- Email: [email protected]
Generalized skew-elliptical distributions are closed under affine transformations
Statistics and Probability Letters 134 (2018) 1–4
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Generalized skew-elliptical distributions are closed under affine transformations Tomer Shushi * Faculty of Natural Sciences, Department of Physics, Ben-Gurion University of the Negev, Beersheba 8410501, Israel Actuarial Research Center, Department of Statistics, University of Haifa, Mount Carmel, Haifa 3498838, Israel
article
info
a b s t r a c t
Article history: Received 26 August 2017 Received in revised form 5 October 2017 Accepted 17 October 2017 Available online 31 October 2017
In this short letter we prove that the family of multivariate generalized skew-elliptical distributions is closed under affine transformations. This fundamental property has many applications in applied and theoretical probability. © 2017 Elsevier B.V. All rights reserved.
Keywords: Characteristic function Generalized skew-elliptical distributions Marginal distributions Skewed distributions
1. Introduction One of the main important properties of the multivariate normal distribution is that it is closed under affine transformations, which means that for an m × n matrix B of rank m, m < n, the random vector BX, X ∽ Nn (µ, Σ ), is also a normal random vector, BX ∽ Nm (Bµ, BΣ BT ). Furthermore, this property also holds for every member of the elliptical family of distributions −1/2
fX (x) = |Σ |
( gn
1 2
(x − µ) Σ T
−1
)
(x − µ) , x ∈ Rn ,
(1)
where X ∽ En (µ, Σ , gn ) is the elliptical random vector with the density generator gn (u) ≥ 0, µ is n × 1 vector of means, Σ is an n × n positive definite matrix. A well-known generalization of the elliptical family, into the world of skewed distributions, is the generalized skewelliptical (GSE) family of distributions which takes the following form (Genton and Loperfido, 2005) fY (y) = 2|Σ |−1/2 gn
(
1 2
)
(y − µ)T Σ −1 (y − µ) π (Σ −1/2 (y − µ)), y ∈ Rn ,
(2)
where π is called the skewing function which is defined as a mapping from the space of random vectors Rn to the real (non-negative) line R+ , i.e.,
π : Rn ∋Y H⇒ π (Y) ≥ 0. Furthermore, π satisfies the following properties 0 ≤ π (y) ≤ 1, ∀y ∈ Rn
*
Correspondence to: Faculty of Natural Sciences, Department of Physics, Ben-Gurion University of the Negev, Beersheba 8410501, Israel. E-mail address: [email protected].
https://doi.org/10.1016/j.spl.2017.10.012 0167-7152/© 2017 Elsevier B.V. All rights reserved.
2
T. Shushi / Statistics and Probability Letters 134 (2018) 1–4
and
π (−y) = 1 − π (y), ∀y ∈ Rn . We say that an n × 1 random vector Y is a GSE random vector if its probability density function (pdf) takes the form of (2), and we write Y ∽ GSEn (µ, Σ , gn , π ). The characteristic function of special members of the GSE family was computed in the literature (see, for instance, Azzalini and Valle, 1996; Loperfido, 2004; Nadarajah and Kotz, 2003; Vernic, 2005, and Shushi, 2016, 2017). In Shushi (2016), it was proved that the characteristic function of any GSE random vector Y takes the following well-defined form c(t) = 2e
itT µ
Ψn
(
1 2
)
t Σ t kn (t), t ∈ Rn , T
where Ψn is the characteristic function of the n × 1 elliptical random vector X, and kn is a function that satisfies the following properties kn (−t) = 1 − kn (t), and 0 ≤ kn (t) ≤ 1. For the sequel, we define a quasi-pdf of a random vector Ztn,Σ ,
(
T
(
)
pt (z) = exp it z gn
1 2
z Σ T
−1
)
z / Ψn
(
1 2
)
t Σ t , z ∈ Rn . T
We now prove that the GSE family of distributions is closed under affine transformations. Theorem 1. Let Y ∽ GSEn (µ, Σ , gn , π ). Then, for any a ∈ Rm , and m × n matrix B of rank m, m < n, the affine transformation a + BY is also a GSE random vector with the characteristic function T (a+Bµ)
cBY (t) = 2eit
Ψm
(
1 2
)
tT BΣ BT t km (t), t ∈ Rm .
Here
( ((
km (t) = E ˜ π
BΣ BT
(
)−1/2
Ztm,BΣ BT
where ˜ π (u) = π Σ −1/2 B† BΣ BT
(
)1/2
))
,
)
u , u ∈ Rm , and B† is the Moore–Penrose pseudoinverse of B from the left side.
Proof. From the definition of the characteristic function of the GSE distributions, and after taking the transformation v = Σ −1/2 (y − µ), we have cBY (t) = 2|Σ |−1/2 eit T (a+Bµ)
= 2eit
Ta
∫ Rn
)T
) ) ( (y − µ)T Σ −1 (y − µ) π Σ −1/2 (y − µ) dy 2 Rn ( ) ( ) 1 T itT BΣ 1/2 v e gn v v π (v) dv, t ∈ Rm .
∫
(
i BT t
e
y
(
gn
1
2
Taking the Moore–Penrose pseudoinverse of A = BΣ 1/2 from the left side, A† , and taking into account the existence and uniqueness of A† , we define a new skewing function π ∗ such that
( ) π ∗ (Av) = π A† Av , ( )† where A† = BΣ 1/2 = Σ −1/2 B† , which is given from the properties of the pseudoinverse matrix. Thus, we conclude that cBY (t) = 2eit
T (a+Bµ)
∫
eit
T (Av)
Rn
( gn
1 2
)
vT v π ∗ (Av) dv = E (Λ (AX)) ,
(3)
where T (AX)
Λ (AX) = eit
π ∗ (AX) ,
and EX (Λ (AX)) is the expected value of Λ (AX) respect to the elliptical distribution, X ∽ En (0,In , gn ), where In is the n × n identity matrix. Then, from the marginal properties of the elliptical distributions (see, Fang et al., 1990), it is clear that AX ∽ Em (0,AAT , gm ).
T. Shushi / Statistics and Probability Letters 134 (2018) 1–4
3
Therefore, Eq. (3) is now taking the following reduced form cBY (t) =
T (a+Bµ)
2eit
∫
√⏐ ⏐ ⏐AAT ⏐
T
eit u gm
(
1 2
Rm
uT AAT
(
)−1
)
u π ∗ (u) du, u ∈ Rm .
Noticing that, AAT = BΣ BT is a covariance matrix, so the inverse of BΣ BT can be written as BΣ BT following skewing function
(
((
˜ π (u) = π ∗
AAT
)−1
, we define the
)1/2 )
u ,
so cBY (t) =
T (a+Bµ)
2eit
∫
√⏐ ⏐ ⏐AAT ⏐
T
eit u gm
(
1 2
Rm
uT AAT
(
)−1
)
u ˜ π
((
AAT
)−1/2 )
u du.
Finally, using Shushi (2016), we conclude that this characteristic function takes the form of the GSE characteristic function cBY (t) = 2e
itT (a+Bµ)
Ψm
(
1 2
)
t BΣ B t km (t), t ∈ Rm . T
T
■
Corollary 1. The weighted-sum R = α T Y, α ∈ Rn , of Y ∽ GSEn (µ, Σ , gn , π ), follows the univariate GSE distribution. Proof. This Corollary immediately follows from Theorem 1, by taking B = α T .
■
Corollary 2. Suppose that the moment generating function (mgf) of Y ∽ GSEn (0,In , gn , π ) exists, and that e−2au gn (u), a, u ≥ 0, is an elliptical density generator, where 0 is the n × 1 null vector and In is the n × n identity matrix. Then, the mgf of the mean–variance transformation f (Y) = α T Y − θ YT Y of Y, α ∈ Rn , θ > 0, takes the form of the mgf of GSE distribution. Proof. Taking the transformation f (Y) , the mgf takes the form mgff (Y) (t ) :=
∫
(
e
(
)
t α T y−θ yT y
gn
1 2
Rn
)
y y π (y) dy. T
Noticing that e−t θ y y gn T
(
1 2
)
yT y ,
is a nonnegative even function of the quadratic form yT y, and by assuming that there exists finite scalar ct > 0 such that
∫ ct ,θ
e
−t θ yT y
( gn
Rn
1 2
)
y y dy = 1, T
we immediately conclude that ct ,θ e−t θ y y gn T
(1 2
)
yT y is an elliptical density function with the density generator
gnt ,θ (u) := e−2t θ u gn (u) ≥ 0. Then, mgff (Y) (t ) can be expressed, as follows: mgff (Y) (t ) =
∫ Rn
T et θ α y gnt
(
1 2
)
y y π (y) dy, T
which takes the form of the GSE moment generating function cα T V (−it θ ) where V ∽ GSEn (0,In , e−2t θ u gn (u), π ). ■ 1.1. Statistical implications The technique of projection pursuit is an important tool that aims to find interesting linear projections from structures of high-dimensional data to a low-dimensional subspace (Naito, 1997; Loperfido, 2010, 2013). Friedman (1987) and also Naito (1997) have argued that elliptical distributions are the least interesting cases, and thus one should investigate projections that are most different from the elliptical projections. Since the GSE distributions have skewed (i.e., non-symmetric) projections, it seems natural to consider them for the projection pursuit technique. In Loperfido (2010), the author has shown very important results in this area. In particular, it was shown that the linear projection with maximal skewness of skewnormal distributions follows a simple analytical form. This result was independently rediscovered in Balakrishnan and Scarpa (2012). In Arevalillo and Navarro (2015) the authors extended the result into the framework of skew-t distributions, and in the recent paper of Kim and Kim (2017), the authors considered the case with scale mixtures of skew-normal distributions.
4
T. Shushi / Statistics and Probability Letters 134 (2018) 1–4
Following Malkovich and Afifi (1973), and Loperfido (2010) with its notations, we define the multivariate skewness and kurtosis for a n-dimensional random vector Y, as follows:
( ) β1D,n (Y) = maxn β1 cT Y
(4)
( ) β2D,n (Y) = maxn β2 cT Y ,
(5)
c∈R0
and c∈R0
respectively. Here β1 (Z ) and β2 (Z ) are defined as the univariate skewness and kurtosis measures, respectively,
( )3 Z − E (Z ) β1 (Z ) = E 2 √ , Var (Z ) and
)4 ( Z − E (Z ) β2 (Z ) = E √ , Var (Z ) for a random variable Z with the expected value E (Z ) and the variance Var (Z ) . The symbol ‘‘D’’ implies on the directional nature of the projection, and Rn0 is the set of all n-dimensional non-zero vectors. Taking into account Corollary 1, we notice that the linear transformation cT Y, c ∈ Rn , of a GSE random vector Y ∽ GSEn (µ, Σ , gn , π ) also possess a GSE distribution. Although it seems that there are no, in general, explicit closed-form solutions to the mentioned maximization problems, since cT Y follows a GSE distribution the optimization problems (4), (5) can be explicitly obtained for any specific Y ∽ GSEn (µ, Σ , gn , π ). Acknowledgments I would like to thank the anonymous referee for the very useful comments. This research was supported by the Israel Science Foundation (Grant No. 1686/17). References Arevalillo, J.M., Navarro, H., 2015. A note on the direction maximizing skewness in multivariate skew-t vectors. Statist. Probab. Lett. 96, 328–332. Azzalini, A., Valle, A.D., 1996. The multivariate skew-normal distribution. Biometrika 83, 715–726. Balakrishnan, N., Scarpa, B., 2012. Multivariate measures of skewness for the skew-normal distribution. J. Multivariate Anal. 104, 73–87. Fang, K.T., Kotz, S., Ng, K.W., 1990. Symmetric Multivariate and Related Distributions. Chapman and Hall. Friedman, J.H., 1987. Exploratory projection pursuit. J. Amer. Statist. Assoc. 82, 249–266. Genton, M.G., Loperfido, N.M., 2005. Generalized skew-elliptical distributions and their quadratic forms. Ann. Inst. Statist. Math. 57, 389–401. Kim, H.M., Kim, C., 2017. Moments of scale mixtures of skew-normal distributions and their quadratic forms. Comm. Statist. Theory Methods 46, 1117–1126. Loperfido, N., 2004. Generalized skew-normal distributions. In: Skew-Elliptical Distributions and their Applications: A Journey beyond Normality. CRC/Chapman & Hall, pp. 65–80. Loperfido, N., 2010. Canonical transformations of skew-normal variates. TEST 19, 146–165. Loperfido, N., 2013. Skewness and the linear discriminant function. Statist. Probab. Lett. 83, 93–99. Malkovich, J.F., Afifi, A.A., 1973. On tests for multivariate normality. J. Amer. Statist. Assoc. 68, 176–179. Nadarajah, S., Kotz, S., 2003. Skewed distributions generated by the normal kernel. Statist. Probab. Lett. 65, 269–277. Naito, K., 1997. A generalized projection pursuit procedure and its significance level. Hiroshima Math. J. 27, 513–554. Shushi, T., 2016. A proof for the conjecture of characteristic function of the generalized skew-elliptical distributions. Statist. Probab. Lett. 119, 301–304. Shushi, T., 2017. Skew-elliptical distributions with applications in risk theory. Eur. Actuar. J. 1–20. Vernic, R., 2005. On the multivariate Skew-Normal distribution and its scale mixtures. An. St. Univ. Ovidius Constanta 13, 83–96.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Generalized skew-elliptical distributions are closed under affine transformations Tomer Shushi * Faculty of Natural Sciences, Department of Physics, Ben-Gurion University of the Negev, Beersheba 8410501, Israel Actuarial Research Center, Department of Statistics, University of Haifa, Mount Carmel, Haifa 3498838, Israel
article
info
a b s t r a c t
Article history: Received 26 August 2017 Received in revised form 5 October 2017 Accepted 17 October 2017 Available online 31 October 2017
In this short letter we prove that the family of multivariate generalized skew-elliptical distributions is closed under affine transformations. This fundamental property has many applications in applied and theoretical probability. © 2017 Elsevier B.V. All rights reserved.
Keywords: Characteristic function Generalized skew-elliptical distributions Marginal distributions Skewed distributions
1. Introduction One of the main important properties of the multivariate normal distribution is that it is closed under affine transformations, which means that for an m × n matrix B of rank m, m < n, the random vector BX, X ∽ Nn (µ, Σ ), is also a normal random vector, BX ∽ Nm (Bµ, BΣ BT ). Furthermore, this property also holds for every member of the elliptical family of distributions −1/2
fX (x) = |Σ |
( gn
1 2
(x − µ) Σ T
−1
)
(x − µ) , x ∈ Rn ,
(1)
where X ∽ En (µ, Σ , gn ) is the elliptical random vector with the density generator gn (u) ≥ 0, µ is n × 1 vector of means, Σ is an n × n positive definite matrix. A well-known generalization of the elliptical family, into the world of skewed distributions, is the generalized skewelliptical (GSE) family of distributions which takes the following form (Genton and Loperfido, 2005) fY (y) = 2|Σ |−1/2 gn
(
1 2
)
(y − µ)T Σ −1 (y − µ) π (Σ −1/2 (y − µ)), y ∈ Rn ,
(2)
where π is called the skewing function which is defined as a mapping from the space of random vectors Rn to the real (non-negative) line R+ , i.e.,
π : Rn ∋Y H⇒ π (Y) ≥ 0. Furthermore, π satisfies the following properties 0 ≤ π (y) ≤ 1, ∀y ∈ Rn
*
Correspondence to: Faculty of Natural Sciences, Department of Physics, Ben-Gurion University of the Negev, Beersheba 8410501, Israel. E-mail address: [email protected].
https://doi.org/10.1016/j.spl.2017.10.012 0167-7152/© 2017 Elsevier B.V. All rights reserved.
2
T. Shushi / Statistics and Probability Letters 134 (2018) 1–4
and
π (−y) = 1 − π (y), ∀y ∈ Rn . We say that an n × 1 random vector Y is a GSE random vector if its probability density function (pdf) takes the form of (2), and we write Y ∽ GSEn (µ, Σ , gn , π ). The characteristic function of special members of the GSE family was computed in the literature (see, for instance, Azzalini and Valle, 1996; Loperfido, 2004; Nadarajah and Kotz, 2003; Vernic, 2005, and Shushi, 2016, 2017). In Shushi (2016), it was proved that the characteristic function of any GSE random vector Y takes the following well-defined form c(t) = 2e
itT µ
Ψn
(
1 2
)
t Σ t kn (t), t ∈ Rn , T
where Ψn is the characteristic function of the n × 1 elliptical random vector X, and kn is a function that satisfies the following properties kn (−t) = 1 − kn (t), and 0 ≤ kn (t) ≤ 1. For the sequel, we define a quasi-pdf of a random vector Ztn,Σ ,
(
T
(
)
pt (z) = exp it z gn
1 2
z Σ T
−1
)
z / Ψn
(
1 2
)
t Σ t , z ∈ Rn . T
We now prove that the GSE family of distributions is closed under affine transformations. Theorem 1. Let Y ∽ GSEn (µ, Σ , gn , π ). Then, for any a ∈ Rm , and m × n matrix B of rank m, m < n, the affine transformation a + BY is also a GSE random vector with the characteristic function T (a+Bµ)
cBY (t) = 2eit
Ψm
(
1 2
)
tT BΣ BT t km (t), t ∈ Rm .
Here
( ((
km (t) = E ˜ π
BΣ BT
(
)−1/2
Ztm,BΣ BT
where ˜ π (u) = π Σ −1/2 B† BΣ BT
(
)1/2
))
,
)
u , u ∈ Rm , and B† is the Moore–Penrose pseudoinverse of B from the left side.
Proof. From the definition of the characteristic function of the GSE distributions, and after taking the transformation v = Σ −1/2 (y − µ), we have cBY (t) = 2|Σ |−1/2 eit T (a+Bµ)
= 2eit
Ta
∫ Rn
)T
) ) ( (y − µ)T Σ −1 (y − µ) π Σ −1/2 (y − µ) dy 2 Rn ( ) ( ) 1 T itT BΣ 1/2 v e gn v v π (v) dv, t ∈ Rm .
∫
(
i BT t
e
y
(
gn
1
2
Taking the Moore–Penrose pseudoinverse of A = BΣ 1/2 from the left side, A† , and taking into account the existence and uniqueness of A† , we define a new skewing function π ∗ such that
( ) π ∗ (Av) = π A† Av , ( )† where A† = BΣ 1/2 = Σ −1/2 B† , which is given from the properties of the pseudoinverse matrix. Thus, we conclude that cBY (t) = 2eit
T (a+Bµ)
∫
eit
T (Av)
Rn
( gn
1 2
)
vT v π ∗ (Av) dv = E (Λ (AX)) ,
(3)
where T (AX)
Λ (AX) = eit
π ∗ (AX) ,
and EX (Λ (AX)) is the expected value of Λ (AX) respect to the elliptical distribution, X ∽ En (0,In , gn ), where In is the n × n identity matrix. Then, from the marginal properties of the elliptical distributions (see, Fang et al., 1990), it is clear that AX ∽ Em (0,AAT , gm ).
T. Shushi / Statistics and Probability Letters 134 (2018) 1–4
3
Therefore, Eq. (3) is now taking the following reduced form cBY (t) =
T (a+Bµ)
2eit
∫
√⏐ ⏐ ⏐AAT ⏐
T
eit u gm
(
1 2
Rm
uT AAT
(
)−1
)
u π ∗ (u) du, u ∈ Rm .
Noticing that, AAT = BΣ BT is a covariance matrix, so the inverse of BΣ BT can be written as BΣ BT following skewing function
(
((
˜ π (u) = π ∗
AAT
)−1
, we define the
)1/2 )
u ,
so cBY (t) =
T (a+Bµ)
2eit
∫
√⏐ ⏐ ⏐AAT ⏐
T
eit u gm
(
1 2
Rm
uT AAT
(
)−1
)
u ˜ π
((
AAT
)−1/2 )
u du.
Finally, using Shushi (2016), we conclude that this characteristic function takes the form of the GSE characteristic function cBY (t) = 2e
itT (a+Bµ)
Ψm
(
1 2
)
t BΣ B t km (t), t ∈ Rm . T
T
■
Corollary 1. The weighted-sum R = α T Y, α ∈ Rn , of Y ∽ GSEn (µ, Σ , gn , π ), follows the univariate GSE distribution. Proof. This Corollary immediately follows from Theorem 1, by taking B = α T .
■
Corollary 2. Suppose that the moment generating function (mgf) of Y ∽ GSEn (0,In , gn , π ) exists, and that e−2au gn (u), a, u ≥ 0, is an elliptical density generator, where 0 is the n × 1 null vector and In is the n × n identity matrix. Then, the mgf of the mean–variance transformation f (Y) = α T Y − θ YT Y of Y, α ∈ Rn , θ > 0, takes the form of the mgf of GSE distribution. Proof. Taking the transformation f (Y) , the mgf takes the form mgff (Y) (t ) :=
∫
(
e
(
)
t α T y−θ yT y
gn
1 2
Rn
)
y y π (y) dy. T
Noticing that e−t θ y y gn T
(
1 2
)
yT y ,
is a nonnegative even function of the quadratic form yT y, and by assuming that there exists finite scalar ct > 0 such that
∫ ct ,θ
e
−t θ yT y
( gn
Rn
1 2
)
y y dy = 1, T
we immediately conclude that ct ,θ e−t θ y y gn T
(1 2
)
yT y is an elliptical density function with the density generator
gnt ,θ (u) := e−2t θ u gn (u) ≥ 0. Then, mgff (Y) (t ) can be expressed, as follows: mgff (Y) (t ) =
∫ Rn
T et θ α y gnt
(
1 2
)
y y π (y) dy, T
which takes the form of the GSE moment generating function cα T V (−it θ ) where V ∽ GSEn (0,In , e−2t θ u gn (u), π ). ■ 1.1. Statistical implications The technique of projection pursuit is an important tool that aims to find interesting linear projections from structures of high-dimensional data to a low-dimensional subspace (Naito, 1997; Loperfido, 2010, 2013). Friedman (1987) and also Naito (1997) have argued that elliptical distributions are the least interesting cases, and thus one should investigate projections that are most different from the elliptical projections. Since the GSE distributions have skewed (i.e., non-symmetric) projections, it seems natural to consider them for the projection pursuit technique. In Loperfido (2010), the author has shown very important results in this area. In particular, it was shown that the linear projection with maximal skewness of skewnormal distributions follows a simple analytical form. This result was independently rediscovered in Balakrishnan and Scarpa (2012). In Arevalillo and Navarro (2015) the authors extended the result into the framework of skew-t distributions, and in the recent paper of Kim and Kim (2017), the authors considered the case with scale mixtures of skew-normal distributions.
4
T. Shushi / Statistics and Probability Letters 134 (2018) 1–4
Following Malkovich and Afifi (1973), and Loperfido (2010) with its notations, we define the multivariate skewness and kurtosis for a n-dimensional random vector Y, as follows:
( ) β1D,n (Y) = maxn β1 cT Y
(4)
( ) β2D,n (Y) = maxn β2 cT Y ,
(5)
c∈R0
and c∈R0
respectively. Here β1 (Z ) and β2 (Z ) are defined as the univariate skewness and kurtosis measures, respectively,
( )3 Z − E (Z ) β1 (Z ) = E 2 √ , Var (Z ) and
)4 ( Z − E (Z ) β2 (Z ) = E √ , Var (Z ) for a random variable Z with the expected value E (Z ) and the variance Var (Z ) . The symbol ‘‘D’’ implies on the directional nature of the projection, and Rn0 is the set of all n-dimensional non-zero vectors. Taking into account Corollary 1, we notice that the linear transformation cT Y, c ∈ Rn , of a GSE random vector Y ∽ GSEn (µ, Σ , gn , π ) also possess a GSE distribution. Although it seems that there are no, in general, explicit closed-form solutions to the mentioned maximization problems, since cT Y follows a GSE distribution the optimization problems (4), (5) can be explicitly obtained for any specific Y ∽ GSEn (µ, Σ , gn , π ). Acknowledgments I would like to thank the anonymous referee for the very useful comments. This research was supported by the Israel Science Foundation (Grant No. 1686/17). References Arevalillo, J.M., Navarro, H., 2015. A note on the direction maximizing skewness in multivariate skew-t vectors. Statist. Probab. Lett. 96, 328–332. Azzalini, A., Valle, A.D., 1996. The multivariate skew-normal distribution. Biometrika 83, 715–726. Balakrishnan, N., Scarpa, B., 2012. Multivariate measures of skewness for the skew-normal distribution. J. Multivariate Anal. 104, 73–87. Fang, K.T., Kotz, S., Ng, K.W., 1990. Symmetric Multivariate and Related Distributions. Chapman and Hall. Friedman, J.H., 1987. Exploratory projection pursuit. J. Amer. Statist. Assoc. 82, 249–266. Genton, M.G., Loperfido, N.M., 2005. Generalized skew-elliptical distributions and their quadratic forms. Ann. Inst. Statist. Math. 57, 389–401. Kim, H.M., Kim, C., 2017. Moments of scale mixtures of skew-normal distributions and their quadratic forms. Comm. Statist. Theory Methods 46, 1117–1126. Loperfido, N., 2004. Generalized skew-normal distributions. In: Skew-Elliptical Distributions and their Applications: A Journey beyond Normality. CRC/Chapman & Hall, pp. 65–80. Loperfido, N., 2010. Canonical transformations of skew-normal variates. TEST 19, 146–165. Loperfido, N., 2013. Skewness and the linear discriminant function. Statist. Probab. Lett. 83, 93–99. Malkovich, J.F., Afifi, A.A., 1973. On tests for multivariate normality. J. Amer. Statist. Assoc. 68, 176–179. Nadarajah, S., Kotz, S., 2003. Skewed distributions generated by the normal kernel. Statist. Probab. Lett. 65, 269–277. Naito, K., 1997. A generalized projection pursuit procedure and its significance level. Hiroshima Math. J. 27, 513–554. Shushi, T., 2016. A proof for the conjecture of characteristic function of the generalized skew-elliptical distributions. Statist. Probab. Lett. 119, 301–304. Shushi, T., 2017. Skew-elliptical distributions with applications in risk theory. Eur. Actuar. J. 1–20. Vernic, R., 2005. On the multivariate Skew-Normal distribution and its scale mixtures. An. St. Univ. Ovidius Constanta 13, 83–96.