e; b log + c c log u, if > e. a b c as u. 4 Applications In this section, we will illustrate our results with application to the Normal, Skew-Normal, Gamma, Variance-Gamma an Skew-Slash istributions. 0

11 4. Stanar Normal From Feller 968 Chapter VII Lemma, the form of g is gx = e x + O, for x <. π x x If we compare this with 6, we have a = π, b =, c = that the approximation y becomes: an = e =. This means yu = { log πu logu } = log u 4π log u 3 an is the same as 3. As in this case we have = e, we can obtain the behaviour of the quantile function by using using 8 an 9 an get hu =yu + O log u = log log u log u + log 4π + log u log u + + O. 8 log u 3 log u 3 If we use only the ominating term of yu in 3 an so put y u = log u then after some algebra we have ygx = x + Oζx where ζx = log x x. As a result, hu = y u + O log u

12 by using Theorem. We can see that y u is a less accurate approximation to hu than yu, an the expression for hu that it gives only improves on y u by giving the correct orer of the ifference hu y u. We are aware that there exists a whole fiel of literature on fining an efficient an accurate approximation in the numerical sense for the Normal quantile functions, such as Abramowitz an Stegun 964, Beasley an Springer 977 to the more recent Voutier 00. Soranzo an Epure 04 provie a substantial bibliography on this subject. But that is not our focus an so our methoology will most likely be outperforme by the more sophisticate approximation in the literature. Take the approximation to the Normal quantile in Section.. in Voutier 00 for example which gives y V u = c 3 log u + c c + log u + c 0 log u + log u + 0 for e 37 / < u < where c 3 = , c 0 = , c = , c = , 0 = , = To see how the expansions perform against the approximations, we plot the stanar Normal quantile function, enote as Φ u, in R against y V, y an y an the results are shown in Fig.. From Fig., we can see that h V gives almost ientical result to the built-in Normal quantile function in R an is better than our expansion yu. We also see that y is an orer worse than the other methos.

13 Figure : Comparison in R of Φ u, y V, y an y 4. Skew-Normal The quantile function of the Skew-Normal istribution is another example that is covere by our result. The istribution was first introuce by Azzalini 985 an has evelope into an extensive theory presente in a recent monograph by Azzalini an Capitanio 04. A ranom variable is sai to have a stanar Skew-Normal istribution if its ensity function is fx = π e x Φλx, x R where λ R controls the skewness of the istribution. When λ = 0, it reuces to the stanar Normal as special case so we shall exclue the case of λ = 0 from our subsequent iscussion. Using Lemma of Capitanio 00 or Azzalini an Capitanio 04, pp. 5 3

14 53, we have e πλ+λ gx = x +λ x + O x, λ > 0; π x e x + O x, λ < 0, x. 4 This means if we compare 4 with 6, we have a = πλ+λ, b =, c = + λ / an = e = when λ > 0; a = / π, b =, c = /, = e = when λ < 0. This means that the approximation y becomes: { yu = = +λ { log [ log u π/ logu +λ uπλ+λ log +λ u ]}, λ > 0; }, λ < 0; log πλu logu/ + λ +λ, if λ > 0; log 5 u 4π log u, if λ < 0. As in this case we have = e, we can obtain the behaviour of the quantile function by using 8 an 9 to get hu =yu + O log u log u + +λ = + +λ +λ log u 3 + O log u + + O + 8 log u 3 log log u + logπλ +λ +λ log u +λ +λ log u log u 3 log log u + log π log u log u log u 3, λ > 0;, λ < 0. 6 The expression 6 justifies the use of expression 5 above in Theorem of Fung an Seneta 06. 4

15 4.3 Gamma Using, one can also etermine the accuracy of the gamma-like upper tail. Suppose that X Γα, β an let its pf be Then fx = βα Γα xα e βx, x > 0. gx = βα Γα xα e βx + O, 7 x as x. This means that if we compare 7 with 0, we have a = βα Γα, b = α, c = β an = e =. By using an approximation to the upper quantile function when u woul be α yu = β β α log log u α β α /Γα β α = β log uγα uβ α α log α α u α As in this case we have = e again, we have for the upper quantile function hu = yu + O log log u log u = β log uγα uβ α α log = β as u by using. α α log u + α β + O log log u log u log log u log log u β log Γα + O log u In Section 4 of Fung an Seneta 0, via their Theorem, the authors obtaine hu = yu + o, u with yu = log u. β 5

16 4.4 Variance-Gamma an Skew-Slash Finally, we propose an approximation to the lower tail quantile function of the Variance- Gamma VG an a Skew-Slash istribution, as they can have similar tail structure. A ranom variable X is sai to have a skew VG istribution introuce in Maan, Carr an Chang 998 an further stuie in Schoutens 003, Seneta 004 an Tjetjep an Seneta 006, if X is efine by a Normal variance-mean mixture as X Y Nµ + θy, σ Y, where µ, θ R an σ > 0. The istribution of Y is Γ ν, ν with ν > 0 such that EY =. When µ = 0 an σ =, from Schoutens 003, we have gx = ν ν Γ ν x ν e ν +θ +θ x ν + θ ν = a x ν e ν +θ +θ x + ν θ + θ + O x + O x 8 as x, where a > 0 is efine by the above.. By comparing with 6, a is efine as above, b = ν, c = ν + θ + θ an = e = which in turn woul suggest the following approximation to the lower quantile function: yu = u log + ν θ + θ a [ log u log u, ν θ + θ + ν θ + θ ν +θ +θ ν ν ] ν as u 0 + by using 7. We can then obtain the behaviour of the quantile function as = hu u log + ν θ + θ a [ log u + ν θ + θ ν +θ +θ ν ν ] ν + O log u 6

17 = log u ν θ + θ ν ν + θ + θ log ν + ν θ + θ a ν ν ν +θ +θ + O log u as u 0 +, by using 8 an 9. A Skew-Slash istribution on the other han was first propose in multivariate form in Arslan 008 an further stuie in Ling an Peng 05. A ranom variable X is sai to have a Skew-Slash istribution if X is efine by a Normal variance-mean mixture as X Y Nµ + θ/y, σ Y, where µ, θ R an σ > 0. The istribution of Y is Betaλ, ; that is, its pf is: fy = λy λ, 0 < y < ; = 0 otherwise. Here we only consier the case of θ > 0 so that the tail behaviour is similar to that of the Variance-Gamma in 8. When µ = 0, σ = an θ > 0, from Ling an Peng 05, Lemma. an equation, we have gx = λθλ x λ+ e θ x + O, x as x. By comparing this with 6, we have a = λθλ, b = λ +, c = θ, an = e =, which in turn suggests the following approximation to the lower quantile function: λ + yu = θ log θ λ+ u λ+ λθ λ / log θ λ+ u λ+ log u θ, as u 0 + by using 7. Ling an Peng 05, Lemma 3., equation 3 use the fact that hu = yu + o, u 0 + with yu = log u/θ. We can obtain the behaviour of the 7

18 quantile function as λ + hu = θ = log u θ log θ λ+ u λ+ λθ λ / log θ + λ + θ λ+ u λ+ as u 0 + by using 8 an 9 once again. log λθ λ θ + O log u + O log u. Acknowlegements The authors thank the referee for a careful reaing of the original version, an a list of suggestions which have resulte in a much improve paper. References Abramowitz, M. an Stegun, I.E. e., Hanbook of Mathematical Functions With Formulas, Graphs an Mathematical Tables. National Bureau of Stanars, Washington. Arslan, O. 008 An alternative multivariate skew-slash istribution. Statistics an Probability Letters 78: Azzalini, A. 985 A class of istributions which inclues the normal ones. Scan. J. Statist. :7 78. Azzalini, A. an Capitanio, A. 04 The Skew-Normal an Relate Families. IMS Monographs. Cambrige University Press, Cambrige. Beasley, J. D. an Springer S. G The percentage points of the Normal Distribution, Applie Statistics 6, 8. Capitanio, A. 00 On the approximation of the tail probability of the scalar skewnormal istribution. Metron 68:

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20 Soranzo, A. an Epure, E. 04. Very simply explicitly invertible approximations of normal cumulative an normal quantile function. Applie Mathematical Sciences 8, Tjetjep, A. an Seneta E. 006 Skewe normal variance-mean moels for asset pricing an the metho of moments. International Statistical Review 74:09 6. Voutier, P.M. 00. A new approximation to the normal istribution quantile function. arxiv: [stat.co] 0