We propose a new approximate skew normal distribution, it is easy to calculate, convenient, mathematically tractable and is in a closed form. It is particularly useful when the probability density function occurs in an expression to be used for further mathematical derivation or in programs for the skew normal distribution. Also, we propose approximate first moment second moment and variance to the skew normal distribution. A numerical comparison between exact and approximate values of pdf and cdf of the skew normal distribution is carried out.
Journal of Advanced Research (2010) 1, 341–350 Cairo University Journal of Advanced Research ORIGINAL ARTICLE Approximate skew normal distribution Samir K Ashour *, Mahmood A Abdel-hameed Department of Mathematics, Institute of Statistical Studies and Research, Cairo University, Egypt Received 23 December 2009; revised 16 February 2010; accepted March 2010 Available online 24 July 2010 KEYWORDS Skew normal distribution; Skewness; Approximation; Cumulative distribution function Abstract We propose a new approximate skew normal distribution, it is easy to calculate, convenient, mathematically tractable and is in a closed form It is particularly useful when the probability density function occurs in an expression to be used for further mathematical derivation or in programs for the skew normal distribution Also, we propose approximate first moment second moment and variance to the skew normal distribution A numerical comparison between exact and approximate values of pdf and cdf of the skew normal distribution is carried out ª 2010 Cairo University Production and hosting by Elsevier B.V All rights reserved Z kz Introduction gzị ẳ pffiffiffiffiffiffi expðÀz2 =2Þ 2p Azzalini [1] defined the skew-normal distribution for a random variable Z with the parameter k be: for any given k R and À1 < z < The term skew normal (SN) refers to a parametric class of probability distributions that extend the normal distribution by an additional shape parameter that regulates the skewness, allowing for a continuous variation from normality to non-normality On the applied side, the skew normal distribution as a generalization of the normal law is a natural choice in all practical situations in which there is some skewness present: a particularly valuable property is the continuity of the passage from the normal case to skewed distributions From a theoretical viewpoint, the SN class has a good number of properties in common with the standard normal distribution: for example, SN densities are unimodal, their support is the real line, and the square of an SN random variable is a Chi-square variable with one degree of freedom The skew normal family of distributions is a fairly recent family of distribution that attracted wide attention in the literature due to its inclusion of the normal distribution, and its mathematical tractability Azzalini [2] gave a comprehensive list of references on the skew normal distribution and related distributions, for skew normal properties and applications see for example Brown [3], Chen et al [4], Gupta and Brown [5], Jammalizadeh et al [6] and Pewsey [7] gzị ẳ 2/ðzÞUðkzÞ ðÀ1 < z < 1Þ; k R ð1:1Þ where /(Ỉ) and U(Ỉ) are the standard normal density and distribution function, respectively, density function (1.1) can be expressed in the following form: * Corresponding author Tel.: +202 25167483; fax: +202 37482533 E-mail address: ashoursamir@hotmail.com (S.K Ashour) 2090-1232 ª 2010 Cairo University Production and hosting by Elsevier B.V All rights reserved Peer review under responsibility of Cairo University doi:10.1016/j.jare.2010.06.004 Production and hosting by Elsevier À1 pffiffiffiffiffiffi expðÀt2 =2Þ dt 2p 342 S.K Ashour and M.A Abdel-hameed For any skew normal random variable Z with skew parameter, k Azzalini [1] introduced the cumulative distribution of Z as follows: Z kz Z x expðÀt2 =2Þ dt dz PðZ xÞ ¼ expðÀz2 =2Þ À1 p À1 and derived the following moment generating function: ! kt t2 Mz tị ẳ 2e U p : ỵ k2 Using the moment generating function it is easy to obtain the following: (1) The first moment of Z is given by ! rffiffiffi k p : EZị ẳ Mz 0ị ẳ p ỵ k2 (2) The second moment of Z is given by Now, using Hoyt’s approximation we shall define an approximate pdf h(x) to the skew normal distribution by replacing the cdf of the standard normal in (1.1) by Hoyt’s approximation (2.1): the approximate function h(x) to the pdf of the skew normal distribution will be: hxị ẳ 2/xịFkxị; À1 < x < 1; À1 < k < where F(kx) is defined by (1.1) on five different intervals depending on the value of the skew factor k So, the approximate pdf of the skew normal h(x) will be x < À3 ; > k > > > Àx2 =2 > 3 p e > 9kx ỵ 3k x ỵ k x ỵ 9ị k x < ; > k 2p > > < Àx2 =2 3 À1 ffie ð3kx À k x ỵ 4ị x < k; k hxị ẳ 4pffiffiffi 2p > > > p1ffiffiffiffi eÀx =2 9kx 3k x2 ỵ k x3 ỵ 7Þ x < ; > k k > 2p > > qffiffi > > : 2ex2 =2 x: p k EZ2 ị ẳ M00z 0ị ẳ 2/0ị ẳ 1: It follows that VarZị ẳ d2 p Clearly h(x) is easier to obtain for any value of the random variable x (positive and negative values) on R depending on the value of the skew factor k R, is convenient, has closed form, and is mathematically tractable k where d ¼ p : ỵ k2 (3) The third moment of Z and the measure of skewness a3 are l3 ¼ 4p fEZịg3 and a3 ẳ l3 l3=2 ¼ À p fEðZÞg VarðZÞ3=2 respectively It can be shown that a3 varies in the interval (À0.9953, 0.9953) The measure of kurtosis is given by !2 l4 EðZÞ2 : a4 ẳ ẳ 2p 3ị l2 Varzị Azzalini [1] concluded that the maximum value of a4 is 0.869 Approximation based on Hoyt’s approximation Lemma Let / be a density function symmetric about and F an absolutely continuous distribution function such that F0 is symmetric about Then h(x) = 2/(x)F(kx) is a proper density function for any real k Proof Let X and Y be independent random variables where X has destiny function F0 and Y has density function / Since X and Y are both symmetric about 0, then X À XY must also be symmetric about 0, So we have P(X À XY < 0) = 1/2 Conditioning on Y, we also have PX XY < 0ị ẳ EY ẵPX < kYjY ẳ yị ẳ EY ẵFkYịjY ẳ y Z ẳ FðkyÞ/ðyÞ dy: À1 Gupta and Chen [8] presented the tables for the cdf of the skew normal distribution with different values of the skew factor k employing Simpson’s rule It is not easy to obtain closed form expressions for the cdf of the skew normal distribution Hoyt [9] suggested approximation for the standard normal distribution using the distribution of the sum of the three mutually independent random variables each uniformly distributed over the interval of À3 to His approximation gave an error not exceeding 0.01, Hoyt’s [9] approximation for the density function of the standard normal is given by: 81 > < 16 ỵ xị x < 1; fxị ẳ x Þ À1 x < 1; > :1 ð3 À xÞ2 x < 3; 16 and corresponding cdf will be: 0 > > > x3 > > > 16 9x ỵ 3x ỵ ỵ > < Fxị ẳ 18 3x x3 ỵ > > > > 9x 3x2 ỵ x3 ỵ > > > : 16 À3 x < À1; x < 3; x: À1 Using density function h(x) the corresponding cumulative distribution function for h(x) will be: x < À3; > > > > > À3 x < À1; > < H1 xị x < 1; Hxị ẳ H2 ðxÞ > > > H3 ðxÞ x < 3; > > > : 13 x; where x < À3; À1 x < 1; It follows that Z 2Fkyị/yị dy ẳ 2:1ị " Z x À9 2 eÀx =2 dx ỵ k ỵ k2 e2k2 H1 xị ẳ p 3k2 ỵ 9ị 2p 3k # 2 2 Àx2 =2 À k þ k þ 3kx þ k x e 3 Approximate skew normal distribution 343 " Z À1 k H2 xị ẳ p 3k2 ỵ 9ị ex =2 dx 2p Àk Z x À9 2 ex =2 dx ỵ k ỵ k2 e2k2 k1 ỵ 2k2 ịe2k2 ỵ8 k # 2 2 x2 =2 ỵ k9 þ 2k þ k x Þe Step (1): For À 3k x < À 1k, let a ¼ À 3k and b ¼ À 1k, then we have: Z b Z b 1 3 2 x2 =2 p xe xhxịdx ẳ 9kx ỵ 3k x ỵ k x ỵ dx a a 2p Z b 2 ex =2 dx ẳ p 9k xex =2 jba ỵ 2p a " Z À1 b k 2 2 x =2 3k x ỵ 2ịe ex =2 dx ẳ p k9 ỵ k2 ị a 2p Àk ! e 2k2 5k2 ỵ ỵ 3k2 e 2k2 : and " Z À1k p 3k2 ỵ 9ị H3 xị ẳ ex =2 dx 2p Àk Z x Z k 2 ex =2 dx ỵ 3k2 ị ex =2 dx ỵ8 1 k k ! 2 92 2k ỵk ỵ k e k27 9kx ỵ k2 x2 ỵ 2k2 ÞeÀx =2 3 Clearly H(À1) = 0, H(1) = h The approximation mean and variance In this section, we will introduce approximate first and second moment as well as approximate variance of the skew normal distribution Step (2): For À 1k x < 1k, let b ¼ À 1k and c ¼ 1k, then we have: Z c Z c 1 xhðxÞ dx ẳ p x 3kx k3 x3 ỵ eÀx =2 dx 2p " b b c Z ! c 2 ¼ pffiffiffiffiffiffi 3k xex =2 ỵ ex =2 dx 2p b b Z c c 2 k x2 ỵ 3ịxex =2 þ eÀx =2 dx b b # c 16 À 12 Àx2 =2 e 2k ị ẳ p 2k ỵ 4e b 2p # Z ỵ k3 k2 Þ k eÀx =2 dx : À1k Theorem Let X be a continuous random variable with density function h(x) Then the expected value of the random variable X will be: " Z c Z k l01 ẳ Exị ẳ p k9 ỵ k2 ị eÀx =2 dx 2p k # Z k À 12 À 92 2 Àx =2 2k 2k e dx ỵ e 3k ị ỵ e 3k ị ỵ 2k3 k ị where À1 < x < 1, À1 < k < Proof To prove Theorem we have four steps as follows: Step (3): For 1k x < 3k, let c ¼ 1k and d ¼ 3k, then we have Z d 1 pffiffiffiffiffiffi xeÀx =2 9kx 3k2 x2 ỵ k3 x3 ỵ dx xhxịdx ẳ c 2p d Z d ¼ pffiffiffiffiffiffi 9k ÀxeÀx =2 ỵ ex =2 dx c 2p c d d 2 x =2 ỵ3k x ỵ 2ịe ỵ k x2 þ 3ÞxeÀx =2 c c " d # Z d 2 eÀx =2 dx 7ex =2 ẳ p k9 ỵ k2 ị þ3 2p c c # Z k Á 40 À 12 À 92 À 2 x2 =2 ỵ e 2k 3k 16 : e dx À e 2k 5k À  k d 0.7 0.7 0.63 0.56 0.49 0.42 exact approximate g (x, ) 0.35 h (x, ) 0.28 0.21 0.14 0.07 3.5 − 3.5 Fig 2.8 2.1 1.4 0.7 x 0.7 1.4 2.1 2.8 The pdf for the skew normal distribution and its approximation for k = 3.5 3.5 344 S.K Ashour and M.A Abdel-hameed 0.7 0.7 0.63 0.56 0.49 0.42 g(x, ) exact approximate 0.35 h( x, 1) 0.28 0.21 0.14 0.07 3.5 2.8 2.1 1.4 0.7 − 3.5 1.4 2.1 2.8 3.5 3.5 The pdf for the skew normal distribution and its approximation for k = Fig k 0.7 x , k Step (4): For x, let d ¼ then we have: 1 rffiffiffi Z rffiffiffi 2 2 xhðxÞ dx ẳ xex =2 dx ẳ ex =2 ị p d p d d 1 rffiffiffi rffiffiffi 2 92 ẳ ex =2 ị ẳ e 2k : 3 p p Z Z c b k From (1), (2), (3) and (4) the first moment (mean) of the approximate skew normal will be " Z 3k Z 1k 2 eÀx =2 dx ỵ 2k3 k2 ị ex =2 dx l1 ẳ p k9 ỵ k2 ị 2p k i ỵe 2k2 3k2 ị þ e 2k2 ð3k2 Þ : Step (2): For b x < c, then we have: Z c 1 3 2 Àx2 =2 x hxị dx ẳ p xe 3kx k x ỵ dx 2p " b c ẳ p 3kx2 ỵ 2ịex =2 b 2p c 3 Àx2 =2 k x ỵ 4x ỵ 8ịe 3 b! Z c c Àx2 =2 Àx2 =2 þ4 Àxe e dx þ b b # Z 1k À 12 À 12 =2 x e dx e 2k ỵ e 2k ỵ k k 1k ! Z 1k À1 eÀx =2 dx À e 2k2 : ¼ pffiffiffiffiffiffi k 2p Àk Step (3): For c x < d, then we have: Theorem Let X be a continuous random variable with density function h(x) Then the second moment of the random variable X equal Z d c Proof To prove Theorem we have the following steps: Step (1): For a x < b, then we have: Z a b 9kx ỵ 3k2 x2 ỵ k3 x3 ỵ dx a b b 2 ẳ p 9kx2 ỵ 2ịex =2 ỵ 3k2 x2 ỵ 3ịxex =2 a a 2p Z b 2 b þ3 eÀx =2 dx þ k3 Àðx4 þ 4x2 þ 8ÞeÀx =2 a a ! Z b b 2 ỵ9 xex =2 ỵ ex =2 dx x2 hxịdx ẳ p 2p Z b =2 x2 eÀx a " ¼ pffiffiffiffiffiffi 9k2 ỵ 1ị 2p ỵe 2k2 a Z 1k 3k # k ỵ 3k eÀx =2 dx À e À 2k2 31 k ỵ k 3 3k Z d 1 x2 hxịdx ẳ p x2 ex =2 9kx 3k2 x2 þ k3 x3 þ dx 2p c d x2 =2 ẳ p 9kx ỵ 2Þe c 2p Z b d 2 x2 =2 ị ỵ ex =2 dx 3k x ỵ 3ịxe c a d ỵ k3 x4 ỵ 4x2 ỵ 8ịex =2 c ! d Z d Àx2 =2 x2 =2 e dx ỵ7 xe ỵ c c " Z 3k 31 40 k ỵ kỵ ẳ p 9k ị ex =2 dx ỵ e 2k2 3 3k 2p k # 48 ỵe 2k2 k3 3k À k Step (4): For d x, then we have: rffiffiffi Z 1 2 x hxị dx ẳ x2 ex =2 dx p d d rffiffiffih ! 1 Z 2 ẳ ex =2 dx xex =2 ỵ d p d # rffiffiffi" Z À 2 =2 x ẳ e dx e 2k ỵ p k k Z Approximate skew normal distribution 345 From (1), (2), (3) and (4) the second moment of the approximate skew normal will be: Z Z a Z b l02 ẳ Ex2 ị ẳ x2 hxịdx ẳ x2 hxịdx ỵ x2 hxịdx 1 a Z d Z Z c x2 hxịdx ỵ x2 hxịdx ỵ x2 hxịdx ỵ b c d " Z À1k 31 À l02 ¼ p 9k2 ỵ 1ị ex =2 dx e 2k2 k3 ỵ k 3 3k 2p 3k "Z # ! k À1 ex =2 dx e 2k2 ỵe 2k2 k3 ỵ 3k ỵ p k 2p 1k Table Comparison between the cumulative distribution function of the skew normal and its approximation Vale of k Maximum absolute errors 0.2 0.4 0.8 10 1.3499 · 10À3 2.8059 · 10À3 5.4105 · 10À3 6.3676 · 10À3 5.8788 · 10À3 7.0604 · 10À3 1.2062 · 10À3 0.0519 0.7 0.7 0.63 0.56 0.49 0.42 exact approximate g( x, 2) 0.35 h( x, 2) 0.28 0.21 0.14 0.07 3.5 2.8 2.1 1.4 0.7 − 3.5 Fig 0.7 1.4 2.1 2.8 3.5 x 3.5 The pdf for the skew normal distribution and its approximation for k = 0.7 0.7 0.63 0.56 0.49 0.42 exact approximate g( x, 3) 0.35 h( x, 3) 0.28 0.21 0.14 0.07 3.5 − 3.5 Fig 2.8 2.1 1.4 0.7 0.7 1.4 2.1 2.8 x The pdf for the skew normal distribution and its approximation for k = 3.5 3.5 346 Table S.K Ashour and M.A Abdel-hameed The approximate cdf of the skew normal distribution x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 k=0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 0.500 0.540 0.579 0.618 0.655 0.691 0.726 0.758 0.788 0.816 0.841 0.864 0.885 0.903 0.919 0.933 0.945 0.955 0.964 0.971 0.977 0.982 0.986 0.989 0.992 0.994 0.995 0.997 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.504 0.544 0.583 0.622 0.659 0.695 0.729 0.761 0.791 0.819 0.844 0.867 0.887 0.905 0.921 0.934 0.946 0.956 0.965 0.972 0.978 0.983 0.986 0.990 0.992 0.994 0.995 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.508 0.548 0.587 0.626 0.663 0.698 0.732 0.764 0.794 0.821 0.846 0.869 0.889 0.907 0.922 0.936 0.947 0.957 0.966 0.973 0.978 0.983 0.987 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.512 0.552 0.591 0.629 0.666 0.702 0.736 0.767 0.797 0.824 0.848 0.871 0.891 0.908 0.924 0.937 0.948 0.958 0.966 0.973 0.979 0.983 0.987 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.516 0.556 0.595 0.633 0.670 0.705 0.739 0.770 0.800 0.826 0.851 0.873 0.893 0.910 0.925 0.938 0.949 0.959 0.967 0.974 0.979 0.984 0.987 0.990 0.993 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.520 0.560 0.599 0.637 0.674 0.709 0.742 0.773 0.802 0.829 0.853 0.875 0.894 0.911 0.926 0.939 0.951 0.960 0.968 0.974 0.980 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.524 0.564 0.603 0.641 0.677 0.712 0.745 0.776 0.805 0.831 0.855 0.877 0.896 0.913 0.928 0.941 0.952 0.961 0.969 0.975 0.980 0.985 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.528 0.567 0.606 0.644 0.681 0.716 0.749 0.779 0.808 0.834 0.858 0.879 0.898 0.915 0.929 0.942 0.953 0.962 0.969 0.976 0.981 0.985 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.532 0.571 0.610 0.648 0.684 0.719 0.752 0.782 0.811 0.836 0.860 0.881 0.900 0.916 0.931 0.943 0.954 0.962 0.970 0.976 0.981 0.985 0.989 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.536 0.575 0.614 0.652 0.688 0.722 0.755 0.785 0.813 0.839 0.862 0.883 0.901 0.918 0.932 0.944 0.954 0.963 0.971 0.977 0.982 0.986 0.989 0.992 0.994 0.995 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 k = 0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 0.441 0.481 0.521 0.561 0.601 0.639 0.676 0.712 0.745 0.777 0.806 0.832 0.856 0.878 0.897 0.914 0.929 0.942 0.953 0.962 0.969 0.445 0.485 0.525 0.565 0.605 0.643 0.680 0.715 0.748 0.780 0.808 0.835 0.859 0.880 0.899 0.916 0.930 0.943 0.953 0.963 0.970 0.449 0.489 0.529 0.569 0.608 0.647 0.684 0.719 0.752 0.783 0.811 0.837 0.861 0.882 0.901 0.917 0.932 0.944 0.954 0.963 0.971 0.453 0.493 0.533 0.573 0.612 0.650 0.687 0.722 0.755 0.785 0.814 0.840 0.863 0.884 0.903 0.919 0.933 0.945 0.955 0.964 0.971 0.457 0.497 0.537 0.577 0.616 0.654 0.691 0.725 0.758 0.788 0.816 0.842 0.865 0.886 0.904 0.920 0.934 0.946 0.956 0.965 0.972 0.461 0.501 0.541 0.581 0.620 0.658 0.694 0.729 0.761 0.791 0.819 0.844 0.867 0.888 0.906 0.922 0.936 0.947 0.957 0.966 0.973 0.465 0.505 0.545 0.585 0.624 0.662 0.698 0.732 0.764 0.794 0.822 0.847 0.870 0.890 0.908 0.923 0.937 0.948 0.958 0.966 0.973 0.469 0.509 0.549 0.589 0.628 0.665 0.701 0.735 0.767 0.797 0.824 0.849 0.872 0.892 0.909 0.925 0.938 0.949 0.959 0.967 0.974 0.473 0.513 0.553 0.593 0.632 0.669 0.705 0.739 0.770 0.800 0.827 0.852 0.874 0.894 0.911 0.926 0.939 0.950 0.960 0.968 0.975 0.477 0.517 0.557 0.597 0.635 0.673 0.708 0.742 0.773 0.803 0.830 0.854 0.876 0.895 0.913 0.928 0.940 0.951 0.961 0.969 0.975 Approximate skew normal distribution 347 Table (continued ) k = 0.2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 0.976 0.981 0.985 0.989 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.976 0.981 0.986 0.989 0.991 0.994 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.977 0.982 0.986 0.989 0.992 0.994 0.995 0.997 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.977 0.982 0.986 0.989 0.992 0.994 0.995 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.978 0.983 0.987 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.978 0.983 0.987 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.979 0.984 0.987 0.990 0.993 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.979 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.980 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.980 0.985 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.385 0.425 0.466 0.508 0.549 0.590 0.630 0.668 0.705 0.740 0.773 0.803 0.830 0.855 0.877 0.897 0.915 0.930 0.943 0.954 0.963 0.970 0.977 0.982 0.986 0.989 0.992 0.994 0.995 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.389 0.429 0.470 0.512 0.553 0.594 0.634 0.672 0.709 0.743 0.776 0.806 0.833 0.858 0.880 0.899 0.916 0.931 0.944 0.955 0.964 0.971 0.977 0.982 0.986 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.393 0.433 0.474 0.516 0.557 0.598 0.638 0.676 0.712 0.747 0.779 0.808 0.835 0.860 0.882 0.901 0.918 0.932 0.945 0.956 0.964 0.972 0.978 0.983 0.987 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.397 0.437 0.479 0.520 0.562 0.602 0.642 0.680 0.716 0.750 0.782 0.811 0.838 0.862 0.884 0.903 0.919 0.934 0.946 0.956 0.965 0.972 0.978 0.983 0.987 0.990 0.993 0.994 0.996 0.998 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.401 0.441 0.483 0.524 0.566 0.606 0.646 0.683 0.719 0.753 0.785 0.814 0.841 0.864 0.886 0.904 0.921 0.935 0.947 0.957 0.966 0.973 0.979 0.984 0.987 0.990 0.993 0.995 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.405 0.446 0.487 0.528 0.570 0.610 0.649 0.687 0.723 0.757 0.788 0.817 0.843 0.867 0.888 0.906 0.922 0.936 0.948 0.958 0.967 0.974 0.979 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.409 0.450 0.491 0.533 0.574 0.614 0.653 0.691 0.726 0.760 0.791 0.820 0.845 0.869 0.890 0.908 0.924 0.938 0.949 0.959 0.968 0.974 0.980 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.413 0.454 0.495 0.537 0.578 0.618 0.657 0.694 0.730 0.763 0.794 0.822 0.848 0.871 0.892 0.910 0.925 0.939 0.950 0.960 0.968 0.975 0.980 0.985 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.417 0.458 0.499 0.541 0.582 0.622 0.661 0.698 0.733 0.766 0.797 0.825 0.850 0.873 0.894 0.911 0.927 0.940 0.952 0.961 0.969 0.976 0.981 0.985 0.989 0.991 0.994 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 (continued on next 0.421 0.462 0.504 0.545 0.586 0.626 0.665 0.702 0.737 0.769 0.800 0.828 0.853 0.875 0.895 0.913 0.928 0.941 0.953 0.962 0.970 0.976 0.981 0.986 0.989 0.992 0.994 0.995 0.997 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 page) k = 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 348 S.K Ashour and M.A Abdel-hameed Table (continued ) k = 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 0.292 0.333 0.376 0.420 0.465 0.511 0.556 0.601 0.644 0.685 0.724 0.760 0.794 0.824 0.852 0.876 0.897 0.916 0.931 0.945 0.956 0.965 0.973 0.979 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.296 0.337 0.380 0.425 0.470 0.515 0.561 0.605 0.648 0.689 0.728 0.764 0.797 0.827 0.854 0.878 0.899 0.917 0.933 0.946 0.957 0.966 0.973 0.979 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.300 0.341 0.384 0.429 0.474 0.520 0.565 0.610 0.652 0.693 0.732 0.767 0.800 0.830 0.857 0.880 0.901 0.919 0.934 0.947 0.958 0.967 0.974 0.980 0.985 0.988 0.991 0.994 0.995 0.997 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.304 0.345 0.389 0.434 0.479 0.525 0.570 0.614 0.656 0.697 0.735 0.771 0.803 0.833 0.859 0.883 0.903 0.921 0.936 0.948 0.959 0.968 0.975 0.981 0.985 0.989 0.992 0.994 0.995 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.308 0.350 0.393 0.438 0.484 0.529 0.574 0.618 0.661 0.701 0.739 0.774 0.806 0.836 0.862 0.885 0.905 0.922 0.937 0.950 0.960 0.968 0.975 0.981 0.985 0.989 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.312 0.354 0.398 0.443 0.488 0.534 0.579 0.623 0.665 0.705 0.743 0.777 0.809 0.838 0.864 0.887 0.907 0.924 0.938 0.951 0.961 0.969 0.976 0.982 0.986 0.989 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.316 0.358 0.402 0.447 0.493 0.538 0.583 0.627 0.669 0.709 0.746 0.781 0.812 0.841 0.867 0.889 0.909 0.925 0.940 0.952 0.962 0.970 0.977 0.982 0.986 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.320 0.363 0.407 0.452 0.497 0.543 0.588 0.631 0.673 0.713 0.750 0.784 0.815 0.844 0.869 0.891 0.910 0.927 0.941 0.953 0.963 0.971 0.977 0.982 0.987 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.324 0.367 0.411 0.456 0.502 0.547 0.592 0.635 0.677 0.716 0.753 0.787 0.818 0.846 0.871 0.893 0.912 0.928 0.942 0.954 0.964 0.971 0.978 0.983 0.987 0.990 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.328 0.371 0.416 0.461 0.506 0.552 0.596 0.640 0.681 0.720 0.757 0.791 0.821 0.849 0.874 0.895 0.914 0.930 0.944 0.955 0.964 0.972 0.978 0.983 0.987 0.990 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.256 0.297 0.341 0.387 0.434 0.482 0.530 0.577 0.623 0.667 0.709 0.748 0.784 0.816 0.845 0.871 0.893 0.913 0.929 0.943 0.260 0.301 0.346 0.392 0.439 0.487 0.535 0.582 0.628 0.672 0.713 0.752 0.787 0.819 0.848 0.873 0.895 0.914 0.931 0.945 0.264 0.306 0.350 0.396 0.444 0.492 0.539 0.587 0.632 0.676 0.717 0.755 0.790 0.822 0.850 0.876 0.897 0.916 0.932 0.946 0.268 0.310 0.355 0.401 0.448 0.496 0.544 0.591 0.637 0.680 0.721 0.759 0.794 0.825 0.853 0.878 0.899 0.918 0.934 0.947 0.272 0.314 0.359 0.406 0.453 0.501 0.549 0.596 0.641 0.684 0.725 0.763 0.797 0.828 0.856 0.880 0.901 0.920 0.935 0.948 0.276 0.319 0.364 0.410 0.458 0.506 0.554 0.600 0.646 0.688 0.729 0.766 0.800 0.831 0.858 0.882 0.903 0.921 0.937 0.949 0.280 0.323 0.368 0.415 0.463 0.511 0.558 0.605 0.650 0.693 0.733 0.770 0.803 0.834 0.861 0.885 0.905 0.923 0.938 0.950 0.285 0.328 0.373 0.420 0.468 0.516 0.563 0.610 0.654 0.697 0.736 0.773 0.807 0.837 0.863 0.887 0.907 0.925 0.939 0.952 0.289 0.332 0.378 0.425 0.472 0.520 0.568 0.614 0.659 0.701 0.740 0.777 0.810 0.840 0.866 0.889 0.909 0.926 0.941 0.953 0.293 0.337 0.382 0.429 0.477 0.525 0.573 0.619 0.663 0.705 0.744 0.780 0.813 0.842 0.868 0.891 0.911 0.928 0.942 0.954 k=1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Approximate skew normal distribution 349 Table (continued ) k=1 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 0.955 0.964 0.972 0.979 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.956 0.965 0.973 0.979 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.957 0.966 0.974 0.980 0.984 0.988 0.991 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.958 0.967 0.974 0.980 0.985 0.989 0.991 0.994 0.995 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 " Z 3k 31 40 ỵ p 9k2 ị ex =2 dx ỵ e 2k2 k3 ỵ k ỵ 3 3k 2p k # ! rffiffiffi" Z 48 À 92 À 92 Àx2 =2 2k 2k ỵ e ỵe ỵ k 3k e dx 3 k p k k "Z # Z Z k k 2 2 ex =2 dx ỵ ex =2 dx ỵ ex =2 dx ¼ pffiffiffiffiffiffi 2p 1k k rffiffiffi Z ! ! rffiffiffi p ffiffiffiffiffi ffi 2 ¼ eÀx =2 dx ¼ 2p ¼ à p p Theorem Let X be a continuous random variable with density function h(x) Then the variance of the random variable X will be: " Z Z k k k2 2 ỵ k2 ị ex =2 dx ỵ 23 k2 ị eÀx =2 dx 32p k #2 À À À3k e 2k2 À e 2k2 0.959 0.968 0.975 0.981 0.985 0.989 0.992 0.994 0.995 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.960 0.969 0.976 0.981 0.986 0.989 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.961 0.969 0.976 0.982 0.986 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.962 0.970 0.977 0.982 0.987 0.990 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.963 0.971 0.977 0.983 0.987 0.990 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.964 0.972 0.978 0.983 0.987 0.990 0.993 0.995 0.996 0.997 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 normal distribution varies appreciably with k only in the neighborhood of zero, while it increases, it is almost constant To compare the exact G(x) and the approximate cdf of the skew normal distribution H(x), a computer program using the Mathcad package was used to obtain the absolute difference between the exact probabilities G(x) and their approximated values H(x) for x = 0, 0.01, ,4.0 and different values for k (see Table 1) Table represents some values of the cdf of the approximate skew normal for different values of k From Table 1, we conclude that the maximum absolute error is not exceeding 0.0070604 when k ranges from to When k ranges from to 10, the maximum error increases from 0.0070604 to 0.0519 Conclusions Varxị ẳ where À1 < x < 1, À1 < k < Comparison between the exact and the approximate skew normal distribution In this section, we will compare the exact and approximate values of the pdf and cdf of the skew normal distribution using different values of the skew factor k numerically Figs 1–4 show the values of exact g(x) and their approximation h(x) for k = 0, 1, and 3, respectively From these figures we see that g(x) is very close to its approximation h(x); therefore, our approximation is very accurate Gupta and Chen [8] present a table of the cdf of the skew normal distribution for x = 0, 0.01, , 4.0, for different values of the skew factor k Their table shows that the cdf of the skew We conclude that new approximation is very close to the exact one, and applicable In future the present work may be extended to suggest another approximation, looking for the one with very small error and easy to use, the new suggestion needs a comprehensive numerical investigation References [1] Azzalini AA Class of distributions which includes the normal ones Scand J Stat 1985;12:171–8 [2] Azzalini A References on the skew-normal distribution and related ones Available from: [accessed 15.02.10] [3] Brown ND Reliability studies of the skew-normal distribution MSc Thesis, University of Maine, USA; 2001 [4] Chen JT, Gupta AK, Nguyen TT The density of the skewnormal sample mean and its applications J Stat Comput Simul 2003;74:487–94 [5] Gupta RC, Brown N Reliability studies of the skew-normal distribution and its application to strength-stress model Commun Stat Theory Methods 2001;30:2427–45 350 [6] Jammalizadeh A, Behboodoan J, Balakrishnan NA Twoparameter generalized skew-normal distribution Stat Probab Lett 2008;78:1722–6 [7] Pewsey A Problems of inference for Azzilini’s skew-normal distribution J Appl Stat 2002;27:859–70 S.K Ashour and M.A Abdel-hameed [8] Gupta AK, Chen T Goodness of fit test for the skew-normal population Commun Stat Comput Simul 2001;30:907–30 [9] Hoyt JP A simple approximation to the standard normal probability density function Am Stat 1986;22(2):25–6 ... the approximate skew normal distribution In this section, we will compare the exact and approximate values of the pdf and cdf of the skew normal distribution using different values of the skew. .. approximate pdf h(x) to the skew normal distribution by replacing the cdf of the standard normal in (1.1) by Hoyt’s approximation (2.1): the approximate function h(x) to the pdf of the skew normal. .. rffiffiffi" Z À 2 =2 x ẳ e dx e 2k ỵ p k k Z Approximate skew normal distribution 345 From (1), (2), (3) and (4) the second moment of the approximate skew normal will be: Z Z a Z b l02 ẳ Ex2 ị ẳ x2