- Email: [email protected]
Skewed distributions generated by the normal kernel
Statistics & Probability Letters 65 (2003) 269 – 277
Skewed distributions generated by the normal kernel Saralees Nadarajaha;∗ , Samuel Kotzb a
Department of Mathematics, University of South Florida, 4202 Fowler Ave., PHY 114, Tampa, FL 33620, USA b Department of Engineering Management and Systems Engineering, The George Washington University, Washington, DC 20052, USA Received January 2003; accepted July 2003
Abstract Following the recent paper by Gupta et al. (Some skew-symmetric models. Random Operators Stochastic Equations 10 (2002) 133) we generate skew probability density functions (pdfs) of the form 2f(u)G(u), where f is taken to be a normal pdf while the cumulative distributive function G is taken to come from one of normal, Student’s t, Cauchy, Laplace, logistic or uniform distribution. The properties of the resulting distributions are studied. In particular, expressions for the nth moment and the characteristic function are derived. We also provide graphical illustrations and quantify the range of possible values of skewness and kurtosis. c 2003 Elsevier B.V. All rights reserved.
1. Introduction Univariate skew-symmetric models have been considered by several authors. A classical example is the skew normal distribution with the probability density function (pdf) f(x) = 2 (x) (x) (where (·) and (·), respectively, denote the pdf and the cumulative distribution function (cdf) of the standard normal distribution). This distribution was introduced by Azzalini (1985). See Gupta et al. (2002) for a most detailed discussion of skew-symmetric models based on the normal, Student’s t, Cauchy, Laplace, logistic and uniform distributions. The main feature of these models is that a new parameter is introduced to control skewness and kurtosis. Thus, for example, the skew normal distribution allows for continuous variation from normality to non-normality, which is useful in many ∗
Corresponding author. Tel.: +1-813-974-9724; fax: +1-813-974-2700. E-mail address: [email protected] (S. Nadarajah).
c 2003 Elsevier B.V. All rights reserved. 0167-7152/$ - see front matter doi:10.1016/j.spl.2003.07.013
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practical situations (Hill and Dixon, 1982; Arnold et al., 1983). Skew-symmetric models have also been used in studying robustness and as priors in Bayesian estimation (O’Hagan and Leonard, 1976; Mukhopadhyay and Vidakovic, 1995). Lemma. Let U ;nd V be two arbitrary absolutely continuous independent random variables symmetric about 0, with pdfs f and g and cdfs F and G, respectively. Then for any ∈ R, the function fX (x) = 2f(x)G(x)
(1)
is a valid pdf of a random variable, say X. The construction of univariate skew-symmetric models is based on the above general result due to Azzalini (1985). For example, the skew normal distribution is obtained by taking f ≡ and G ≡ in (1). The models in Gupta et al. (2002) are obtained by taking both f and G to belong to one of normal, Student’s t, Cauchy, Laplace, logistic or uniform family. See also Balakrishnan and Ambagaspitiya (1994) and Arnold and Beaver (2000a,b) for similar constructions. Mukhopadhyay and Vidakovic (1995) pointed out an extension of the above approaches by suggesting that one takes f and G in (1) to belong to diJerent families. This idea has not been followed up by anyone (to the best of our knowledge). In this paper, we follow it up by taking f to be the pdf of a normal distribution with zero mean and variance 2 , but let g be normal, Student’s t, Cauchy, Laplace, logistic or uniform. Consequently, we have the following skewed models generated by a normal kernel: the skew normal–normal model (Section 2), the skew normal–t model (Section 3), the skew normal–Cauchy model (Section 4), the skew normal–Laplace model (Section 5), the skew normal–logistic model (Section 6) and the skew normal–uniform model (Section 7). We study moment properties (including characteristic function) of each of these models and provide graphical illustrations. We also quantify the range of possible values of skewness (1 ) and kurtosis (2 ) measures for each of the models. We assume without loss of generality that ¿ 0 in (1) since the corresponding properties for ¡ 0 can be obtained by using the fact G(x) = 1 − G(−x). Also note that the even-order moments of X are the same as those of U (this follows from Lemma 2 in Gupta et al., 2002). Since f ≡ , we see that n+1 2n=2 n n (2) E(X ) = √ 2 for n even. The calculations of this paper can be repeated by changing the kernel (the pdf f) to be either Student’s t, Cauchy, Laplace, logistic or uniform. These calculations will be the subject of a subsequent paper (we are unable to produce them here because of the limit to the number of pages). Besides the applications mentioned above, the model (1) can be motivated stochastically by one of the following representations (due to Azzalini (1986)): • X = SU U , where, conditionally on U = u, SU = +1 with probability G(u) and SU = −1 with probability 1 − G(u). • X = SU |U |, where, conditionally on |U | = |u|, SU = +1 with probability G(|u|) and SU = −1 with probability 1 − G(|u|).
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271
Both these representations have clear physical meanings. Model (1) can also be interpreted as the conditional pdf of U given U ¿ V , where U and V are two absolutely continuous independent random variables symmetric about 0, with pdfs f and g. Because of the limited space, we shall not give details of the derivations in this paper. Our calculations make use of the following special functions: the incomplete gamma function and the complementary incomplete gamma function deKned by x (a; x) = t a−1 exp (−t) dt; 0
and
(a; x) =
x
∞
t a−1 exp (−t) dt;
respectively; the beta function and the incomplete beta function ratio deKned by (a)(b) B(a; b) = (a + b) and 1 Ix (a; b) = B(a; b)
0
x
t a−1 (1 − t)b−1 dt;
respectively; and, the generalized hypergeometric function and the conLuent hypergeometric function deKned by ∞ (a1 )k (a2 )k · · · (ap )k xk p Fq (a1 ; : : : ; ap ; b1 ; : : : ; bq ; x) = (b1 )k (b2 )k · · · (bq )k k! k=0
and !(a; b; x) =
(b − 1) 1−b (1 − b) x 1 F1 (1 + a − b; 2 − b; x); 1 F1 (a; b; x) + (1 + a − b) (a)
respectively, where (c)k = c(c + 1) · · · (c + k − 1) denotes the ascending factorial. Details of the above special functions can be read from Prudnikov et al. (1990, Vol. 1). 2. Skew normal–normal model Take g to be a normal pdf with zero mean and variance #2 . Then (1) yields the pdf: x x2 2 (3) exp − 2
fX (x) = √ 2 # 2 for −∞ ¡ x ¡ ∞. This contains Azzalini’s (1985) skew normal distribution as a particular case for = # = 1. The moments and the characteristic function of X can be derived using properties in Section 2.8 of Prudnikov et al. (1990, vol. 2). The nth moment of X about zero turns out to be −n=2 (n 2 2 k −1)=2 n+2 2 2 ((1 − n)=2)k 2(n+2)=2 n+1 n 1+ 2 E(X ) = − 2 # 2 # (3=2)k # k=0
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if n is odd. In particular, the Krst and third moments are √ 22 2 2 1+ 2 E(X ) = √ # # and
√ 24 3#2 + 22 2 E(X ) = √ ; (#2 + 2 2 )3=2 3
respectively. We have already mentioned that E(X n ) for n even is given by (2). The characteristic function of X is 2 2 it 1 2 t
: + E[exp(itX )] = 2 exp − 2 # 2 #2
3. Skew normal–t model Take g to be the pdf of the Student’s t distribution, i.e. −(1+$)=2 ((1 + $)=2) x2 g(x) = √ 1+ ; −∞ ¡ x ¡ ∞: $ $($=2)
(4)
Consider the following properties of the incomplete beta function ratio: Ix (a; b) =
b (a + l − 1) l=1
(a)(l)
xa (1 − x)l−1
for integer b;
and Ix
1 1 ;j − 2 2
2 = arctan
j −1
x + cl ; 1−x l=1
where cl =
(l) x1=2 (1 − x)l−1=2 : (1=2)(l + 1=2)
Using these properties, it can be shown that the cdf corresponding to (4) is ($−1)=2 x 1 (l)$l−1=2 1 1 x + √ G(x) = + arctan √ 2 2 l=1 (l + 1=2) ($ + x2 )l $
(5)
if $ is odd, and G(x) =
$=2 x 1 1 (l − 1=2)$l−1 + √ 2 2 (l) ($ + x2 )l−1=2 l=1
(6)
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if $ is even. Substituting (5) and (6) into (1), we obtain the pdf of X for the skewed normal–t model. The corresponding moments follow from properties in Sections 2.3 and 2.7 of Prudnikov et al. (1990, Vol. 1): √ (n+1)=2 $ n+1 n+3 2$ =cos((n + 1)=2) n+1 2n=2 n n √ − ; ; E(X ) = √ 1 F1 2 2 2 22 2 (n + 1)n+1 √ 3 2−n $ 1 2(n+1)=2 n−1 $ n
; 1; ; ; 2 2 − 2 F2 3=2 2 2 2 2 2 +
($−1)=2 n $(n+1)=2 (n=2 + 1) (l) n $
√ ! + 1; − l + 2; 2 2 ; (l + 1=2) 2 2 2 2n+1 l=1
if n is odd and $ is odd, and $=2
$(n+1)=2 ((n + 1)=2) (l − 1=2) √ E(X ) = ! (l) 2n+1 n
l=1
n n 5 $ + 1; − l + ; 2 2 2 2 2 2
if n is odd and $ is even. In particular, the Krst moment is √ √ ($−1)=2 $ √$ 3 5 $ 2$ (l) 2 2 + √ ! ; − l; 2 2
− E(X ) = √ exp 22 2 2 2 2 4 2 l=1 (l + 1=2) if $ is odd and
$=2 3 $ (l − 1=2) $ ! ; 3 − l; 2 2 E(X ) = √ (l) 2 2 22 l=1
if $ is even 4. Skew normal–Cauchy model If g is taken to be the Cauchy pdf 2 −1 x 1 1+ g(x) = ; −∞ ¡ x ¡ ∞ then from (1) we obtain the skew normal–Cauchy model for X . The pdf of X becomes √ x 2 1 1 x2 + arctan fX (x) = √ exp − 2 2 2
(7)
for −∞ ¡ x ¡ ∞. Using properties in Section 2.7 of Prudnikov et al. (1990, Vol. 1), one can obtain √ (n+1) n + 1 n + 3 2 n+1 2n=2 n 2 =cos((n + 1)=2) n √ − ; ; 2 2 E(X ) = √ 1 F1 2 2 2 2 (n + 1)n+1 3 2 − n 2 1 2(n+1)=2 n−1 n
; 1; ; ; 2 2 F − 2 2 3=2 2 2 2 2 2
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If n is odd. In particular, the Krst and the third moments are √ 2
2 2
− E(X ) = √ exp 22 2 and
√
2 i 1 2 2 23 2
−1 : − exp 1− 2 2 + √ E(X ) = √ 2 22 2 i 2 3
5. Skew normal–Laplace model If g is the pdf of a Laplace distribution given by 1=(2 ) exp(x= ) if x 6 0; g(x) = 1=(2 ) exp(−x= ) if x ¿ 0 then substituting into (1) we obtain the skew normal–Laplace distribution for X . The pdf of X is: √ if x 6 0; 1=( 2) exp{x= − x2 =(22 )} (8) fX (x) = √ 1=( 2)[2 − exp(−x= )] exp{−x2 =(22 )} if x ¿ 0: Using properties in Section 2.3 of Prudnikov et al. (1990, Vol. 1), 2 2 @n q n+1 2n=2 n + n 2 exp
(−q) E(X n ) = √ 2 @q 2 q== if n is odd. In particular, the Krst and the third moments are 2 2 22 exp
− E(X ) = 2 2 and
√ 2 5 2 2 2 2 23 6 2 64 exp + :
−
− exp E(X ) = − √ 2 + 2 2 3 2 2 3
Using properties in Section 2.5 of Prudnikov et al. (1990, Vol. 1), the characteristic function of X can be worked out as 2 2 2 2 t
(it) − exp + it + it
− E[exp(itX )] = 2 exp − 2 2 2 2 − it − it :
− + exp 2
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275
6. Skew normal–logistic model If g denotes the pdf of a logistic distribution exp(x=() 1 ; −∞ ¡ x ¡ ∞ g(x) = ( {1 + exp(x=()}2 then from (1) we obtain the skew normal–logistic distribution for X . The pdf of X is given by √ x x2 2 √ (9) 1 + exp − exp − 2 fX (x) = 2 ( for −∞ ¡ x ¡ ∞. Using the Taylor series expansion for (1 + z)−1 , one can obtain the following series representations for (9): √ ∞ 2 x2 −1 jx fX (x) = √ exp − 2 exp − 2 ( j j=0 for x ¿ 0, and
√ ∞ x 2 x2 −1 jx fX (x) = √ exp − 2 exp ( 2 ( j j=0
for x ¡ 0. Using these representations and properties in Sections 2.3 of Prudnikov et al. (1990, Vol. 1), one can obtain
2 2 ∞ −1 q @n n
(−q) E(X ) = 2 exp n @q 2 j j=0 q=(1+j)=(
2 2 @n q
(−q) + (−1)n n exp @q 2 q=j=( if n is odd. In particular, the expectation is 2 2 2 ∞ 2 −1 j 2 j
− j E(X ) = − exp ( 2(2 ( j j=0 −
∞ j=0
(j + 1)
−1 j
exp
2 2
(j + 1) 2(2
2
−
(j + 1) : (
The characteristic function of X is 2 ∞ −1 j 2 j − it − it
− E[exp(itX )] = 2 exp 2 ( ( j j=0 2 (j + 1) 2 (j + 1) + it + it :
− + exp 2 ( (
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7. Skew normal–uniform model Taking g to be the pdf of a uniform distribution on [ − h; h], we obtain the skew normal–uniform model given by the pdf fX (x) = 0
(10)
if x 6 − h, by
x2 x + h exp − 2 fX (x) = √ 2 2h
(11)
if −h 6 x 6 h, and by
x2 2 exp − 2 fX (x) = √ 2 2
(12)
if x ¿ h. Directly using the deKnitions of the incomplete gamma function and the complementary incomplete gamma function, h2 2n=2 n n n + 1 h2 2(n+1)=2 n+1 n √ + 1; 2 2 + √ ; E(X ) = 2 2 2 22 2 h if n is odd. In particular, the Krst and the third moments are
h 2 2
−1 E(X ) = h and
√ 3
h 34 2 h2 2
−1 : E(X ) = − √ exp − 2 2 + 2 h 3
8. Discussion Fig. 1 below illustrates the possible shapes of the pdf (10)–(12) for the skew normal–uniform model. The table below summarizes the range of possible values of the skewness and kurtosis measures for all the six models considered. Model
Range of skewness (1 )
Range of kurtosis (2 )
Normal–normal Normal–t Normal–Cauchy Normal–Laplace Normal–logistic Normal–uniform
(−0:995; 0:995) (−0:995; 0:995) (−0:995; 0:995) (−0:995; 0:995) (−0:995; 0:995) (−0:995; 0:995)
(3, 3.869) (3, 4.124) (3, 4.124) (3, 3.869) (3, 3.869) (2.817, 3.869)
S. Nadarajah, S. Kotz / Statistics & Probability Letters 65 (2003) 269 – 277 0.7
λ λ λ λ
0.6
= = = =
277
0.5 1 2 10
PDF
0.5 0.4 0.3 0.2 0.1 0.0
−2
−1
0
1
2
x
Fig. 1. The skew normal–uniform pdf (10)–(12) for = 0:5; 1; 2; 10, = 1 and h = 5.
Clearly each model exhibits both positive and negative skewness with −0:995 6 1 6 0:995. The amount of peakedness is in general sharper than that of the normal pdf. The only exception is the normal–uniform model, where for some values of around 0 the pdf is more Lat topped than the normal pdf (this is probably due to ‘discreteness’ of the uniform distribution). The two models associated with polynomial tails (the normal–t and the normal–Cauchy) appear to attain a higher degree of sharpness than the remaining models. If model (1) is extended to include a second parameter–in the manner suggested by Azzalini (1986)—then wider range of values for skewness and kurtosis will be realized. Acknowledgements The authors are grateful to the editor and to a referee for their most useful comments and suggestions. References Arnold, B.C., Beaver, R.J., 2000a. Some skewed multivariate distributions. Amer. J. Math. Management Sci. 20, 27–38. Arnold, B.C., Beaver, R.J., 2000b. The skew-Cauchy distribution. Statist. Probab. Lett. 49, 285–290. Arnold, B.C., Beaver, R.J., Groeneveld, R.A., Meeker, W.Q., 1983. The nontruncated marginal of a truncated bivariate normal distribution. Psychometrika 58, 471–488. Azzalini, A., 1985. A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178. Azzalini, A., 1986. Further results on a class of distributions which includes the normal ones. Statistica 46, 199–208. Balakrishnan, N., Ambagaspitiya, R.S., 1994. On skew-Laplace distributions. Technical Report, Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada. Gupta, A.K., Chang, F.C., Huang, W.J., 2002. Some skew-symmetric models. Random Operators Stochastic Equations 10, 133–140. Hill, M.A., Dixon, W.J., 1982. Robustness in real life: a study of clinical laboratory data. Biometrics 38, 377–396. Mukhopadhyay, S., Vidakovic, B., 1995. ESciency of linear Bayes rules for a normal mean: skewed priors class. The Statistician 44, 389–397. O’Hagan, A., Leonard, T., 1976. Bayes estimation subject to uncertainty about parameter constraints. Biometrika 63, 201–203. Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I., 1990. Integrals and Series, Vols. 1–3. Gordon and Breach Science Publishers, Amsterdam.
Skewed distributions generated by the normal kernel Saralees Nadarajaha;∗ , Samuel Kotzb a
Department of Mathematics, University of South Florida, 4202 Fowler Ave., PHY 114, Tampa, FL 33620, USA b Department of Engineering Management and Systems Engineering, The George Washington University, Washington, DC 20052, USA Received January 2003; accepted July 2003
Abstract Following the recent paper by Gupta et al. (Some skew-symmetric models. Random Operators Stochastic Equations 10 (2002) 133) we generate skew probability density functions (pdfs) of the form 2f(u)G(u), where f is taken to be a normal pdf while the cumulative distributive function G is taken to come from one of normal, Student’s t, Cauchy, Laplace, logistic or uniform distribution. The properties of the resulting distributions are studied. In particular, expressions for the nth moment and the characteristic function are derived. We also provide graphical illustrations and quantify the range of possible values of skewness and kurtosis. c 2003 Elsevier B.V. All rights reserved.
1. Introduction Univariate skew-symmetric models have been considered by several authors. A classical example is the skew normal distribution with the probability density function (pdf) f(x) = 2 (x) (x) (where (·) and (·), respectively, denote the pdf and the cumulative distribution function (cdf) of the standard normal distribution). This distribution was introduced by Azzalini (1985). See Gupta et al. (2002) for a most detailed discussion of skew-symmetric models based on the normal, Student’s t, Cauchy, Laplace, logistic and uniform distributions. The main feature of these models is that a new parameter is introduced to control skewness and kurtosis. Thus, for example, the skew normal distribution allows for continuous variation from normality to non-normality, which is useful in many ∗
Corresponding author. Tel.: +1-813-974-9724; fax: +1-813-974-2700. E-mail address: [email protected] (S. Nadarajah).
c 2003 Elsevier B.V. All rights reserved. 0167-7152/$ - see front matter doi:10.1016/j.spl.2003.07.013
270
S. Nadarajah, S. Kotz / Statistics & Probability Letters 65 (2003) 269 – 277
practical situations (Hill and Dixon, 1982; Arnold et al., 1983). Skew-symmetric models have also been used in studying robustness and as priors in Bayesian estimation (O’Hagan and Leonard, 1976; Mukhopadhyay and Vidakovic, 1995). Lemma. Let U ;nd V be two arbitrary absolutely continuous independent random variables symmetric about 0, with pdfs f and g and cdfs F and G, respectively. Then for any ∈ R, the function fX (x) = 2f(x)G(x)
(1)
is a valid pdf of a random variable, say X. The construction of univariate skew-symmetric models is based on the above general result due to Azzalini (1985). For example, the skew normal distribution is obtained by taking f ≡ and G ≡ in (1). The models in Gupta et al. (2002) are obtained by taking both f and G to belong to one of normal, Student’s t, Cauchy, Laplace, logistic or uniform family. See also Balakrishnan and Ambagaspitiya (1994) and Arnold and Beaver (2000a,b) for similar constructions. Mukhopadhyay and Vidakovic (1995) pointed out an extension of the above approaches by suggesting that one takes f and G in (1) to belong to diJerent families. This idea has not been followed up by anyone (to the best of our knowledge). In this paper, we follow it up by taking f to be the pdf of a normal distribution with zero mean and variance 2 , but let g be normal, Student’s t, Cauchy, Laplace, logistic or uniform. Consequently, we have the following skewed models generated by a normal kernel: the skew normal–normal model (Section 2), the skew normal–t model (Section 3), the skew normal–Cauchy model (Section 4), the skew normal–Laplace model (Section 5), the skew normal–logistic model (Section 6) and the skew normal–uniform model (Section 7). We study moment properties (including characteristic function) of each of these models and provide graphical illustrations. We also quantify the range of possible values of skewness (1 ) and kurtosis (2 ) measures for each of the models. We assume without loss of generality that ¿ 0 in (1) since the corresponding properties for ¡ 0 can be obtained by using the fact G(x) = 1 − G(−x). Also note that the even-order moments of X are the same as those of U (this follows from Lemma 2 in Gupta et al., 2002). Since f ≡ , we see that n+1 2n=2 n n (2) E(X ) = √ 2 for n even. The calculations of this paper can be repeated by changing the kernel (the pdf f) to be either Student’s t, Cauchy, Laplace, logistic or uniform. These calculations will be the subject of a subsequent paper (we are unable to produce them here because of the limit to the number of pages). Besides the applications mentioned above, the model (1) can be motivated stochastically by one of the following representations (due to Azzalini (1986)): • X = SU U , where, conditionally on U = u, SU = +1 with probability G(u) and SU = −1 with probability 1 − G(u). • X = SU |U |, where, conditionally on |U | = |u|, SU = +1 with probability G(|u|) and SU = −1 with probability 1 − G(|u|).
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271
Both these representations have clear physical meanings. Model (1) can also be interpreted as the conditional pdf of U given U ¿ V , where U and V are two absolutely continuous independent random variables symmetric about 0, with pdfs f and g. Because of the limited space, we shall not give details of the derivations in this paper. Our calculations make use of the following special functions: the incomplete gamma function and the complementary incomplete gamma function deKned by x (a; x) = t a−1 exp (−t) dt; 0
and
(a; x) =
x
∞
t a−1 exp (−t) dt;
respectively; the beta function and the incomplete beta function ratio deKned by (a)(b) B(a; b) = (a + b) and 1 Ix (a; b) = B(a; b)
0
x
t a−1 (1 − t)b−1 dt;
respectively; and, the generalized hypergeometric function and the conLuent hypergeometric function deKned by ∞ (a1 )k (a2 )k · · · (ap )k xk p Fq (a1 ; : : : ; ap ; b1 ; : : : ; bq ; x) = (b1 )k (b2 )k · · · (bq )k k! k=0
and !(a; b; x) =
(b − 1) 1−b (1 − b) x 1 F1 (1 + a − b; 2 − b; x); 1 F1 (a; b; x) + (1 + a − b) (a)
respectively, where (c)k = c(c + 1) · · · (c + k − 1) denotes the ascending factorial. Details of the above special functions can be read from Prudnikov et al. (1990, Vol. 1). 2. Skew normal–normal model Take g to be a normal pdf with zero mean and variance #2 . Then (1) yields the pdf: x x2 2 (3) exp − 2
fX (x) = √ 2 # 2 for −∞ ¡ x ¡ ∞. This contains Azzalini’s (1985) skew normal distribution as a particular case for = # = 1. The moments and the characteristic function of X can be derived using properties in Section 2.8 of Prudnikov et al. (1990, vol. 2). The nth moment of X about zero turns out to be −n=2 (n 2 2 k −1)=2 n+2 2 2 ((1 − n)=2)k 2(n+2)=2 n+1 n 1+ 2 E(X ) = − 2 # 2 # (3=2)k # k=0
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if n is odd. In particular, the Krst and third moments are √ 22 2 2 1+ 2 E(X ) = √ # # and
√ 24 3#2 + 22 2 E(X ) = √ ; (#2 + 2 2 )3=2 3
respectively. We have already mentioned that E(X n ) for n even is given by (2). The characteristic function of X is 2 2 it 1 2 t
: + E[exp(itX )] = 2 exp − 2 # 2 #2
3. Skew normal–t model Take g to be the pdf of the Student’s t distribution, i.e. −(1+$)=2 ((1 + $)=2) x2 g(x) = √ 1+ ; −∞ ¡ x ¡ ∞: $ $($=2)
(4)
Consider the following properties of the incomplete beta function ratio: Ix (a; b) =
b (a + l − 1) l=1
(a)(l)
xa (1 − x)l−1
for integer b;
and Ix
1 1 ;j − 2 2
2 = arctan
j −1
x + cl ; 1−x l=1
where cl =
(l) x1=2 (1 − x)l−1=2 : (1=2)(l + 1=2)
Using these properties, it can be shown that the cdf corresponding to (4) is ($−1)=2 x 1 (l)$l−1=2 1 1 x + √ G(x) = + arctan √ 2 2 l=1 (l + 1=2) ($ + x2 )l $
(5)
if $ is odd, and G(x) =
$=2 x 1 1 (l − 1=2)$l−1 + √ 2 2 (l) ($ + x2 )l−1=2 l=1
(6)
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273
if $ is even. Substituting (5) and (6) into (1), we obtain the pdf of X for the skewed normal–t model. The corresponding moments follow from properties in Sections 2.3 and 2.7 of Prudnikov et al. (1990, Vol. 1): √ (n+1)=2 $ n+1 n+3 2$ =cos((n + 1)=2) n+1 2n=2 n n √ − ; ; E(X ) = √ 1 F1 2 2 2 22 2 (n + 1)n+1 √ 3 2−n $ 1 2(n+1)=2 n−1 $ n
; 1; ; ; 2 2 − 2 F2 3=2 2 2 2 2 2 +
($−1)=2 n $(n+1)=2 (n=2 + 1) (l) n $
√ ! + 1; − l + 2; 2 2 ; (l + 1=2) 2 2 2 2n+1 l=1
if n is odd and $ is odd, and $=2
$(n+1)=2 ((n + 1)=2) (l − 1=2) √ E(X ) = ! (l) 2n+1 n
l=1
n n 5 $ + 1; − l + ; 2 2 2 2 2 2
if n is odd and $ is even. In particular, the Krst moment is √ √ ($−1)=2 $ √$ 3 5 $ 2$ (l) 2 2 + √ ! ; − l; 2 2
− E(X ) = √ exp 22 2 2 2 2 4 2 l=1 (l + 1=2) if $ is odd and
$=2 3 $ (l − 1=2) $ ! ; 3 − l; 2 2 E(X ) = √ (l) 2 2 22 l=1
if $ is even 4. Skew normal–Cauchy model If g is taken to be the Cauchy pdf 2 −1 x 1 1+ g(x) = ; −∞ ¡ x ¡ ∞ then from (1) we obtain the skew normal–Cauchy model for X . The pdf of X becomes √ x 2 1 1 x2 + arctan fX (x) = √ exp − 2 2 2
(7)
for −∞ ¡ x ¡ ∞. Using properties in Section 2.7 of Prudnikov et al. (1990, Vol. 1), one can obtain √ (n+1) n + 1 n + 3 2 n+1 2n=2 n 2 =cos((n + 1)=2) n √ − ; ; 2 2 E(X ) = √ 1 F1 2 2 2 2 (n + 1)n+1 3 2 − n 2 1 2(n+1)=2 n−1 n
; 1; ; ; 2 2 F − 2 2 3=2 2 2 2 2 2
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If n is odd. In particular, the Krst and the third moments are √ 2
2 2
− E(X ) = √ exp 22 2 and
√
2 i 1 2 2 23 2
−1 : − exp 1− 2 2 + √ E(X ) = √ 2 22 2 i 2 3
5. Skew normal–Laplace model If g is the pdf of a Laplace distribution given by 1=(2 ) exp(x= ) if x 6 0; g(x) = 1=(2 ) exp(−x= ) if x ¿ 0 then substituting into (1) we obtain the skew normal–Laplace distribution for X . The pdf of X is: √ if x 6 0; 1=( 2) exp{x= − x2 =(22 )} (8) fX (x) = √ 1=( 2)[2 − exp(−x= )] exp{−x2 =(22 )} if x ¿ 0: Using properties in Section 2.3 of Prudnikov et al. (1990, Vol. 1), 2 2 @n q n+1 2n=2 n + n 2 exp
(−q) E(X n ) = √ 2 @q 2 q== if n is odd. In particular, the Krst and the third moments are 2 2 22 exp
− E(X ) = 2 2 and
√ 2 5 2 2 2 2 23 6 2 64 exp + :
−
− exp E(X ) = − √ 2 + 2 2 3 2 2 3
Using properties in Section 2.5 of Prudnikov et al. (1990, Vol. 1), the characteristic function of X can be worked out as 2 2 2 2 t
(it) − exp + it + it
− E[exp(itX )] = 2 exp − 2 2 2 2 − it − it :
− + exp 2
S. Nadarajah, S. Kotz / Statistics & Probability Letters 65 (2003) 269 – 277
275
6. Skew normal–logistic model If g denotes the pdf of a logistic distribution exp(x=() 1 ; −∞ ¡ x ¡ ∞ g(x) = ( {1 + exp(x=()}2 then from (1) we obtain the skew normal–logistic distribution for X . The pdf of X is given by √ x x2 2 √ (9) 1 + exp − exp − 2 fX (x) = 2 ( for −∞ ¡ x ¡ ∞. Using the Taylor series expansion for (1 + z)−1 , one can obtain the following series representations for (9): √ ∞ 2 x2 −1 jx fX (x) = √ exp − 2 exp − 2 ( j j=0 for x ¿ 0, and
√ ∞ x 2 x2 −1 jx fX (x) = √ exp − 2 exp ( 2 ( j j=0
for x ¡ 0. Using these representations and properties in Sections 2.3 of Prudnikov et al. (1990, Vol. 1), one can obtain
2 2 ∞ −1 q @n n
(−q) E(X ) = 2 exp n @q 2 j j=0 q=(1+j)=(
2 2 @n q
(−q) + (−1)n n exp @q 2 q=j=( if n is odd. In particular, the expectation is 2 2 2 ∞ 2 −1 j 2 j
− j E(X ) = − exp ( 2(2 ( j j=0 −
∞ j=0
(j + 1)
−1 j
exp
2 2
(j + 1) 2(2
2
−
(j + 1) : (
The characteristic function of X is 2 ∞ −1 j 2 j − it − it
− E[exp(itX )] = 2 exp 2 ( ( j j=0 2 (j + 1) 2 (j + 1) + it + it :
− + exp 2 ( (
276
S. Nadarajah, S. Kotz / Statistics & Probability Letters 65 (2003) 269 – 277
7. Skew normal–uniform model Taking g to be the pdf of a uniform distribution on [ − h; h], we obtain the skew normal–uniform model given by the pdf fX (x) = 0
(10)
if x 6 − h, by
x2 x + h exp − 2 fX (x) = √ 2 2h
(11)
if −h 6 x 6 h, and by
x2 2 exp − 2 fX (x) = √ 2 2
(12)
if x ¿ h. Directly using the deKnitions of the incomplete gamma function and the complementary incomplete gamma function, h2 2n=2 n n n + 1 h2 2(n+1)=2 n+1 n √ + 1; 2 2 + √ ; E(X ) = 2 2 2 22 2 h if n is odd. In particular, the Krst and the third moments are
h 2 2
−1 E(X ) = h and
√ 3
h 34 2 h2 2
−1 : E(X ) = − √ exp − 2 2 + 2 h 3
8. Discussion Fig. 1 below illustrates the possible shapes of the pdf (10)–(12) for the skew normal–uniform model. The table below summarizes the range of possible values of the skewness and kurtosis measures for all the six models considered. Model
Range of skewness (1 )
Range of kurtosis (2 )
Normal–normal Normal–t Normal–Cauchy Normal–Laplace Normal–logistic Normal–uniform
(−0:995; 0:995) (−0:995; 0:995) (−0:995; 0:995) (−0:995; 0:995) (−0:995; 0:995) (−0:995; 0:995)
(3, 3.869) (3, 4.124) (3, 4.124) (3, 3.869) (3, 3.869) (2.817, 3.869)
S. Nadarajah, S. Kotz / Statistics & Probability Letters 65 (2003) 269 – 277 0.7
λ λ λ λ
0.6
= = = =
277
0.5 1 2 10
0.5 0.4 0.3 0.2 0.1 0.0
−2
−1
0
1
2
x
Fig. 1. The skew normal–uniform pdf (10)–(12) for = 0:5; 1; 2; 10, = 1 and h = 5.
Clearly each model exhibits both positive and negative skewness with −0:995 6 1 6 0:995. The amount of peakedness is in general sharper than that of the normal pdf. The only exception is the normal–uniform model, where for some values of around 0 the pdf is more Lat topped than the normal pdf (this is probably due to ‘discreteness’ of the uniform distribution). The two models associated with polynomial tails (the normal–t and the normal–Cauchy) appear to attain a higher degree of sharpness than the remaining models. If model (1) is extended to include a second parameter–in the manner suggested by Azzalini (1986)—then wider range of values for skewness and kurtosis will be realized. Acknowledgements The authors are grateful to the editor and to a referee for their most useful comments and suggestions. References Arnold, B.C., Beaver, R.J., 2000a. Some skewed multivariate distributions. Amer. J. Math. Management Sci. 20, 27–38. Arnold, B.C., Beaver, R.J., 2000b. The skew-Cauchy distribution. Statist. Probab. Lett. 49, 285–290. Arnold, B.C., Beaver, R.J., Groeneveld, R.A., Meeker, W.Q., 1983. The nontruncated marginal of a truncated bivariate normal distribution. Psychometrika 58, 471–488. Azzalini, A., 1985. A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171–178. Azzalini, A., 1986. Further results on a class of distributions which includes the normal ones. Statistica 46, 199–208. Balakrishnan, N., Ambagaspitiya, R.S., 1994. On skew-Laplace distributions. Technical Report, Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada. Gupta, A.K., Chang, F.C., Huang, W.J., 2002. Some skew-symmetric models. Random Operators Stochastic Equations 10, 133–140. Hill, M.A., Dixon, W.J., 1982. Robustness in real life: a study of clinical laboratory data. Biometrics 38, 377–396. Mukhopadhyay, S., Vidakovic, B., 1995. ESciency of linear Bayes rules for a normal mean: skewed priors class. The Statistician 44, 389–397. O’Hagan, A., Leonard, T., 1976. Bayes estimation subject to uncertainty about parameter constraints. Biometrika 63, 201–203. Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I., 1990. Integrals and Series, Vols. 1–3. Gordon and Breach Science Publishers, Amsterdam.