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Journal Article•DOI•

Flexible Class of Skew‐Symmetric Distributions

Yanyuan Ma1, Marc G. Genton1•
North Carolina State University1
01 Sep 2004-Scandinavian Journal of Statistics (Blackwell)-Vol. 31, Iss: 3, pp 459-468
TL;DR: In this paper, a flexible class of skew-symmetric distributions for which the probab- ility density function has the form of a product of a symmetric density and a skewing function is proposed.
Abstract: We propose a flexible class of skew-symmetric distributions for which the probab- ility density function has the form of a product of a symmetric density and a skewing function. By constructing an enumerable dense subset of skewing functions on a compact set, we are able to consider a family of distributions, which can capture skewness, heavy tails and multimodality systematically. We present three illustrative examples for the fibreglass data, the simulated data from a mixture of two normal distributions and the Swiss bills dlata.
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A Flexible Class of Skew-Symmetri Distributions
(running head: exible skew-symmetri distributions)
YANYUAN MA
North Carolina State University
MARC G. GENTON
North Carolina State University
ABSTRACT. We prop ose a exible lass of skew-symmetri distributions for whih the
probability density funtion has the form of a pro dut of a symmetri density and a skewing
funtion. By onstruting an enumerable dense subset of skewing funtions on a ompat
set, we are able to onsider a family of distributions whih an apture skewness, heavy
tails, and multimo dality systematially. We present three illustrative examples for the
b er-glass data, simulated data from a mixture of two normal distributions, and Swiss
bills data.
Key Words:
dense subset; generalized skew-elliptial; multimodality; skewness; skew-normal.
1 Intro dution
A popular approah to ahieve departures from normality onsists of modifying the probability density
funtion (p df ) of a random vetor in a multipliative fashion. Wang, Boyer, & Genton (2004) showed
that any
p
-dimensional multivariate pdf
g
(
x
) admits, for any xed loation parameter
2
R
p
, a unique
skew-symmetri (SS) representation:
g
(
x
) = 2
f
(
x
)
(
x
)
;
(1)
where
f
:
R
p
!
R
+
is a symmetri p df and
:
R
p
!
[0
;
1℄ is a skewing funtion satisfying
(
x
) =
1
(
x
). Vie-versa, any funtion
g
of the type dened by (1) is a valid pdf. By symmetri, we mean
f
(
x
) =
f
(
x
) and we will use \symmetri pdf " and the prop erty
f
(
x
) =
f
(
x
) interhangeably in
the sequel. Throughout this pap er, we restrit our interest on funtions
f
2
C
0
(
R
p
) and ontinuous
skewing funtions
(
x
), where
C
0
(
R
p
) denotes ontinuous funtions on
R
p
with the prop erty
f
(
x
)
!
0
when
k
x
k
2
! 1
, and
k k
2
denotes the
L
2
norm. Genton & Lop erdo (2002) onsidered the subfamily
of generalized skew-elliptial (GSE) distributions for whih the p df
f
in (1) is elliptially ontoured
rather than only symmetri. Many denitions of skewed distributions found in the literature an be
written in the form of a skew-symmetri distribution (1). For instane, Azzalini & Dalla Valle's (1996)
multivariate skew-normal distribution orresp onds to
f
(
x
) =
p
(
x
;
0
;
) and
(
x
) = (
T
x
), where
p
(
x
;
;
) is the
p
-dimensional multivariate normal pdf with mean vetor
and orrelation matrix ,
1
is the standard normal umulative distribution funtion (df ), and
is a shap e parameter ontrolling
skewness. Similarly, multivariate distributions suh as skew-
t
(Brano & Dey, 2001; Azzalini & Capitanio,
2003; Jones & Faddy, 2003; Sahu, Brano, & Dey, 2003), skew-Cauhy (Arnold & Beaver, 2000) and
other skew-elliptial ones (Azzalini & Capitanio, 1999; Brano & Dey, 2001; Sahu
et al.
, 2003) an be
represented by the skew-symmetri distribution (1) with appropriate hoies of
f
and
.
In this artile, we prop ose a exible lass of distributions (1) by onstruting an enumerable dense
subset of the skewing funtions
on a ompat set. The result is a family of distributions whih
an apture skewness, heavy tails, and multimodality systematially. The onstrution of the subset is
through p olynomials, whih has a similar avor as the seminonparametri (SNP) representation prop osed
by Gallant & Nyhka (1987). The latter is dened as the pro dut of the standard normal p df and the
square of a polynomial. The SNP distribution requires the oeÆients in the polynomial to b e onstrained
in order to yield a valid density. It also relies on rejetion sampling shemes to simulate random samples.
These diÆulties do not o ur with our onstrution.
The ontent of the pap er is organized as follows. In Setion 2, we desribe a subset of skewing
funtions based on o dd p olynomials and prove that it results in a dense subset of the skew-symmetri
distributions. In partiular, we dene exible skew-normal and skew-
t
distributions that an have more
than one mode. This is an essential property for some situations and provides an alternative to modeling
with mixtures of distributions. The exibility and p ossible multimodality of the new lass of distributions
is illustrated in Setion 3. We present three illustrative examples in Setion 4, and a disussion in Setion
5.
2 A dense subset of skew-symmetri distributions
In this setion, we onstrut a dense subset of skew-symmetri distributions through approximating the
skewing funtion
on a ompat set. Any ontinuous skewing funtion
an be written as:
(
x
) =
H
(
w
(
x
))
;
(2)
where
H
:
R
!
[0
;
1℄ is the df of a ontinuous random variable symmetri around 0, and
w
:
R
p
!
R
is an o dd ontinuous funtion, that is
w
(
x
) =
w
(
x
). In fat, for a hosen
H
suh that
H
1
exists,
w
(
x
) =
H
1
(
(
x
)) is a ontinuous odd funtion. This representation has been used by Azzalini &
Capitanio (2003) to dene ertain distributions by p erturbation of symmetry. Note however that the
representation (2) is not unique due to the many possible hoies of
H
.
Let
P
K
(
x
) b e an o dd p olynomial of order
K
. A p olynomial of order
K
in
R
p
is dened as a linear
ombination of terms of the form
Q
p
i
=1
x
r
i
i
, where
k
=
P
p
i
=1
r
i
K
. If eah term has an odd order (all
k
's are o dd), then the polynomial is alled an odd p olynomial, whereas if eah term has an even order
(all
k
's are even), it is alled an even polynomial. We dene exible skew-symmetri (FSS) distributions
2
by restriting (1) to:
2
f
(
x
)
K
(
x
)
;
(3)
where
K
(
x
) =
H
(
P
K
(
x
)) and
H
is any df of a ontinuous random variable symmetri around 0. Note
that there are no onstraints on the oeÆients of the p olynomial
P
K
in order to make (3) a valid
pdf. In partiular, (3) denes exible generalized skew-elliptial (FGSE) distributions when the pdf
f
is
elliptially ontoured. For instane, exible generalized skew-normal (FGSN) distributions are dened
by:
2
p
(
x
;
;
)(
P
K
(
A
(
x
)))
;
(4)
and exible generalized skew-
t
(FGST) distributions are dened by:
2
t
p
(
x
;
;
;
)
T
(
P
K
(
A
(
x
));
)
;
(5)
where we use the Choleski deomposition
1
=
A
T
A
,
t
p
denotes a
p
-dimensional multivariate
t
pdf,
and
T
denotes a univariate
t
df, both with degrees of freedom
. Note that we ould use , or any
other symmetri df, instead of
T
for the skewing funtion in (5). In pratie, a popular hoie for the
df
H
would b e or the univariate df orresponding to the symmetri p df
f
. Eetively, the following
proposition shows that FSS distributions an approximate skew-symmetri distributions arbitrarily well.
Prop osition 1
Let the lass of exible skew-symmetri (FSS) distributions onsist of distributions with
pdf given in (3) and the lass of skew-symmetri (SS) distributions of distributions with pdf given in (1),
where
f
2
C
0
(
R
p
)
in both lasses and
is ontinuous. Then the lass of FSS distributions is dense in
the lass of SS distributions under the
L
1
norm.
Pro of
: An arbitrary distribution in the SS lass an be written as 2
f
(
x
)
H
(
w
(
x
)), where
f
and
H
are ontinuous,
H
1
exists, and
w
is a ontinuous o dd funtion. Beause
f
2
C
0
(
R
p
), for any arbitrary
>
0, we an nd a ompat set
D
whih is symmetri around
(if
x
2
D
then
x
2
D
), suh that
for any
x
=
2
D
,
f
(
x
)
< =
4. Thus, for any
x
=
2
D
,
j
2
f
(
x
)
(
x
)
2
f
(
x
)
H
(
P
((
x
))
j
<
for any odd p olynomial
P
.
Sine
f
is ontinuous,
f
is bounded on
D
. We denote the bound by
C
, i.e.
f
(
x
)
C
for any
x
2
D
. We use
D
1
to denote the image spae of
w
, i.e.
D
1
=
f
w
(
x
)
j
x
2
D
g
. Beause of the
ontinuity of
w
, whih is a result of the ontinuity of b oth
H
and
,
D
1
is also ompat. The ontinuous
funtion
H
is uniformly ontinuous on the ompat set
D
1
. Hene there exists
>
0 suh that for
any
y
1
,
y
2
2
D
1
and
j
y
1
y
2
j
<
, we get
j
H
(
y
1
)
H
(
y
2
)
j
< =
(2
C
). From the Stone-Weierstrass
theorem (see e.g. Rudin, 1973, p. 115), there exists a polynomial
P
suh that
j
w
(
x
)
P
(
x
)
j
<
for any
x
2
D
. We deomp ose
P
into an even term
P
e
and an odd term
P
o
, i.e.
P
=
P
e
+
P
o
.
Then
j
w
(
x
)
P
e
(
x
)
P
o
(
x
)
j
<
and
j
w
(
x
)
P
e
(
x
)
P
o
(
x
)
j
<
. Beause
w
and
P
o
are odd, and
P
e
is even, we get
j
w
(
x
)
P
e
(
x
) +
P
o
(
x
)
j
<
. Notie that
2
j
w
(
x
)
P
o
(
x
)
j j
w
(
x
)
P
e
(
x
)
P
o
(
x
)
j
+
j
w
(
x
)
P
e
(
x
) +
P
o
(
x
)
j
<
2
,
3
so
j
w
(
x
)
P
o
(
x
)
j
<
. Combining these results, we know that for an arbitrary member
2
f
(
x
)
H
(
w
(
x
)) in SS and an arbitrary
>
0, we an nd a member 2
f
(
x
)
H
(
P
o
(
x
)) in
FSS suh that
j
2
f
(
x
)
H
(
w
(
x
))
2
f
(
x
)
H
(
P
o
(
x
))
j
<
for any
x
2
D
.
Hene FSS is dense in SS with resp et to the
L
1
norm.
Remark 1
The requirement
f
2
C
0
(
R
p
)
in proposition 1 an be relaxed to al low that
f
has a nite
number,
m
say, of poles. In this ase, FSS is dense in SS with respet to almost uniform onvergene
(uniform in a set whose omplement is of measure arbitrarily smal l). Indeed, let
R
p
(
r
)
denote
R
p
minus
the union of
m
open bal ls of radius
r
entered at the
m
poles. Then FSS is dense in SS on
R
p
(
r
)
under
the
L
1
norm. Letting
r
!
0
, the result fol lows.
Proposition 1 shows in partiular that the lass of generalized skew-elliptial, skew
t
, and skew-
normal distributions an b e approximated arbitrarily well by their exible versions.
3 Flexibility and multimodality
In Figure 1, we illustrate the shap e exibility of the FGSN distribution in the univariate ase. Its pdf
for
K
= 3 is dened by:
2
1
(
x
;
;
2
)(
(
x
)
=
+
(
x
)
3
=
3
)
:
(6)
Figure 1 should b e here.
Figure 1(a) depits the p df of the FGSN model for
= 0,
2
= 1,
= 4, and
= 0, i.e. it redues
to Azzalini's (1985) univariate skew-normal distribution. However, when
6
= 0, the p df (6) an exhibit
bimodality as shown in Figure 1(b) with
= 1, and
=
1. In general, as the degree
K
of the o dd
polynomial in the skewing funtion beomes large, the number of mo des allowed in the p df inreases,
thus induing a greater exibility in the available shapes. Unfortunately, the number of modes depends
on the degree
K
of the o dd p olynomial, on the symmetri pdf
f
, and on the df
H
of the skewing
funtion
K
in a omplex fashion. Indeed, even for the univariate situation given by
p
= 1, the mo des
are determined by zeros of the rst derivative of the FSS distribution (3) given by:
2
f
0
(
x
)
H
(
P
K
(
x
)) + 2
f
(
x
)
H
0
(
P
K
(
x
))
P
0
K
(
x
)
;
(7)
for whih the number of zeros annot b e easily omputed. Even with restritions to some sp ei
f
and
H
funtions, a general statement on the relation between the number of mo des and the order of the
polynomial seems not available. However, in the univariate ase, if we onsider a normal pdf
f
=
1
and
a standard normal df
H
= with an o dd p olynomial of order
K
= 3, we have the following proposition.
Prop osition 2
The lass of exible generalized skew-normal (FGSN) distributions with pdf
2
1
(
x
;
;
2
)(
(
x
)
=
+
(
x
)
3
=
3
)
has at most 2 modes.
4
Pro of
: Without loss of generality, we an set
= 0,
= 1, assume
>
0, and only need to prove that
(
x
) = 2
(
x
)(
x
+
x
3
) has at most two modes. We prove this by ontradition. If
(
x
) has more
than two mo des, then
0
(
x
) has at least ve zeros. In the following pro of, we show that this annot b e
the ase. We have
0
(
x
) = 2
(
x
)((
+ 3
x
2
)
(
x
+
x
3
)
x
(
x
+
x
3
)) and need to onsider three
ases:
ase 1:
= 0
We write
0
(
x
) = 2
x
(
x
)
(
x
), where
(
x
) = 3
x
(
x
3
)
(
x
3
). We an verify that
0
(
x
) =
3
(
x
3
)
1
(
y
) where
y
=
x
2
and
1
(
y
) = 1
y
3
2
y
3
. Sine
1
(
y
) is a dereasing funtion on
y
0,
0
(
x
) has at most two zeros. Thus,
(
x
) has at most three zeros, hene
0
(
x
) has at most four
zeros.
ase 2:
>
0
Notie that
0
(
x
)
>
0 for
x
0. For
1
(
x
) =
0
(
x
)
=
(2
x
(
x
)) =
(
x
+
x
3
)(
+ 3
x
2
)
=x
(
x
+
x
3
),
we get
0
1
(
x
) =
(
x
+
x
3
)
=
(
9
x
2
)
2
(
y
), where
y
=
+ 3
x
2
>
0 and
2
(
y
) =
y
4
+
y
3
+ (3
2
2
)
y
2
(3
+ 9
)
y
+ 18

. Sine
00
2
(
y
) = 12
y
2
+ 6
y
+ (6
4
2
) has at most 1 positive zero, and
0
2
(
y
) = 4
y
3
+ 3
y
2
+ (6
4
2
)
y
(3
+ 9
)
<
0 at
y
= 0, we know that
0
2
(
y
) has at most one positive
zero. Thus
2
(
y
) has at most 2 positive zeros. This means
0
1
(
x
) has at most two p ositive zeros, so
0
(
x
)
has at most three (p ositive) zeros.
ase 3:
<
0
Notie that
0
(
x
)
<
0 for
x
2
[0
;
p
=
(3
) ℄ and
0
(
x
)
>
0 for
x
2
(
1
;
p
=
(3
) ℄. So we only
look for solutions
x
2
(
p
=
(3
)
;
1
) and
x
2
(
p
=
(3
)
;
0). Let
y
=
+ 3
x
2
, then there is a one
to one mapping b etween the
x
in the ab ove range and
y
2
(
;
1
). Let
1
(
x
) and
2
(
y
) have the same
expressions as in ase 2. We have that
2
(
y
) has at most four zeros sine it is a fourth order p olynomial.
Notie that
2
(
)
<
0
;
2
(
1
)
>
0, so
2
(
y
) has at most three zeros in (
;
1
). This means
0
1
(
x
) has
at most three zeros, hene
0
(
x
) has at most four zeros.
Figure 1 illustrates the result of prop osition 2 by depiting a unimo dal and a bimo dal pdf from the
univariate FGSN with
K
= 3. For
K
= 1, the p df is always unimodal as was already noted by Azzalini
(1985) for the univariate skew-normal distribution.
Next we investigate the exibility of the FGSN distribution in the bivariate ase. Its pdf for
K
= 3,
=
0
, and =
I
2
is given by:
2
2
(
x
1
; x
2
;
0
; I
2
)(
1
x
1
+
2
x
2
+
1
x
3
1
+
2
x
3
2
+
3
x
2
1
x
2
+
4
x
1
x
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2
)
:
(8)
Figure 2 should b e here.
Figure 2 depits the ontours of four dierent pdfs (8) for various ombinations of values of the
skewness parameters
1
,
2
,
1
,
2
,
3
, and
4
. In partiular, for
1
=
2
=
3
=
4
= 0, the
pdf is exatly the bivariate skew-normal proposed by Azzalini & Dalla Valle (1996), and known to be
unimodal, see Figure 2(a). However, Figures 2(b)-(d) show that many dierent distributional shap es an
be obtained with the parameters
1
; : : : ;
4
, in partiular bimodal and trimo dal distributions. Additional
5
Citations
More filters
Journal Article•DOI•
Conditional ASGT-GARCH Approach to Value-at-Risk

[...]

Emrah Altun1, Hüseyin Tatlidil1, Gamze Ozel1•
Hacettepe University1
01 Feb 2019-Iranian Journal of Science and Technology Transaction A-science
TL;DR: In this article, the alpha-skew generalized T (ASGT) distribution was used to forecast the value-at-risk (VaR) by employing the GARCH model.
Abstract: Most of the Value-at-Risk (VaR) models assume that asset returns are normally distributed, despite the fact that they are commonly known to be left skewed, fat-tailed and excess kurtosis. Forecasting VaR with misspecified model leads to the underestimation or overestimation of the true VaR. This paper proposes a new conditional model to forecast VaR by employing the alpha-skew generalized T (ASGT) distribution to GARCH models. ASGT distribution, introduced by Acitas et al. (Revista Colombiana de Estadistica 38(2):353–370, 2015), allows to model skewness, leptokurtosis and fat tail properties of conditional distribution of asset returns. ISE-100 index is used to examine the one-day-ahead VaR forecasting ability of the GARCH model under normal, Student’s t, generalized error, generalized T, skewed generalized T and ASGT innovation distributions. Empirical results show that the ASGT provides a superior fit to the conditional distribution of the log-returns followed by normal, Student’s t, generalized error, generalized T and skewed generalized T distributions. Moreover, for all confidence levels, all models tend to underestimate real market risk. Furthermore, the GARCH-based model, with ASGT error distribution, generates the most reliable VaR forecasts followed by other competitive models for a long position. As a result of this study, we conclude that the effects of skewness and fat-tails are more important in terms of forecasting true VaR than only the effect of fat-tails on VaR forecasts.

5 citations

Univariate and multivariate symmetry: statistical inference and distributional aspects

[...]

Christophe Ley, Davy Paindaveine
26 Nov 2010
TL;DR: In this paper, the authors propose a new univariate skew-normal skew-symmetric mechanism for higher dimensions, which can be extended to any dimension, and prove that the surjectivity property holds in dimensions k>1.
Abstract: This thesis deals with several statistical and probabilistic aspects of symmetry and asymmetry, both in a univariate and multivariate context, and is divided into three distinct parts.The first part, composed of Chapters 1, 2 and 3 of the thesis, solves two conjectures associated with multivariate skew-symmetric distributions. Since the introduction in 1985 by Adelchi Azzalini of the most famous representative of that class of distributions, namely the skew-normal distribution, it is well-known that, in the vicinity of symmetry, the Fisher information matrix is singular and the profile log-likelihood function for skewness admits a stationary point whatever the sample under consideration. Since that moment, researchers have tried to determine the subclasses of skew-symmetric distributions who suffer from each of those problems, which has led to the aforementioned two conjectures. This thesis completely solves these two problems.The second part of the thesis, namely Chapters 4 and 5, aims at applying and constructing extremely general skewing mechanisms. As such, in Chapter 4, we make use of the univariate mechanism of Ferreira and Steel (2006) to build optimal (in the Le Cam sense) tests for univariate symmetry which are very flexible. Actually, their mechanism allowing to turn a given symmetric distribution into any asymmetric distribution, the alternatives to the null hypothesis of symmetry can take any possible shape. These univariate mechanisms, besides that surjectivity property, enjoy numerous good properties, but cannot be extended to higher dimensions in a satisfactory way. For this reason, we propose in Chapter 5 different general mechanisms, sharing all the nice properties of their competitors in Ferreira and Steel (2006), but which moreover can be extended to any dimension. We formally prove that the surjectivity property holds in dimensions k>1 and we study the principal characteristics of these new multivariate mechanisms.Finally, the third part of this thesis, composed of Chapter 6, proposes a test for multivariate central symmetry by having recourse to the concepts of statistical depth and runs. This test extends the celebrated univariate runs test of McWilliams (1990) to higher dimensions. We analyze its asymptotic behavior (especially in dimension k=2) under the null hypothesis and its invariance and robustness properties. We conclude by an overview of possible modifications of these new tests./Cette these traite de differents aspects statistiques et probabilistes de symetrie et asymetrie univariees et multivariees, et est subdivisee en trois parties distinctes.La premiere partie, qui comprend les chapitres 1, 2 et 3 de la these, est destinee a la resolution de deux conjectures associees aux lois skew-symetriques multivariees. Depuis l'introduction en 1985 par Adelchi Azzalini du plus celebre representant de cette classe de lois, a savoir la loi skew-normale, il est bien connu qu'en un voisinage de la situation symetrique la matrice d'information de Fisher est singuliere et la fonction de vraisemblance profile pour le parametre d'asymetrie admet un point stationnaire quel que soit l'echantillon considere. Des lors, des chercheurs ont essaye de determiner les sous-classes de lois skew-symetriques qui souffrent de chacune de ces problematiques, ce qui a mene aux deux conjectures precitees. Cette these resoud completement ces deux problemes.La deuxieme partie, constituee des chapitres 4 et 5, poursuit le but d'appliquer et de proposer des mechanismes d'asymetrisation tres generaux. Ainsi, au chapitre 4, nous utilisons le mechanisme univarie de Ferreira and Steel (2006) pour construire des tests de symetrie univariee optimaux (au sens de Le Cam) qui sont tres flexibles. En effet, leur mechanisme permettant de transformer une loi symetrique donnee en n'importe quelle loi asymetrique, les contre-hypotheses a la symetrie peuvent prendre toute forme imaginable. Ces mechanismes univaries, outre cette propriete de surjectivite, possedent de nombreux autres attraits, mais ne permettent pas une extension satisfaisante aux dimensions superieures. Pour cette raison, nous proposons au chapitre 5 des mechanismes generaux alternatifs, qui partagent toutes les proprietes de leurs competiteurs de Ferreira and Steel (2006), mais qui en plus sont generalisables a n'importe quelle dimension. Nous demontrons formellement que la surjectivite tient en dimension k > 1 et etudions les caracteristiques principales de ces nouveaux mechanismes multivaries.Finalement, la troisieme partie de cette these, composee du chapitre 6, propose un test de symetrie centrale multivariee en ayant recours aux concepts de profondeur statistique et de runs. Ce test etend le celebre test de runs univarie de McWilliams (1990) aux dimensions superieures. Nous en analysons le comportement asymptotique (surtout en dimension k = 2) sous l'hypothese nulle et les proprietes d'invariance et de robustesse. Nous concluons par un apercu sur des modifications possibles de ces nouveaux tests.

4 citations

Posted Content•
Extensions of the Burr Type XII distribution and Applications

[...]

Meitner Cadena
29 May 2017-arXiv: Methodology
TL;DR: The extended Burr Type XII (EBXIID) family as discussed by the authors is a family of distributions that allows the analysis not only of extreme values as the BXII distribution, but also of light-tailed data.
Abstract: The Burr type XII (BXII) distribution has been largely used in different fields due to its great flexibility for fitting data. These applications have typically involved data showing heavy-tailed behaviors. In order to give more flexibility to the BXII distribution, in this paper, modifications to this distribution through the use of parametric functions are introduced. For instance, members of this new family of distributions allow the analysis not only of data containing extreme values as the BXII distribution, but also of light-tailed data. We refer to this new family of distributions as the extended Burr Type XII distribution (EBXIID) family. Statistical properties of members of the EBXIID family are discussed. The maximum likelihood method is proposed for estimating model parameters. The performance of the new family of distributions is studied using simulations. Applications of the new models to real data sets coming from different domains show that models of the EBXIID family are an alternative to other known distributions.

4 citations

Cites methods from "Flexible Class of Skew‐Symmetric Di..."

  • ...[5] applied to these data the beta generalized exponential (BGE) and the beta exponential (BE) distributions; Jones and Pewsey [29] modeled them using the sinh-arcsinh (SHASH) distribution and its normal (SHASH-N), normal-tailed (SHASH-NT) and symmetric (SHASH-S) sub-models; and, Ma and Genton [38] fitted them using flexible generalized skew-normal (FGSN) distributions by varying a parameter K ....

    [...]

  • ...On the other hand, Barreto et al. [5] applied to these data the beta generalized exponential (BGE) and the beta exponential (BE) distributions; Jones and Pewsey [29] modeled them using the sinh-arcsinh (SHASH) distribution and its normal (SHASH-N), normal-tailed (SHASH-NT) and symmetric (SHASH-S) sub-models; and, Ma and Genton [38] fitted them using flexible generalized skew-normal (FGSN) distributions by varying a parameter K ....

    [...]

Journal Article•DOI•
On Properties of the Bimodal Skew-Normal Distribution and an Application

[...]

David Elal Olivero, Juan F. Olivares Pacheco, Osvaldo Venegas, Heleno Bolfarine, Héctor W. Gómez 
02 May 2020
TL;DR: In this article, an alternative construction for the bimodal skew-normal distribution is proposed based on a study of the mixture of skew normal distributions, and parameters are estimated using the maximum likelihood estimation method.
Abstract: The main object of this paper is to develop an alternative construction for the bimodal skew-normal distribution. The construction is based upon a study of the mixture of skew-normal distributions. We study some basic properties of this family, its stochastic representations and expressions for its moments. Parameters are estimated using the maximum likelihood estimation method. A simulation study is carried out to observe the performance of the maximum likelihood estimators. Finally, we compare the efficiency of the new distribution with other distributions in the literature using a real data set. The study shows that the proposed approach presents satisfactory results.

4 citations

Skewed Bernstein-von Mises theorem and skew-modal approximations

[...]

Daniele Durante, Francesco Pozza, Botond Szabo
08 Jan 2023
TL;DR: In this article , the authors provide rigorous theoretical arguments for such a behavior by deriving a novel limiting law that coincides with a closed-form and tractable skewed generalization of Gaussian densities, and yields a total variation distance from the exact posterior whose convergence rate can be shown to be of order 1 /n , up to a logarithmic factor.
Abstract: Deterministic Gaussian approximations of intractable posterior distributions are common in Bayesian inference. From an asymptotic perspective, a theoretical justification in regular parametric settings is provided by the Bernstein–von Mises theorem. However, such a limiting behavior may require a large sample size before becoming visible in practice. In fact, in situations with small–to–moderate sample size, even simple parametric models often yield posterior distributions which are far from resembling a Gaussian shape, mainly due to skewness. In this article, we provide rigorous theoretical arguments for such a behavior by deriving a novel limiting law that coincides with a closed–form and tractable skewed generalization of Gaussian densities, and yields a total variation distance from the exact posterior whose convergence rate can be shown to be of order 1 /n , up to a logarithmic factor. Such a result provides a substantial accuracy improvement over the classical Bernstein–von Mises theorem whose convergence rate, under similar conditions, is of order 1 / √ n , again up to a logarithmic factor. In contrast to higher–order approximations based on, e.g., Edgeworth expansions, which require finite truncations for inference, possibly leading to even negative densities, our theory characterizes the limiting behavior of Bayesian posteriors with respect to a sequence of valid and tractable densities. These advancements further motivate a practical plug–in version which replaces the unknown model parameters with the corresponding maximum–a–posteriori estimate to obtain a novel skew–modal approximation achieving the same improved rate of convergence of our skewed Bernstein–von Mises theorem. Extensive simulations and a real–data application confirm that our new theory closely matches the empirical behavior observed in practice even in finite, possibly small, sample regimes. The proposed skew–modal approximation further exhibits improved accuracy not only relative to classical Laplace approximation, but also with respect to state–of–the–art Gaussian and non– Gaussian approximations from mean–field variational Bayes and expectation–propagation.

4 citations

References
More filters
Book•
Functional analysis

[...]

Walter Rudin
01 Jan 1973

14,545 citations

Journal Article•
A class of distributions which includes the normal ones

[...]

Adelchi Azzalini
01 Jan 1985-Scandinavian Journal of Statistics
TL;DR: In this paper, a nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard.
Abstract: On introduit une nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard

2,470 citations

"Flexible Class of Skew‐Symmetric Di..." refers background or methods in this paper

  • ...This representation has been used by Azzalini & Capitanio (2003) to define certain distributions by perturbation of symmetry....

    [...]

  • ...For K = 1, the pdf is always unimodal as was already noted by Azzalini (1985) for the univariate skew-normal distribution....

    [...]

  • ...The case K = 1 corresponds to Azzalini & Dalla Valle's (1996) bivariate skew-normal distribution, which cannot capture the bimodality....

    [...]

  • ...For K ¼ 1, the pdf is always unimodal as was already noted by Azzalini (1985) for the univariate skew-normal distribution....

    [...]

  • ...In particular, for ,il = ,B2 = /33 = & = 0, the pdf is exactly the bivariate skew-normal proposed by Azzalini & Dalla Valle (1996), and known to be unimodal (see Fig....

    [...]

Book•
Smoothing Methods in Statistics

[...]

Jeffrey S. Simonoff1•
University of Wisconsin-Madison1
06 Jun 1996
TL;DR: In this article, a nonparametric/parametric Compromise is used to improve the kernel density estimator, and the effect of simple Density Estimators is discussed.
Abstract: 1. Introduction.- 1.1 Smoothing Methods: a Nonparametric/Parametric Compromise.- 1.2 Uses of Smoothing Methods.- 1.3 Outline of the Chapters.- Background material.- Computational issues.- Exercises.- 2. Simple Univariate Density Estimation.- 2.1 The Histogram.- 2.2 The Frequency Polygon.- 2.3 Varying the Bin Width.- 2.4 The Effectiveness of Simple Density Estimators.- Background material.- Computational issues.- Exercises.- 3. Smoother Univariate Density Estimation.- 3.1 Kernel Density Estimation.- 3.2 Problems with Kernel Density Estimation.- 3.3 Adjustments and Improvements to Kernel Density Estimation.- 3.4 Local Likelihood Estimation.- 3.5 Roughness Penalty and Spline-Based Methods.- 3.6 Comparison of Univariate Density Estimators.- Background material.- Computational issues.- Exercises.- 4. Multivariate Density Estimation.- 4.1 Simple Density Estimation Methods.- 4.2 Kernel Density Estimation.- 4.3 Other Estimators.- 4.4 Dimension Reduction and Projection Pursuit.- 4.5 The State of Multivariate Density Estimation.- Background material.- Computational issues.- Exercises.- 5. Nonparametrie Regression.- 5.1 Scatter Plot Smoothing and Kernel Regression.- 5.2 Local Polynomial Regression.- 5.3 Bandwidth Selection.- 5.4 Locally Varying the Bandwidth.- 5.5 Outliers and Autocorrelation.- 5.6 Spline Smoothing.- 5.7 Multiple Predictors and Additive Models.- 5.8 Comparing Nonparametric Regression Methods.- Background material.- Computational issues.- Exercises.- 6. Smoothing Ordered Categorical Data.- 6.1 Smoothing and Ordered Categorical Data.- 6.2 Smoothing Sparse Multinomials.- 6.3 Smoothing Sparse Contingency Tables.- 6.4 Categorical Data, Regression, and Density Estimation.- Background material.- Computational issues.- Exercises.- 7. Further Applications of Smoothing.- 7.1 Discriminant Analysis.- 7.2 Goodness-of-Fit Tests.- 7.3 Smoothing-Based Parametric Estimation.- 7.4 The Smoothed Bootstrap.- Background material.- Computational issues.- Exercises.- Appendices.- A. Descriptions of the Data Sets.- B. More on Computational Issues.- References.- Author Index.

1,719 citations

Journal Article•DOI•
The multivariate skew-normal distribution

[...]

Adelchi Azzalini, A. Dalla Valle
01 Dec 1996-Biometrika
TL;DR: In this article, a multivariate parametric family such that the marginal densities are scalar skew-normal is introduced, and its properties are studied with special emphasis on the bivariate case.
Abstract: SUMMARY The paper extends earlier work on the so-called skew-normal distribution, a family of distributions including the normal, but with an extra parameter to regulate skewness. The present work introduces a multivariate parametric family such that the marginal densities are scalar skew-normal, and studies its properties, with special emphasis on the bivariate case.

1,478 citations

Journal Article•DOI•
Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution

[...]

Adelchi Azzalini1, Antonella Capitanio1•
University of Padua1
01 May 2003-Journal of The Royal Statistical Society Series B-statistical Methodology
TL;DR: In this paper, a fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities.
Abstract: Summary. A fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is sufficiently general to encompass some recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew t-density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.

1,215 citations

"Flexible Class of Skew‐Symmetric Di..." refers background in this paper

  • ...Finally, note that the stochastic representation of FSSdistributions follows from the stochastic representation of SS distributions described byWang et al. (2004), see also Azzalini & Capitanio (2003)....

    [...]

  • ...Similarly, multivariate distributions such as skew-t (Branco & Dey, 2001; Azzalini & Capitanio, 2003; Jones & Faddy, 2003; Sahu et al., 2003), skew-Cauchy (Arnold & Beaver, 2000) and other skewelliptical ones (Azzalini & Capitanio, 1999; Branco & Dey, 2001; Sahu et al., 2003) can be represented by…...

    [...]

  • ...Jones & Faddy (2003) and Azzalini & Capitanio (2003) fit two forms of skew-t distributions to these data....

    [...]

  • ...If each term has an odd order (all ks are odd), then the polynomial is called an odd polynomial, whereas if each term has an even order (all ks are even), it is called an even polynomial....

    [...]

Related Papers (5)
A class of distributions which includes the normal ones

[...]

01 Jan 1985-Scandinavian Journal of Statistics
Adelchi Azzalini
Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution

[...]

01 May 2003-Journal of The Royal Statistical Society Series B-statistical Methodology
Adelchi Azzalini, Antonella Capitanio
The multivariate skew-normal distribution

[...]

01 Dec 1996-Biometrika
Adelchi Azzalini, A. Dalla Valle
Skew-Elliptical Distributions and Their Applications : A Journey Beyond Normality

[...]

27 Jul 2004
Marc G. Genton
A General Class of Multivariate Skew-Elliptical Distributions

[...]

01 Oct 2001-Journal of Multivariate Analysis
Márcia D. Branco, Dipak K. Dey