Journal of Statistical and Econometric ethods vol. no. 3 - ISS: 5-557 print version 5-565online Scienpress Ltd 3 Tail Approimation of the Skew-ormal by the Skew-ormal Laplace: Application to Owen s T Function and the Bivariate ormal Distribution Werner ürlimann Abstract By equal mean two skew-symmetric families with the same kernel are quite similar and the tails are often very close together. We use this observation to approimate the tail distribution of the skew-normal by the skew-normal-laplace and accordingly obtain a normal function approimation to Owen s T function which determines the survival function of the skew-normal distribution. Our method is also used to derive skew-normal-laplace approimations to the bivariate standard normal distribution valid for large absolute values of the arguments and correlation coefficients a situation difficult to handle with traditional numerical methods. athematics Subject Classification: 6E5 6E7 6E Keywords: Skew-symmetric skew-normal skew-normal-laplace Owen s T function normal approimation asymptotic approimation bivariate normal Introduction There are many known methods available to etend symmetric families of probability densities to asymmetric models. An important general method is due to [3]. Consider a continuous density f called the kernel which is symmetric about zero and let G be an absolutely continuous distribution called the skewing distribution that is symmetric about zero. Then for any skewing parameter the function Wolters Kluwer Financial Services Switzerland AG e-mail: whurlimann@bluewin.ch Article Info: Received : December 3. Revised : December 3. ublished online : arch 3
Werner ürlimann g f G defines a probability density. Skew families of distributions have been studied in [3] and [4] for the normal and Cauchy kernels and in [] and [] for the gamma and Laplace kernels. It can be asked how different two skew-symmetric families with the same kernel can be. This main question has been studied by [] who argues that the shape of the family of skew-symmetric distributions with fied kernel is not very sensitive to the choice of the skewing distribution. In fact by equal mean two such distributions will be quite similar because they have the first and all even moments in common. While the maimum absolute deviation is often attained at zero which implies that the distributions will differ significantly only in a neighborhood of zero the tails are often very close together. Applied to statistical approimation this observation motivates the search of an analytically simple choice of skewing distribution in order to approimate a given skew-symmetric family without closed-form distribution epression. A result of this type is used to derive a normal and an elementary function approimation to Owen s T function introduced in [6] which determines the survival function of the skew-normal distribution. The use of the Laplace skewing distribution turns out to be appropriate for this. The paper is organized as follows. Section presents a brief account of the results from [] on the similarity of the shape of skew-symmetric families with the same kernel which motivates our study as eplained above. Section 3 derives our normal approimation to Owen s T function by analyzing the signed difference between the skew-normal and the skew-normal-laplace distribution. Given a small maimum absolute error and some technical conditions there eists a common threshold parameter such that the signed difference between Owen s T function and its normal analytical approimation remains doubly uniformly bounded for all parameter values eceeding this threshold. umerical eamples illustrate our findings. Section 4 applies our method to the bivariate normal distribution and derives double uniform bounds especially useful for large absolute values of the arguments and correlation coefficients a situation difficult to handle with traditional numerical methods. Skew-symmetric Families and their Shape The density function of the skew-symmetric family with continuous zero symmetric density kernel f absolutely continuous and zero symmetric skewing distribution G and skewing parameter is defined by g f G. This family has been introduced in [3] [4] and includes the skew normal by [5]. Skew families of distributions have been studied in [3] and [4] for the normal and Cauchy kernels and in [] and [] for the gamma and Laplace kernels. It can be asked how different two skew-symmetric families with the same kernel can be. This main question has been studied in []. In general consider besides the with distribution G the density h f for another skewing distribution with skewing parameter and let be its distribution. Assume the kernel has a first moment. Then the means of these skew-symmetric distributions are: g
Tail Approimation of the Skew-ormal by the Skew-ormal Laplace 3 g G h d d f G d f d.. It is shown in [] that both G and are strictly increasing odd functions such that and lim lim f d G G is the mean of the folded distribution. oreover it is well-known that the even moments are the same as those of the folded kernel whatever the parameters are. Therefore every member of the families g } and h } has the same set of even moments independently { { of the choice of the skewing distribution. By equal mean the distributions G and will be quite similar because they have the first and all even moments in common. ow the above properties about the mean guarantee the eistence of a unique value such that G for each. One notes that for and for. Under these conditions it is natural to consider the following measure of discrepancy between two skew-symmetric families with the same kernel []: d G ma G. Since the difference in distributions G is an even function whatever the parameters are [] Theorem p.663 the search of the maimum discrepancy can be restricted either to the interval or. By construction symmetry of the kernel about zero it is clear that is a critical value such that ' ' G g h. we conclude that From the property lim G G yields the actual maimum ma G G in case the difference G has no other critical points. In the light of these findings we say that a pair g } of skew-symmetric families with the same kernel identical mean and { h even moments is a regular pair provided the following condition is fulfilled: g h.3 It follows that for regular pairs the distance between the skew-symmetric families defined by: d G ma G ma G.4 is a meaningful measure of discrepancy [].
4 Werner ürlimann In general while the maimum absolute deviation in.4 is attained at and the distributions will differ significantly only in a neighborhood of the left and right tails of G and are often very close together. In particular this means that the survival functions G G and for pairs with small discrepancy are epected to differ only very slightly by increasing value of. ore precisely it is possible to investigate whether for a sufficiently high threshold and a parameter value the tail difference remains doubly uniformly bounded by a given small maimum absolute error such that: G.5 Applied to statistical approimation this observation motivates the search for an analytically simple choice of skewing distribution in order to approimate a skew-symmetric family without closed-form distribution epression. A result of this type is used in the net Section to derive a normal and an elementary function approimation to Owen s T function introduced in [6] which determines the survival function of the skew-normal distribution. The use of the Laplace skewing distribution turns out to be appropriate for this. 3 Approimations to Owen s T Function Consider the skew-normal density g f and the skew-normal-laplace density h f with e e []. Eample B shows that d G. 87 is attained at. 47. 867 so that these two families are quite similar. The survival function of the skew-normal reads: G T 3. where is the standard normal survival function and T Owen s T function defined by: ep t T t t dt dt t ' is 3. The skew-normal-laplace survival function is given by: e 3.3 e with / the ill s ratio. To derive a double uniform approimation of the type.5 we proceed as follows. Let
Tail Approimation of the Skew-ormal by the Skew-ormal Laplace 5 G be the signed difference between these survival functions. To analyze its monotonic properties with respect to the parameter we consider its partial derivative: 3.4 G ' The derivative ' ' in 3.4 is obtained as follows. By definition the function solves the mean equation G apply partial integration or [3] Section 5 for the skew-normal-laplace mean: Use ill s ratio to rewrite this as: ep 3.5 Through differentiation and noting that ' we obtain the 3/ relationship / 3.6 ' which by 3.6 implies that solves the differential equation: ' 3/ 3.7 As a sufficient condition which implies a double uniform bound.5 we show that for all sufficiently high values of. Indeed since is strictly monotonic decreasing in both arguments if suffices to determine thresholds and for given absolute error the value if it eists that solves the signed difference equation then automatically for all. To derive this key result we use 3.4 and note first that: t G t t t dt t t e dt 3.8 Inserted into 3.4 we obtain the integral epression: t t t t ' e dt 3.9 ow we certainly have provided the curly bracket is negative for all
6 Werner ürlimann t for some appropriate threshold which is equivalent with the inequality: t t ln ' t t 3. Let ln ' D be the discriminant of the quadratic form in 3.. If D the inequality t holds t. Otherwise the inequality t t ma t where t denotes the greatest holds zero of the quadratic equation t. By 3.5-3.6 and since is uniquely defined we can epress t as function of only namely: t t D A graphical numerical analysis shows that: D ln 3ln ln. t t *.4357 *. 465 ma D. 3. It remains to show that for all provided is sufficiently high. A partial differentiation yields h g e for all. A sufficient condition for is the inequality e. Since by the inequality of [9] [5] and it suffices that e which is equivalent to the quadratic inequality: t ln q 3. ow we argue as above. Let ln ' D be the discriminant of the quadratic form in 3.. If D the inequality 3. will hold for all. Otherwise the inequality 3. is fulfilled provided there eists such that ma z where z denotes the greatest zero of the quadratic equation q. Similarly to 3. we can epress z as function of only namely:
Tail Approimation of the Skew-ormal by the Skew-ormal Laplace 7. ln ln ln D D z z 3.3 umerical analysis shows that 4357. ma z z D for.57 which yields 9639. / by the relationship 3.6. The following result has been shown. Theorem 3.. Given are the skew-normal and the skew-normal-laplace distribution functions G with equal mean. Then for given the signed tail difference satisfies the double uniform bound:.4357 G 3.4 provided the signed difference equation has a solution.9639. This result implies the following normal approimation to Owen s T function. Corollary 3.. Given and the definitions of Theorem 3. Owen s T function satisfies the double uniform normal approimation:. / e A A T A 3.5 roof. This follows from Theorem 3. using the representations 3. 3.3 and the fact that solves the equation 3.6. Since Owen s T function has a probabilistic meaning and noting that A we obtain the following probabilistic interpretation of Corollary 3.: SL SL 3.6 where are independent standard normal random variables and SL is a skew-normal-laplace random variable. In other words as the following asymptotic equivalence holds:. ~ SL 3.7 [7] mention that some approimations to Owen s T function cannot be used for accurate calculations [9] [8] and especially emphasize their inaccuracy when [6]. Corollary 3. yields a useful accurate analytical asymptotic approimation which in
8 Werner ürlimann contrast to all previous numerical approimations has the advantage of a stochastic interpretation. From a numerical elementary perspective it is possible to simplify further the obtained approimation using the eponential function only. For this purpose we propose to use the simple analytical approimation by [] of the standard normal tail probability function given by: ep.77.46. 3.8 Corollary 3.. Given the definitions of Theorem 3. Lin s eponential approimation E to the signed difference A yields the eponential approimation: T ~ / E 4 ep ep 3.9 with fied.77.46. Table 3. illustrates numerically the obtained elementary approimations. We note that the absolute accuracy of the normal approimation over the range is worst at for the given level of accuracy. By increasing the relative accuracy of the normal approimation increases very rapidly as illustrated in the Graph 3.. In contrast the relative accuracy of the eponential approimation remains small over a finite interval only for eample 3 in Graph 3.. Therefore the eponential approimation is not a useful asymptotic approimation. Owen s T function is calculated using numerical integration as implemented in ATCAD for eample. Table 3.: Double uniform approimation bounds to Owen s T function 3 3 T A E 4.64.7465 3.7338 3.6338 3.576 5.73.9954 3.74 3.74 3.68645 6.344 3.7979 3.77 3.797 3.698 7.9364 3.967 3.77 3.77 3.6995 8 3.533 4.86 3.77 3.76 3.6997 3
Tail Approimation of the Skew-ormal by the Skew-ormal Laplace 9.4.3 AR.. 3 4 5 6 Graph 3.: Relative signed error A / T for by fied.64.5 A_LinR 3 4 5 6 Graph 3.: Relative signed error E / T for.64 by fied 4 Application to the Bivariate ormal Distribution Owen s T function has originally been introduced to simplify the calculation of bivariate normal probabilities. The bivariate standard normal distribution with correlation coefficient is defined and denoted by: y; y; y u v; dudv ep y y y. Recall the skew-normal distribution 3. which in suggestive changed notation reads: 4. T. 4. In terms of 4. the formula. in [6] can be rewritten as: s
Werner ürlimann with ; y y s y s y y y y y 4.3 the indicator function. Since ; s the formula 4.3 is equivalent to the reduction formula: y; ;sgn y y y y;sgn y y y yy yy 4.4 whose special case y is given in []. It is also useful to note that 4.4 is equivalent to the normal copula formula 3.6 in []. Usually the numerical implementation of the bivariate normal distribution is done with the tetrachoric series an infinite bivariate epansion based on the ermite polynomials consult [] for a comprehensive review. owever this epansion converges only slightly faster than a geometric series with quotient which is not very practical to use when is large in absolute value. The author [] improves on this numerical evaluation and derives for ; another series that converges approimately as a geometric series with quotient. Alternatively to the latter the use of Theorem 3. yields a simple normal analytical approimation via the skew-normal Laplace which is valid for sufficiently large values of the arguments. Theorem 4.. Assume that. 4357 and / where and. 9639 as in Theorem 3.. Then the bivariate standard normal distribution satisfies the following inequalities: ; ; 4.5 where denotes the skew-normal-laplace distribution 3.3 with parameter determined by the equation. roof. In the notations of Section 3 set the parameter of the skew-normal equal to /. We have ; G G if and ; G if. In the first case we use the fact that the difference G is an even function [] p.663 to see that G G. The first inequality in 4.5 follows from 3. by noting that the condition is equivalent to
Tail Approimation of the Skew-ormal by the Skew-ormal Laplace. If then G G /. The second inequality in 4.5 follows again from 3.. Finally we note that Owen s T function has further applications in mathematical statistics including non-central t-probabilities e.g. [7] [3] and multivariate normal probabilities [8] among others. References []. Ali. al and J. Woo Skewed reflected distributions generated by reflected gamma kernel akistan J. Statist. 4 8 77-86. []. Ali. al and J. Woo Skewed reflected distributions generated by the Laplace kernel Austrian J. Statist. 38 9 45-58. [3] A. Azzalini A class of distributions which includes the normal ones Scand. J. Statist. 985 7-78. [4] A. Azzalini Further results on a class of distributions which includes the normal ones Statistica Bologna 46 986 99-8. [5] Z.W. Birnbaum An inequality for ill s ratio Ann. ath. Statist. 3 94 45-46. [6] B.E. Cooper Algorithm AS4 An auiliary function for distribution integrals Appl. Statist. 7 968 9-9 Corrigenda 8 969 8 9 97 4. [7] B.E. Cooper Algorithm AS5 The integral of non-central t-distribution Appl. Statist. 7 968 93-94. [8] T.G. Donelly Algorithm 46 Bivariate normal distribution Comm. Assoc. Comput. ach 6 973 638. [9] R.D. Gordon Values of ill s ratio of area to bounding ordinate of the normal probability integral for large values of the argument Ann. ath. Statist. 94 364-366. [] S.S. Gupta robability integrals of multivariate normal and multivariate t Ann. ath. Statist. 34 963 79-88. [] J.-T. Lin Approimating the normal tail probability and its inverse for use on a pocket calculator Appl. Statist. 38 989 69-7. [] C. eyer The bivariate normal copula reprint 9 URL: arxiv:9.86v [math.r] [3] S. adarajah and S. Kotz Skewed distributions generated by the normal kernel Statist. robab. Lett. 65 3 69-77. [4] S. adarajah and S. Kotz Skewed distributions generated by the Cauchy kernel Braz. J. robab. Statist. 9 5 39-5. [5] A. O agan and T. Leonard Bayes estimation subject to uncertainty about parameter constraints Biometrika 63 976-3. [6] D.B. Owen Tables for computing bivariate normal probabilities Ann. ath. Statist. 7 956 75-9. [7]. atefield and D. Tandy Fast and accurate calculation of Owen s T function J. Statist. Software 55-5. [8].. Schervish ultivariate normal probabilities with error bound Appl. Statist. 33 984 8-94.
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