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Flexible Class of Skew‐Symmetric Distributions

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
2
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
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Journal ArticleDOI
TL;DR: In this article, a class of distributions which are useful in making inferences about the sum of values from a normal and a doubly truncated normal distribution is proposed, which is associated with the conditional distributions of truncated bivariate normal.
Abstract: This paper proposes a class of distributions which is useful in making inferences about the sum of values from a normal and a doubly truncated normal distribution. It is seen that the class is associated with the conditional distributions of truncated bivariate normal. The salient features of the class are mathematical tractability and strict inclusion of the normal and the skew-normal laws. Further it includes a shape parameter, to some extent, controls the index of skewness so that the class of distributions will prove useful in other contexts. Necessary theories involved in deriving the class of distributions are provided and some properties of the class are also studied.

2 citations

Journal ArticleDOI
TL;DR: Monte Carlo simulations show that the proposed estimating procedures can provide desirable asymptotic properties of the ML estimates and demand less computational burden in comparison with other existing algorithms based on convolution representations.
Abstract: A scale-shape mixtures of flexible generalized skew normal (SSMFGSN) distributions is proposed as a novel device for modeling asymmetric data. Computationally feasible EM-type algorithms derived from the selection mechanism are presented to compute maximum likelihood (ML) estimates of SSMFGSN distributions. Some characterizations and probabilistic properties of the SSMFGSN distributions are also studied. Monte Carlo simulations show that the proposed estimating procedures can provide desirable asymptotic properties of the ML estimates and demand less computational burden in comparison with other existing algorithms based on convolution representations. The usefulness of the proposed methodology is illustrated by analyzing a real dataset.

2 citations

DOI
01 Jan 2007
TL;DR: A flexible parametric approach is suggested for adjusting the measurement error bias while estimating the response-covariate relationship through logistic regression model and this method is applicable to the other members of the generalized linear models, and other types of non-linear regression models too.
Abstract: In many fields of statistical application the fundamental task is to quantify the association between some explanatory variables or covariates and a response or outcome variable through a suitable regression model. The accuracy of such quantification depends on how precisely we measure the relevant covariates. In many instances, we can not measure some of the covariates accurately, rather we can measure noisy versions of them. In statistical terminology this is known as measurement errors or errors in variables. Regression analyses based on noisy covariate measurements lead to biased and inaccurate inference about the true underlying response-covariate associations. In this thesis we investigate some aspects of measurement error modelling in the case of binary logistic regression models. We suggest a flexible parametric approach for adjusting the measurement error bias while estimating the response-covariate relationship through logistic regression model. We investigate the performance of the proposed flexible parametric approach in comparison with the other flexible parametric and nonparametric approaches through extensive simulation studies. We also compare the proposed method with the other competitive methods with respect to a real-life data set. Though emphasis is put on the logistic regression model the proposed method is applicable to the other members of the generalized linear models, and other types of non-linear regression models too. Finally, we develop a new computational technique to approximate the large sample bias that my arise due to exposure model misspecification in the estimation of the regression parameters in a measurement error scenario.

2 citations

Journal ArticleDOI
TL;DR: In this article , the pricing of American options whose asset price dynamics follow Azzalini Itô-McKean skew Brownian motions is considered, and the corresponding optimal stopping time problem is then formulated and the main properties of its value function are provided.

1 citations

Journal ArticleDOI
23 Jun 2021-Symmetry
TL;DR: In this article, a new family of distributions for modelling censored multimodal data is introduced, which extends the widely known tobit model by introducing two parameters that control the shape and the asymmetry of the distribution.
Abstract: This paper introduces a new family of distributions for modelling censored multimodal data. The model extends the widely known tobit model by introducing two parameters that control the shape and the asymmetry of the distribution. Basic properties of this new family of distributions are studied in detail and a model for censored positive data is also studied. The problem of estimating parameters is addressed by considering the maximum likelihood method. The score functions and the elements of the observed information matrix are given. Finally, three applications to real data sets are reported to illustrate the developed methodology.

1 citations

References
More filters
Book
01 Jan 1973

14,545 citations

Journal Article
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
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 ArticleDOI
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 ArticleDOI
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....

    [...]