Statistics and Probability Letters 81 (2011) 1813–1821
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On a generalized mixture of standard normal and skew normal distributions C. Satheesh Kumar ∗ , M.R. Anusree Department of Statistics, University of Kerala, Trivandrum-695 581, India
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info
Article history: Received 17 February 2011 Received in revised form 6 July 2011 Accepted 6 July 2011 Available online 14 July 2011 Keywords: Characteristic function Method of maximum likelihood Skew normal distribution Plurimodality
abstract Here we propose a new class of distributions as a generalized mixture of standard normal and skew normal distributions (GMNSND) and study some of its properties by deriving its characteristic function, mean, variance, coefficient of skewness etc. Further, certain reliability aspects of GMNSND are studied and a location scale extension of GMNSND is considered. The estimation of the parameters of this extended GMNSND by the method of maximum likelihood is discussed. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The normal distribution is the basis of many statistical work and has a unique position in probability theory. It is an unavoidable tool for the analysis and interpretation of basic data. It can be noted that the unrestricted usage of normal distribution to model a data in many real life applications leads to an error in the result. This may be due to the effect of certain unknown variables giving rise to skewness in the data. To overcome this difficulty, Azzalini (1985) introduced a generalized version of normal distribution, namely skew normal distribution and this distribution has been studied by several authors such as Azzalini (1986), Henze (1986), Azzalini and Dalla-Valle (1996) and Branco and Dey (2001). Azzalini (1985) defined skew normal distribution as in the following. A random variable Z is said to have skew normal distribution with parameter λ ∈ R = (−∞, ∞) if its probability density function (p.d.f.) g (z ; λ) is of the following form. For z ∈ R, g (z ; λ) = 2f (z )F (λz )
(1.1)
where f (.) and F (.) are, respectively, the p.d.f. and cumulative distribution function (c.d.f.) of a standard normal variate. Hereafter, we denote a distribution with p.d.f. (1.1) as SND(λ). The density given by (1.1) is appropriate for the data exhibiting unimodal density with some skewness present in it. Also it has certain formal properties which hold for normal distribution. Buccianti (2005) remarked that normal and skew normal models are not adequate to describe the situations of plurimodality. He investigated the shape of the frequency distribution of the logratio ln(cl− /Na+ ) whose components are related to water composition for 26 wells. Samples have been collected around the active center of Vulcano Island from 1977. Data of the logratio have been tentatively modeled by evaluating the performance of the skew normal model for each well. Value of λ for the wells of Vulcano Island appear to cover a wide range corresponding to (1) a more or less good symmetry, (2) the presence of a moderate skewness, (3) the presence of plurimodality. For the first and second situation, he noted
∗
Corresponding author. Tel.: +91 04712418905. E-mail addresses:
[email protected] (C. Satheesh Kumar),
[email protected] (M.R. Anusree).
0167-7152/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2011.07.009
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C. Satheesh Kumar, M.R. Anusree / Statistics and Probability Letters 81 (2011) 1813–1821
that normal or skew normal model is better and for the third case these models are not adequate. To accommodate such plurimodal situations, we modify the skew normal density and the distribution corresponding to this modified density is termed the generalized mixture of standard normal and skew normal distributions (GMNSND). The organization of this paper is as follows. In Section 2 we present the definition of the GMNSND and derive some of its important properties. In Section 3 we obtain expression for certain reliability measures such as reliability function, failure rate and mean residual life function. A location scale extension of the GMNSND is considered in Section 4 and derived explicit expressions for its characteristic function, mean, variance, measure of skewness etc. Further the parameters of this extended family are estimated by method of maximum likelihood. 2. The generalized mixture of standard normal and skew normal distributions Here we define a new class of generalized skew normal distribution as a generalized mixture of standard normal and skew normal distributions and derive some of its important properties. Definition 2.1. A random variable Z is said to have a generalized skew normal distribution if its p.d.f. is of the following form, in which z ∈ R, λ ∈ R and α > −2. h(z ; λ, α) =
2
α+2
f (z )[1 + α F (λz )].
(2.1)
A distribution with p.d.f. (2.1) we denoted as GMNSN(α, λ). Note that (1) when α = 0 and/or when λ = 0GMNSN(α, λ) reduces to standard normal distribution, (2) when α → ∞, GMNSN(α, λ) tends to SND(λ), (3) when α = −1, GMNSN(α, λ) reduces to SND(−λ) and (4) GMNSN (−λ, −1) is SND(λ). For some other choices of α and λ the p.d.f. given in (2.1) of GMNSND is plotted and the figures obtained are presented in Appendix. Now we have the following results. Result 2.1. If Z has GMNSN(α, λ), then Y1 = −Z has GMNSN(α, −λ). Proof. The p.d.f. h1 (y1 ) of Y1 is the following, for y1 ∈ R, λ ∈ R and α > −2.
dz h1 (y1 ) = h(−y1 ; λ, α) dy
1
2
f (−y1 )[1 + α F (−λy1 )] α+2 = h(y1 ; −λ, α),
=
since f (.) is the p.d.f. of a standard normal variate. Hence Y1 follows GMNSN(α, −λ).
Result 2.2. If Z has GMNSN(α, λ), then Y2 = Z follows Chi-square distribution with one degrees of freedom. 2
Proof. The p.d.f. h2 (y2 ) of Y2 = Z 2 is the following, for y2 > 0.
dz + h(−√y2 ; λ, α) dz dy dy
√
h2 (y2 ) = h( y2 ; λ, α)
=
2
α+2
2
2
√ √ 1 f ( y2 )[1 + α F (λ y2 )] √
2 y2
+
2
α+2
√ √ 1 f (− y2 )[1 + α F (−λ y2 )] √ 2 y2
√ 1 = √ f ( y2 ). y2
Then we have the following for y2 > 0. y2 1 1 h2 (y2 ) = √ y2 2 −1 e− 2 , 2π
which is the p.d.f. of a chi-square variate with one degrees of freedom.
Result 2.3. If Z follows GMNSN(α, λ), then Y3 = |Z | follows standard half normal distribution. Proof. For z > 0, the p.d.f. h3 (z ) of Y3 is h3 (z ) = h(z ; λ, α) + h(−z ; λ, α) 2 2 = f (z )[1 + α F (λz )] + f (−z )[1 + α F (−λz )] α+2 α+2 = 2f (z ). This shows that Y3 follows half normal distribution.
C. Satheesh Kumar, M.R. Anusree / Statistics and Probability Letters 81 (2011) 1813–1821
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Result 2.4. The cumulative distribution function(c.d.f.) H (z ) of GMNSN(α, λ) with p.d.f. (2.1) is the following, for z ∈ R. 2α κ(z , λ), α+2
H (z ) = F (z ) −
∞ λt
where κ(z , λ) =
0
z
(2.2)
f (u)f (t )dudt.
Proof. The c.d.f. H (z ) of GMNSN(α, λ) is z
∫
H (z ) =
h(t ; λ, α)dt −∞
] [ ∫ ∞ ∫ λt α f (u)f (t )du.dt = F (z ) + F (z ) − 2 α+2 α+2 0 z 2α κ(z , λ). = F (z ) − α+2 Note that Owen (1956) gives an expression for κ(z , λ) in terms of infinite series. 2
Now to derive the characteristic function of GMNSN(α, λ) we need the following lemma. Lemma 2.1 (Ellison, 1964). For a standard normal variable Z with distribution function F, we have the following for all a, b ∈ R.
E {F (aZ + b)} = F
b
. ( 1 + a2 )
Result 2.5. The characteristic function φZ (t ) of GMNSN(α, λ) with p.d.f. (2.1) is the following, for t ∈ R and i = 2e
φZ (t ) = where δ =
2 − t2
α+2 λ
[1 + α F (iδ t )],
√ −1. (2.3)
.
1
(1+λ2 ) 2
Proof. Let Z follows GMNSN (α, λ) with p.d.f. (2.1). Then by the definition of characteristic function, we have the following, √ for any t ∈ R and i = −1.
φZ (t ) = E (eitZ ) (α + 2) 2
=
∞
∫
2
=
(α + 2)
eitz f (z )dz + α −∞
e
2 − t2
∞
∫
eitz f (z )F (λz )dz
−∞
1+α
∫
∞
1
√ −∞
2π
e−
(z −it )2 2
F (λz )dz .
(2.4)
On substituting z − it = y in (2.4) we obtain
φZ (t ) =
2e−
t2 2
(α + 2)
{1 + α E [F (λ(y + it ))]},
which implies (2.3) in the light of Lemma 2.1.
Result 2.6. The nth raw moment µ′n of GMNSN(α, λ) with p.d.f. (2.1) is the following, for n ≥ 0.
µ′n =
n − n
2
α+2
r =0
r
ξr ϕn−r
(2.5)
where for r=0,1,2, . . . ,n
ξr =
0,
if r is odd
r! r 2r , if r is even ( 2 )!2 r −1 (−1) 2 αδ r (r − 1)! √ r −1 r −1 , 2 ϕr = ! 2 2 π 2 0, if r is even.
if r is odd
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Proof. From Result 2.5 the characteristic function of GMNSN(α, λ) can be written as 2
φZ (t ) =
α+2
I (t )J (t ),
(2.6)
−t 2
in which I (t ) = e 2 and J (t ) = [1 + α F (iδ t )]. On differentiating (2.6) with respect to t ‘n’ times and putting t = 0 we get the nth moment of Z as
µn =
2
n −
α+2
r =0
′
I
(r )
(t )J
(n−r )
(t )
,
(2.7)
t =0
in which I (r ) (t ) and J (n−r ) (t ) respectively denote the rth and (n − r )th derivative of I (t ) and J (t ) which are obtained as,
t2 [ 2r ] r −2j − 2 t r !e I ( r ) (t ) = j (r − 2j)! j ! 2 j =0
and
(2.8)
n−r −1 2 n −r n−r −1−k n−r −2k − αδ (−1) t (n − r − 1)!f (δ t ) J (n−r ) (t ) = . k (n − r − 1 − 2k)! k ! 2 j =0
(2.9)
If we put t = 0 in (2.8) and (2.9) and using the notation ξr = I (r ) (0) and ϕn−r = J (n−r ) (0), we get (2.5) from (2.7). Using Result 2.6, we prove the following results. Result 2.7. The mean and variance of GMNSN(α, λ) with p.d.f. (2.1) is
12
αδ 2 α2 δ2 and Variance = 1 − . α+2 π (α + 2)2
2
Mean =
π
Result 2.8. The measure of skewness (β1 ) and the measure of kurtosis (β2 ) of GMNSN(α, λ) with p.d.f. (2.1) are respectively given by, d2 2d2 − δ 2
β1 =
(1 − 2 12 αδ
where d = π
)
d2 3
α+2
2 and β2 =
3 + 4dδ 2 − 6d2 − 3d4
(1 − d2 )2
,
.
Remark 2.1. The GMNSN(α, λ) is positively skewed when −1 < d < 0 or 0 < d < 1 and negatively skewed when d > 1. Remark 2.2. As δ tends to 1, skewness depends only on α . Result 2.9. Let X1 and X2 be two independent and identically distributed standard normal variates with p.d.f. f (x) and c.d.f. F (x) and let X(1) = min{x1 , x2 } and X(2) = max{x1 , x2 }, then (i) X(1) follows GMNSN(1, −1) and (ii) the generalized mixture of X1 and X(2) is GMNSN(α, 1), for any α > −2. Proof. The p.d.f. of the ith order statistics is fi:n (x) =
n!
(i − 1)!(n − i)!
F (x)i−1 [1 − F (x)]n−i f (x),
for 1 ≤ i ≤ n.
(2.10)
Using (2.10), we have the p.d.f. of X(1) as f1:2 (x) = 2f (x)(1 − F (x)) which is the p.d.f. of GMNSN(−1, 1) variate. By applying (2.10), we obtain the p.d.f. of X(2) as f2:2 (x) = 2f (x)F (x) and therefore the p.d.f. of the generalized mixture of X1 and X(2) is the following, for any α > −2. f ∗ (x) =
2
α+2
f ( x) +
α f2:2 (x) α+2
which is the p.d.f of GMNSN(α, 1).
3. Reliability measures Here we investigate some properties of GMNSN(α, λ) with p.d.f. (2.1) useful in reliability studies. Let Z follows GMNSN(α, λ) with p.d.f. (2.1). Now from the definition of reliability function R(t ), failure rate r (t ) and mean residual life function µ(t ) of Z we obtain the following results.
C. Satheesh Kumar, M.R. Anusree / Statistics and Probability Letters 81 (2011) 1813–1821
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Result 3.1. The reliability function R(t ) of Z is the following, in which κ(t , λ) is as defined in Result 2.4. R(t ) = 1 − F (t ) +
2α
α+2
κ(t , λ).
Result 3.2. The failure rate r (t ) of Z is given by, r (t ) =
2f (t )[1 + α F (λt )]
(α + 2)[1 − F (t )] + 2ακ(t , λ)
.
Result 3.3. The mean residual life function of GMNSN(α, λ) is
2
µ(t ) =
1
e
√
(α + 2)R(t )
2
2π
− t2
+√
αλ
1 + λ2
+ α f (t )F (λt ) −
αλ 2π (1 + λ2 )
F (t
1+λ ) 2
− t.
(3.1)
Proof. By definition, the mean residual life function (MRLF) of Z is given by
µ(t ) = E (Z − t /Z > t ) = E (Z /Z > t ) − t , where E (Z /Z > t ) =
=
∞
[∫
2
(α + 2)R(t ) 2
(α + 2)R(t )
xf (x) + α t
∫
∞
xf (x)F (λx)dx
]
t
[I1 + α I2 ],
(3.2)
where ∞
∫
e−
t2 2
xf (x)dx = √ 2π
I1 = t
(3.3)
∞
∫
xf (x)F (λx)dx
I2 = t
∞
∫
f ′ (x)F (λx)dx
= − t
= F (λt )f (t ) + λ
∞
∫
f (x)F (λx)dx t
√ λ[1 − F (t 1 + λ2 )] . = F (λt )f (t ) + 2π (1 + λ2 )
(3.4)
Now by applying Eqs. (3.3) and (3.4) in (3.2), we get (3.1). The functions R(t ), r (t ) and µ(t ) are equivalent in the sense that if one of them is given the other two can be uniquely determined. Remark 3.1. GMNSN(α, λ) has increasing failure rate for all α and λ and hence decreasing mean residual life. Result 3.4. The p.d.f. of GMNSN(α, λ) is log concave (i) for all λz ≥ 0 and α > 0, (ii) for all λz < 0 and −1 < α < 0, (iii) for α f (λz ) all λz ≥ 0, −1 < α < 0 when T < 0 and (iv) for all λz < 0, α > 0 when T > 0, in which T = λz + 1+α F (λz ) . Proof. In order to show that log[h(z ; λ, α)] is a concave function of z, it is sufficient to prove that its second derivative is negative for all z. d dz
{log[h(z ; λ, α)]} = −z +
d2 dz 2
αλf (λz ) 1 + α F (λz )
{log[h(z ; λ, α)]} = −1 − Λ(z ; λ, α),
where
Λ(z ; λ, α) =
[ ] αλ2 f (λz ) α f (λz ) λz + . 1 + α F (λz ) 1 + α F (λz )
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α f (λz )
Note that f (λz ) and F (λz ) are positive for all z ∈ R and hence 1+α F (λz ) is positive for α > 0 and negative for all α < 0. α f (λz ) Further, if λz ≥ 0, then λz + 1+α F (λz ) is positive for all α > 0 and hence Λ(z ; λ, α) is positive. Also if λz < 0, let t = −λz, α f (λz )
then f (λz ) = f (−λz ) = f (t ) and F (λz ) = 1 − F (t ), thus λz + 1+α F (λz ) is negative for all −1 < α < 0 and hence Λ(z ; λ, α) is positive. If λz ≥ 0 and −1 < α < 0, then [1 + α F (λz )] is positive and hence Λ(z ; λ, α) is positive when T < 0. In a similar way, if λz < 0 and α > 0, then [1 + α F (λz )] is positive so that Λ(z ; λ, α) is positive only when T > 0. Thus in all these cases Λ(z ; λ, α) is positive. Thus the proof of the result follows. Result 3.5. GMNSN(α, λ) density is strongly unimodal (i) for all λz ≥ 0 and α > 0, (ii) for all λz < 0 and −1 < α < 0, (iii) for all λz ≥ 0, −1 < α < 0 when T < 0 and (iv) for all λz < 0, α > 0 when T > 0. Since by Result 3.4, h(z ; λ, α) is a log concave function of z and hence h(z ; λ, α) is strongly unimodal under these conditions. As a consequence of Result 3.5, we have the following remark. Remark 3.2. GMNSN(α, λ) is plurimodal for (i) λz > 0 and α ∈ (−1, 0) if Λ(z ; λ, α) < −1 and (ii) λz < 0, and α > 0 if Λ(z ; λ, α) < −1. 4. Location scale extension In this section we discuss an extended form of GMNSN(α, λ) by introducing the location parameter µ and scale parameter σ . Definition 4.1. Let Z ∼ GMNSN(α, λ) with p.d.f. (2.1). Then X = µ + σ Z is said to have an extended GMNSND with parameter µ, σ , λ and α with the following p.d.f. h∗ (x, µ, σ ; λ, α) =
2
σ (α + 2)
f
x−µ
σ
[
1 + λF λ
x−µ
]
σ
(4.1)
in which x ∈ R, µ ∈ R, λ ∈ R, σ > 0, α > −2. A distribution with p.d.f. (4.1) is denoted as EGMNSN(µ, σ ; α, λ). Clearly when α = 0 and/or when λ = 0 EGMNSN(µ, σ ; α, λ) reduces to N (µ, σ 2 ). Now we have the following results. The proof of these results are similar to Results 2.5 and 2.6 and hence omitted. Result 4.1. The characteristic function of EGMNSN(µ, σ ; α, λ) is given by
ψX (t ) =
2
α+2
eiµt −
t2σ 2 2
[1 + α F (iδσ t )].
Result 4.2. Mean and variance of EGMNSN(µ, σ ; α, λ) are respectively
Mean = µ +
2 αδσ
π α+2
and Variance = σ 2 −
2 α2 δ2 σ 2
π (α + 2)2
.
Result 4.3. The coefficient of skewness of EGMNSN(µ, σ ; α, λ) is
2 4α ασ 3 δ 3 π2 (α+ − 1 2 2) β1 = 23 µαδσ (α + 2) µ2 + σ 2 + √24π(α+ 2)
and
the coefficient of kurtosis is
β2 = −3d4 + d2 b1 + dc1 − d1 , where b1 = −11µ2 − 6σ 2 + 4σ 2 δ 2 , c1 = −4µ3 − 24µσ 2 + µσ 2 δ 2 and d1 = −µ4 . Result 4.4. If X follows EGMNSN(µ, σ ; α, λ) then Z1 = −X follows EGMNSN(µ, σ ; α, −λ). Result 4.5. Let X follows EGMNSN(µ, σ ; α, λ) then Z2 =| X | follows half normal distribution HN (µ, σ ).
C. Satheesh Kumar, M.R. Anusree / Statistics and Probability Letters 81 (2011) 1813–1821
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Proof. The p.d.f. of Z2 = |X | is the following for x > 0 hZ2 (x) = fX (x) + fX (−x)
=
2
α+2 = 2f (x),
f (x)[1 + α F (λx)] +
which is the p.d.f. of HN (µ, σ ).
2
α+2
f (−x)[1 + α F (−λx)]
5. Maximum likelihood estimation The loglikelihood function, lnL of the random sample of size n from a population following EGMNSN(µ, σ ; α, λ) is the following, in which c = 2n ln π2 . ln L = c −
n
ln σ − n ln(α + 2) − 2
2
n 1 − (xi − µ)2
2 i=1
σ2
+
n − i=1
] xi − µ ln 1 + α F λ . σ [
(5.1)
On differentiating (5.1) with respect to parameters µ, σ , λ and α and then equating to zero, we obtain the following normal equations.
n f σλ (xi − µ) αλ − = 0 σ2 σ i=1 1 + α F λ xi −µ i =1 σ n n f σλ (xi − µ) (xi − µ) −n 1 − (xi − µ)2 αλ − = 0 + − 3 2σ 2 2 i=1 σ4 σ i=1 1 + α F λ xi −µ σ n − α f σλ (xi − µ) (xi − µ) xi − µ xi −µ =0 α σ 1 + αF λ σ i=1 (xi −µ) n F λ − σ −n = 0. (α + 2) i=1 1 + α F λ xi −µ σ n − (xi − µ)
−
(5.2)
(5.3)
(5.4)
(5.5)
Let
λ (xi −µ) σ . w(xi ) = (x −µ) 1 + αF λ i σ f
Then the equations from (5.2) to (5.5) becomes
n n − αλ − xi − µ w(xi ) = , σ i=1 σ2 i=1 n
σ2
=
n − (xi − µ)2 i=1
σ4
−
n αλ − w(xi )(xi − µ), σ 3 i =1
(5.6)
(5.7)
n α− w(xi )(xi − µ) = 0 σ i=1
(5.8)
x −µ n − w(xi ) F λ iσ xi −µ − + = 0. α+2 f λ σ i =1
(5.9)
and n
On solving the Eqs. (5.6)–(5.9) we get the maximum likelihood estimators of the parameters of EGMNSN(µ, σ ; α, λ). For numerical illustration, we consider the IQ data set for 52 non-white males hired by a large insurance company in 1971 given in Roberts (1988). The data have been previously analyzed in Shaafi and Behboodian (2008). We obtained the MLE of the parameters by using the data and MATHCAD software. Kolmogorov–Smirnov statistic (KSS) is obtained for comparing the new model EGMNSN(µ, σ ; α, λ) against the existing models—the normal and skew normal distributions. After doing so it is seen that EGMNSN(µ, σ ; α, λ) is more appropriate for the data analyzed. The results obtained are presented in Table 1. From Table 1 it is clear that the EGMNSN(µ, σ ; α, λ) is a more appropriate model to the data, compared to normal distribution N (µ, σ 2 ) and skew normal distribution of Azzalini (1985).
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C. Satheesh Kumar, M.R. Anusree / Statistics and Probability Letters 81 (2011) 1813–1821 Table 1 Estimated values of the parameters for the models: Normal, Skew Normal and EGMNSND with respective KSS values. Distribution
Normal (µ, σ 2 )
Skew normal (µ, σ 2 , λ)
EGMNSN (µ, σ 2 ; λ, α)
µ ˆ σˆ λˆ µ ˆ
106.653 8.229 – – 0.1244
98.79 11.38 1.71 – 0.1014
104.5448 8.49552 0.6260425 3.314856 0.084222
KSS
Fig. 1. Probability plot of GMNSND for α = −1.1 and λ = 1.
Fig. 2. Probability plot of GMNSND for α = −0.6 and λ = 12.
Fig. 3. Probability plot of GMNSND for α = 1.5 and λ = 3.
C. Satheesh Kumar, M.R. Anusree / Statistics and Probability Letters 81 (2011) 1813–1821
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Fig. 4. Probability plot of GMNSND for α = −0.6 and λ = 20.
Conclusions From Table 1 it is clear that for the data analyzed here the EGMNSN(µ, σ ; α, λ) gives a better fit than the existing models like normal and skew normal distributions. Thus the model discussed in this paper provides more flexibility in modeling due to the presence of the extra parameter. Also the present model open up the suitability of modeling certain plurimodal data for some values of α by Remark 3.2. For example, when α = −0.6 and λ = 12 or 20, the distribution will be plurimodal (see, Appendix). Acknowledgments The authors are grateful to the Chief Editor and the referee for their valuable comments on an earlier version of the paper. Appendix See Figs. 1–4. References Azzalini, A., 1985. A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12, 171–178. Azzalini, A., 1986. Further results on a class of distributions which includes the normal. Statistica 46, 199–208. Azzalini, A., Dalla-Valle, A., 1996. The multivariate skew normal distribution. Biometrica 83, 175–726. Branco, M., Dey, D., 2001. A general class of multivariate elliptical distributions. Journal of Multivariate Analysis 79, 99–113. Buccianti, A., 2005. Meaning of the λ parameter of skew normal and log skew normal distributions in fluid geo chemistry. In: CODAWORK’05, October 19–21, pp. 1–15. Ellison, B., 1964. Two theorems for inference about the normal distribution with applications in acceptance sampling. Journal of American Statistical Association 59, 89–95. Henze, N., 1986. A probabilistic representation of the skew normal distribution. Scandinavian Journal of Statistics 13, 271–275. Owen, D.B., 1956. Tables for computing bivariate normal probabilities. Annals of Mathematical Statistics 27, 1075–1090. Roberts, H.V., 1988. Data Analysis for Managers with Minitab. Scientific Press, Redwood City, CA. Shaafi, M., Behboodian, J., 2008. The Balakrishnan skew-normal density. Statistical Papers 49, 769–778.