truncated-exponential skew-symmetric distributions · vahid nassiri and adel mohammadpour...

Truncated-Exponential Skew-SymmetricDistributions

Saralees Nadarajah, Vahid Nassiri & Adel Mohammadpour

First version: 15 December 2009

Research Report No. 19, 2009, Probability and Statistics Group

School of Mathematics, The University of Manchester

Truncated–exponential skew–symmetric distributions

by

Saralees Nadarajah

School of Mathematics

University of Manchester

Manchester M13 9PL, UK

Vahid Nassiri and Adel Mohammadpour

Department of Statistics

Amirkabir University of Technology (Tehran Polytechnic)

Tehran 15914, IRAN

Abstract: The family of skew distributions introduced by Azzalini and extended by others has

received widespread attention. However, it suffers from complicated inference procedures. In this

paper, a new family of skew distributions that overcomes the difficulties is introduced. This new

family belongs to the exponential family. Many properties of this family are studied, inference

procedures developed and simulation studies performed to assess the procedures. Some particular

cases of this family, evidence of its flexibility and a real data application are presented. At least

ten advantages of the new family over Azzalini’s distributions are established.

Keywords: Azzalini skew distributions; Estimation; Exponential family; Skewness.

1 Introduction

The need for skew distributions arises in every area of the sciences, engineering and medicine.

The most common approach for the construction of skew distributions is to introduce skewness

into known symmetric distributions. Ferreira and Steel (2006) presented a unified approach for

constructing such distributions. Let X be a symmetric random variable about zero with fX(·) and

FX(·) denoting its probability density function (pdf) and cumulative distribution function (cdf),

respectively. Define Y as a new random variable with the pdf

fY (y) = fX(y)ω (FX(y)) , y ∈ R, (1)

where ω(·) is a pdf on the unit interval (0, 1). Then Y is said to be a skew version of the symmetric

random variable X (Definition 1, Ferreira and Steel, 2006).

The unified family in (1) contains many of the known families of skew distributions. If ω(·) is

a beta pdf then (1) yields the family of distributions studied by Jones (2004). If ω(x) = 2λ/(λ2 +

1

1)fX(λsign(1/2−x)F−1X (x))/fX(F−1

X (x)), 0 < λ < ∞ then (1) yields the family of distributions

studied by Fernandez and Steel (1998). The most popular version of (1) are the skew distributions

introduced by Azzalini (1985). Take ω(x) = 2FX(λF−1X (x)). Then (1) reduces to

fY (y) = 2fX(y)FX (λy), y ∈ R, λ ∈ R. (2)

We shall refer to distributions with the pdf (2) as Azzalini skew distributions. A particular case of

(2) is the class of skew–normal distributions obtained by setting fX(·) = φ(·), the standard normal

pdf, and FX(·) = Φ(·), the standard normal cdf. The family given by (2) and the skew–normal class

have been studied and extended by many authors, see Azzalini (1986), Azzalini and Dalla Valle

(1996), Azzalini and Capitanio (1999), Arnold and Beaver (2000), Pewsey (2000), Loperfido (2001),

Arnold and Beaver (2002), Nadarajah and Kotz (2003), Gupta and Gupta (2004), Behboodian et

al. (2006), Nadarajah and Kotz (2006), Huang and Chen (2007) and Sharafi and Behboodian

(2007).

In this paper, we shall focus on the family of Azzalini skew distributions. One of the main

difficulties of this family is about making inferences on its skewness parameter λ. It is known, for

example, that the maximum likelihood estimator for λ for Azzalini skew–normal distributions does

not always exist, see Pewsey (2000).

The aim of this paper is to introduce a new family of distributions as a competitor to (2). This

new family belongs to the exponential family, so inferences on the skewness parameter become

much easier. Its pdf is a particular case of (1) with ω(·) corresponding to a truncated exponential

distribution: this particular choice is made because it yields a natural extension of (2) to an

exponential family. We shall refer to the new family as truncated–exponential skew–symmetric

distributions. We establish at least ten advantages of the new family over (2).

The contents of the rest of this paper are organized as follows. In Section 2 we introduce the

new family of distributions and study its mathematical properties. Section 3 provides inference

procedures for maximum likelihood estimation, moments estimation, hypotheses testing and simu-

lation. Some particular cases and evidence of flexibility of the new family are presented in Section

4. A simulation study is performed in Section 5 to compare the performances of the methods of

maximum likelihood and moments. A real data application to illustrate the usefulness of the new

family is presented in Section 6. Finally, some conclusions including a list of ten advantages of the

proposed family are noted in Section 7.

2 Truncated–exponential skew–symmetric distributions and their

properties

In this section we introduce the truncated–exponential skew–symmetric random variable and study

its properties.

2

Definition 1. A random variable Y has the truncated–exponential skew–symmetric distribution

with parameter λ, TESS(λ), if its pdf has the following form:

fY (y;λ) =λ

1 − exp(−λ)fX(y) exp {−λFX(y)} , y ∈ R, λ ∈ R, (3)

where fX(·) and FX(·) are, respectively, the pdf and the cdf of a symmetric random variable X

about zero.

We shall refer to λ in (3) as the shape parameter. From (3) an explicit expression for the cdf

of Y is obtained as:

FY (y;λ) =1 − exp {−λFX(y)}

1 − exp(−λ), y ∈ R, λ ∈ R. (4)

The inverse cdf is:

F−1Y (y;λ) = F−1

X (−(1/λ) ln {1 − y (1 − exp(−λ))}) , y ∈ R, λ ∈ R. (5)

We can use (5) for several purposes, e.g. for finding quantiles or generating random numbers. If

P (Y ≤ qp) = p, then the pth quantile, qp, can be obtained using (5). We can also use (5) and the

inversion method to generate a random sample, Y1, . . . , Yn, from TESS(λ).

Note that (3) is a particular case of (1) for ω(x) = λ exp(−λx)/{1 − exp(−λ)}, a truncated

exponential pdf. By introducing the exponential function and replacing FX(λy) by λFX(y), we

have in (3) a natural extension of (2) to an exponential family. Note also that (3) is symmetric

with respect to λ in the sense that f(y;λ) = f(−y;−λ). Furthermore, in the limit, as λ → 0,

Y ∼ TESS(λ) has the same distribution as X. Note that (3) is undefined at λ = 0, so λ = 0

should be interpreted as the limit λ → 0. If λ → ±∞ then Y ∼ TESS(λ) reduces to degenerate

random variables. If λ → ∞ then FY (y) = 0 if FX(y) = 0 and FY (y) = 1 for all other values of y.

If λ → −∞ then FY (y) = 1 if FX(y) = 1 and FY (y) = 0 for all other values of y.

The shape parameter, λ, in (3) satisfies the first three ordering properties of van Zwet (1964).

In particular, if λ is considered as a function of Y , then: 1) λ(aY + b) = λ(Y ) for all a > 0 and

b ∈ R; 2) λ(Y ) = 0 for symmetric Y ; 3) λ(−Y ) = −λ(Y ). The fourth and the last ordering of

van Zwet (1964) is: suppose λ1(Y ) ≤ λ2(Y ) and let fi, Fi and F−1i denote the pdf, the cdf and

the inverse cdf corresponding to λi(Y ), i = 1, 2; then λ1(Y ) is said to be smaller than λ2(Y ) in

convex order if and only if F−12 (F1(y)) is convex in y or equivalently f1(F

−11 (u))/f2(F

−12 (u)) is

increasing in u ∈ [0, 1]. Unfortunately, this ordering property does not appear to be satisfied by

(3): see Figure 1 showing the plot of f1(F−11 (u))/f2(F

−12 (u)) versus u when X is a standard normal

random variable, λ1 = −10 and λ2 = 5.

[Figure 1 about here.]

The modes of (3) are the roots of the equation

f′

X(y)

fX(y)= λfX(y). (6)

3

Note that the roots are to the left (respectively, right) of zero as λ > 0 (respectively, λ < 0). The

root, say y = y0, corresponds to a maximum if

f′′

X(y0)

fX(y0)− λf

′

X(y0) <

(f

′

X(y0)

fX(y0)

)2

. (7)

The root corresponds to a minimum if

f′′

X(y0)

fX(y0)− λf

′

X(y0) >

(f

′

X(y0)

fX(y0)

)2

. (8)

The root corresponds to a point of inflexion if

f′′

X(y0)

fX(y0)− λf

′

X(y0) =

(f

′

X(y0)

fX(y0)

)2

. (9)

The mode corresponding to a maximum is unique if the y0 is such that f ′

X(y) > λf2X(y) for all

y < y0 and f ′

X(y) < λf2X(y) for all y > y0. The mode corresponding to a minimum is unique if

the y0 is such that f ′

X(y) < λf2X(y) for all y < y0 and f ′

X(y) > λf2X(y) for all y > y0. The mode

corresponding to a point of inflexion is unique if the y0 is such that either f ′

X(y) < λf2X(y) for all

y 6= y0 or f ′

X(y) > λf2X(y) for all y 6= y0. Note that (3) can be multi–modal if, for example, fX(·)

is multi–modal.

Note that the tails of Y ∼ TESS(λ) have the same behavior as the tails of X because fY (y) ∼λ/{exp(λ)−1}fX (y) as y → ∞ and fY (y) ∼ λ/{1− exp(−λ)}fX(y) as y → −∞. Also 1−FY (y) ∼λ/{exp(λ) − 1}{1 − FX(y)} as y → ∞ and FY (y) ∼ λ/{1 − exp(−λ)}FX (y) as y → −∞.

Let Y ∼ TESS(λ) and T = FX(Y ). Then it is straightforward to show that

FT (t) =1 − exp(−λt)

1 − exp(−λ), t ∈ [0, 1], λ ∈ R,

fT (t) =λ exp(−λt)

1 − exp(−λ), t ∈ [0, 1], λ ∈ R

and

MT (s) =λ

λ − s

1 − exp(s − λ)

1 − exp(−λ), λ ∈ R,

where MT (s) is the moment generating function of T . In particular,

E(T ) =1 − exp(−λ)(λ + 1)

λ {1 − exp(−λ)} , λ ∈ R. (10)

Note that the pdf of T is the same as the ω(·) chosen to construct (3).

4

Entropies are measures of variation of the uncertainty. Let sY and sX denote Shannon entropies

(Shannon, 1951) corresponding to fY (·) and fX(·), respectively. We have by (10):

sY = ln1 − exp(−λ)

λ+

1 − exp(−λ)(λ + 1)

1 − exp(−λ)− λ

1 − exp(−λ)I, (11)

where

I =

∫∞

−∞

ln fX(y)fX(y) exp {−λFX(y)} dy.

By the series expansion for exponential, one can express

I = −sX +

∞∑

k=1

(−1)kλk

k!Ik,

where

Ik =

∫∞

−∞

ln fX(y)fX(y)F kX (y)dy.

Let rY (γ) and rX(γ) denote Renyi entropies (Renyi, 1961) corresponding to fY (·) and fX(·),respectively. Similar calculations using the series expansion for exponential show that

rY (γ) =1

1 − γln

[∫∞

−∞

λγ

{1 − exp(−λ)}γ fγX(y) exp {−γλFX(y)} dy

]

=1

1 − γln

[λγ

{1 − exp(−λ)}γ

∞∑

k=0

(−λγ)k

k!

∫∞

−∞

fγX(y)F k

X(y)dy

]

=1

1 − γln

{∫λγ

{1 − exp(−λ)}γ

[

exp {(1 − γ)rX(γ)} +

∞∑

k=1

(−λγ)k

k!Jk

]}

, (12)

where

Jk =

∫∞

−∞

fγX(y)fX(y)F k

X (y)dy.

The Shannon entropy in (11) can also be obtained as the limit of (12) as γ ↑ 1.

Theorems 1 and 2 consider the moments of Y ∼ TESS(λ).

Theorem 1. Let Y ∼ TESS(λ). If E|X|r, r > 0, exists, then E|Y |r exists.

Proof. We have

E|Y |r =

∫ +∞

−∞

|y|r λ

1 − exp(−λ)fX(y) exp {−λFX(y)} dy

=λ

1 − exp(−λ)

∫ +∞

−∞

|y|r exp {−λFX(y)} fX(y)dy

=λ

1 − exp(−λ)E [|X|r exp {−λFX(X)}] .

The result follows by noting that E[|X|r exp{−λFX(X)}] ≤ max(1, exp(−λ))E[|X|r ].

5

Theorem 2. Let Xk:n denote the kth order statistic for a random sample of size n from fX(·). Let

Y ∼ TESS(λ). If the conditions of Theorem 1 hold then

E (Y r) =λ

1 − exp(−λ)

∞∑

k=0

(−λ)k

(k + 1)!E(Xr

k+1:k+1

).

Proof. By the series expansion for exponential, one can express

E (Y r) =λ

1 − exp(−λ)

∫∞

−∞

yrfX(y) exp {−λFX(y)} dy

=λ

1 − exp(−λ)

∞∑

k=0

(−λ)k

k!

∫∞

−∞

yrfX(y)F kX (y)dy

=λ

1 − exp(−λ)

∞∑

k=0

(−λ)k

k!

1

(k + 1)E(Xr

k+1:k+1

)

So, the result follows.

Let Mk:n(t) = E[exp(tXk:n)] and φk:n(t) = E[exp(itXk:n)] denote, respectively, the moment

generating function (mgf) and the characteristic function (chf) of Xk:n, where i =√−1. It then

follows from Theorem 2 that the mgf and the chf of Y ∼ TESS(λ) are

E[exp(tY )] =λ

1 − exp(−λ)

∞∑

k=0

(−λ)k

(k + 1)!Mk+1:k+1(t)

and

E[exp(itY )] =λ

1 − exp(−λ)

∞∑

k=0

(−λ)k

(k + 1)!φk+1:k+1(t),

respectively.

Let Yk:n denote the kth order statistic for a random sample of size n from fY (·). Write Y = Y (λ)

when Y ∼ TESS(λ). Then the pdf and the cdf of Yk:n can be expressed as

fYk:n(y) =

n! {1 − exp(−λ)}−n

(k − 1)!(n − k)!

k−1∑

l=0

n−k∑

m=0

(k − 1

l

)(n − k

m

)(−1)l+n−k−m

l + m + 1

× exp {−(n − k − m)λ} [1 − exp {−(l + m + 1)λ}] fY ((l+m+1)λ)(y)

and

FYk:n(y) =

n! {1 − exp(−λ)}−n

(k − 1)!(n − k)!

k−1∑

l=0

n−k∑

m=0

(k − 1

l

)(n − k

m

)(−1)l+n−k−m

l + m + 1

× exp {−(n − k − m)λ} [1 − exp {−(l + m + 1)λ}]FY ((l+m+1)λ)(y),

6

respectively. Also the rth moment, the mgf and the chf of Yk:n can be expressed as

E (Y rk:n) =

n! {1 − exp(−λ)}−n

(k − 1)!(n − k)!

k−1∑

l=0

n−k∑

m=0

(k − 1

l

)(n − k

m

)(−1)l+n−k−m

l + m + 1

× exp {−(n − k − m)λ} [1 − exp {−(l + m + 1)λ}] E (Y r((l + m + 1)λ)) ,

E [exp (tYk:n)] =n! {1 − exp(−λ)}−n

(k − 1)!(n − k)!

k−1∑

l=0

n−k∑

m=0

(k − 1

l

)(n − k

m

)(−1)l+n−k−m

l + m + 1

× exp {−(n − k − m)λ} [1 − exp {−(l + m + 1)λ}]MY ((l+m+1)λ)(t)

and

E [exp (itYk:n)] =n! {1 − exp(−λ)}−n

(k − 1)!(n − k)!

k−1∑

l=0

n−k∑

m=0

(k − 1

l

)(n − k

m

)(−1)l+n−k−m

l + m + 1

× exp {−(n − k − m)λ} [1 − exp {−(l + m + 1)λ}]φY ((l+m+1)λ)(t),

respectively. In particular, the rth L-moment (due to Hoskings (1990)) of Y ∼ TESS(λ) can be

expressed as

λr =r−1∑

j=0

(−1)r−1−j

(r − 1

j

)(r − 1 + j

j

)βj ,

where βr = (1/(r+1))E(Yr+1:r+1). The L moments have several advantages over ordinary moments:

for example, they apply for any distribution having finite mean; no higher-order moments need be

finite (Hoskings, 1990).

3 Inference

In this section we draw some inferences for a truncated–exponential skew–symmetric random vari-

able with additional location and scale parameters.

Definition 2. A random variable Y has the truncated–exponential skew–symmetric distribution

with parameters (λ, κ, δ), TESS(λ, κ, δ), if its pdf has the following form:

fY (y;λ, κ, δ) =λ

δ {1 − exp(−λ)}fX

(y − κ

δ

)exp

{−λFX

(y − κ

δ

)}(13)

for y ∈ R, λ ∈ R, κ ∈ R and δ > 0, where fX(·) and FX(·) are, respectively, the pdf and the cdf of

a symmetric random variable X about zero.

Note that the TESS pdf in (13) can be written as h(y)c(λ) exp{w(λ)t(y)}, where h(y) = fX((y−κ)/δ) ≥ 0, c(λ) = λ/{δ(1 − exp(−λ))} > 0, w(λ) = −λ (λ ∈ R) and t(y) = FX((y − κ)/δ) (y ∈ R).

7

So, the pdf belongs to the exponential family with respect to λ (see Lehmann and Casella (1998),

page 23) and∑n

i=1 FX(Zi), where Zi = (Yi − κ)/δ, is a complete sufficient statistic for λ provided

that κ and δ are assumed known.

For estimating (λ, κ, δ), we find their moments and maximum likelihood estimators. Suppose

y1, y2, . . . , yn is a random sample from (13). Let mk = (1/n)∑n

j=1 ykj for k = 1, 2, 3. By equating

the theoretical moments of (13) with the sample moments, we obtain the equations:

λ

1 − exp(−λ)

k∑

l=0

(k

l

)κk−lδl

∞∑

m=0

(−λ)m

(m + 1)!E(X l

m+1:m+1

)= mk

for k = 1, 2, 3. The moments estimators are the simultaneous solutions of these three equations.

Now consider estimation by the method of maximum likelihood. The likelihood function of the

three parameters is

L(λ, κ, δ) =

(λ

δ(1 − exp(−λ))

)n{

n∏

i=1

fX (zi)

}exp

{−λ

n∑

i=1

FX (zi)

},

where zi = (yi − κ)/δ. So, the maximum likelihood estimators are the simultaneous solutions of

the equations

n

(1

λ− 1

exp(λ) − 1

)=

n∑

i=1

FX (zi) , (14)

n∑

i=1

f′

X (zi)

fX (zi)= λ

n∑

i=1

fX (zi) (15)

and

n∑

i=1

zif

′

X (zi)

fX (zi)= λ

n∑

i=1

zifX (zi) − n, (16)

where zi = (yi − κ)/δ.

Theorem 3 shows that (14) always has a root and is unique. The proof of this theorem requires

the following lemma which is straightforward.

Lemma 1. Consider g(x) = 1/x−1/{exp(x)−1}. Then g(x) is strictly decreasing and g(x) ∈ (0, 1).

Theorem 3. If κ and δ are assumed known then the maximum likelihood estimator of λ given by

(14) always exists and is unique.

Proof. Note that (14) can be re–expressed as ng(λ) − r(z) = 0, where r(z) =∑n

i=1 FX(zi). By

Lemma 1, ng(λ) is strictly decreasing and lies in (0, n). So, ng(λ) − r(z) is strictly decreasing and

lies in (−r(z), n − r(z)). Note that n − r(z) > 0 since r(z) < n. So, the theorem follows by the

intermediate value theorem (see Rudin (1976), page 93).

8

The elements of the Fisher information matrix of the maximum likelihood estimators can be

calculated as:

E

(−∂2 ln L

∂λ2

)=

n

λ2− n exp(λ)

{exp(λ) − 1}2 ,

E

(−∂2 ln L

∂λ∂κ

)= −n

δE [fX(Z)] ,

E

(−∂2 ln L

∂λ∂δ

)= −n

δE [ZfX(Z)] ,

E

(−∂2 ln L

∂κ2

)= − n

δ2E

[f

′′

X(Z)

fX(Z)

]+

n

δ2E

(

f′

X(Z)

fX(Z)

)2

+nλ

δ2E[f

′

X(Z)],

E

(−∂2 ln L

∂κ∂δ

)= − n

δ2E

[f

′

X(Z)

fX(Z)

]− n

δ2E

[Z

f′′

X(Z)

fX(Z)

]+

n

δ2E

Z

(f

′

X(Z)

fX(Z)

)2

+nλ

δ2E [fX(Z)]

+nλ

δ2E[Zf

′

X(Z)]

and

E

(−∂2 ln L

∂δ2

)= − n

δ2− 2n

δ2E

[

Zf

′

X(Z)

fX(Z)

]

− n

δ2E

[

Z2 f′′

X(Z)

fX(Z)

]

+n

δ2E

Z2

(f

′

X(Z)

fX(Z)

)2

+2nλ

δ2E [ZfX(Z)] +

nλ

δ2E[Z2f

′

X(Z)],

where Z = (Y − κ)/δ and Y ∼ TESS(λ, κ, δ). It follows that the standard error for λ has a closed

form if κ and δ are assumed known.

In general, the elements of the Fisher information matrix will have to be computed numerically.

If λ = 0 then the last three elements reduce to

E

(−∂2 ln L

∂κ2

)= − n

δ2E

[f

′′

X(Z)

fX(Z)

]

+n

δ2E

(

f′

X(Z)

fX(Z)

)2

,

E

(−∂2 ln L

∂κ∂δ

)= − n

δ2E

[f

′

X(Z)

fX(Z)

]

− n

δ2E

[

Zf

′′

X(Z)

fX(Z)

]

+n

δ2E

Z

(f

′

X(Z)

fX(Z)

)2

and

E

(−∂2 ln L

∂δ2

)= − n

δ2− 2n

δ2E

[Z

f′

X(Z)

fX(Z)

]− n

δ2E

[Z2 f

′′

X(Z)

fX(Z)

]+

n

δ2E

Z2

(f

′

X(Z)

fX(Z)

)2

,

9

where Z = (Y − κ)/δ and Y ∼ TESS(0, κ, δ). If in addition X is standard normal then

E

(−∂2 ln L

∂κ2

)=

n

δ2,

E

(−∂2 ln L

∂κ∂δ

)=

2n

δ2E [Z]

and

E

(−∂2 ln L

∂δ2

)=

n

δ2

{3E[Z2]− 1}

,

where Z = (Y − κ)/δ and Y ∼ TESS(0, κ, δ).

We noted earlier that T =∑n

i=1 FX(Zi) is a sufficient statistic for λ provided that κ and δ

are assumed known. It can be noted further that the TESS family has the monotone likelihood

ratio (MLR) property with respect to T . So, by using Karlin–Rubin’s theorem, one can find the

uniformly most powerful (UMP) level (size) α test for testing H0 : λ ≤ λ0 versus H1 : λ > λ0, i.e.

if

ϕ(X) =

{1, if T ≥ t0,

0, if T < t0

then ϕ(X) is a UMP size α test provided that κ and δ are known, where P (T ≥ t0) = α. By (5),

t0 = −(1/λ) ln{1 − (1 − α)(1 − exp(−λ))} for the case n = 1.

For testing H0 : F = F0, where F0 is a known TESS cdf, against H1 : F 6= F0, one can use

nonparametric goodness of fit tests such as ones based on the chi-square test or the Kolmogrov–

Smirnov test. These tests can still be used if F0 is unknown and estimated from some data. In this

case, the critical values for the Kolmogrov–Smirnov test can be obtained by simulation. Since the

TESS cdf has a closed form, performing these tests is straightforward. A quantile–quantile plot or

a probability–probability plot can also be used as informal checks for H0 : F = F0.

Finally, consider simulating truncated–exponential skew–symmetric variates. As mentioned in

Section 2, the inversion method can be applied since the inverse cdf of TESS(λ) is given by (5).

Another method for simulation is the rejection method. If λ > 0 then the following scheme holds:

1. simulate X = x from the pdf fX(·);

2. simulate Y = UMfX(x), where U is a uniform variate on the unit interval [0, 1] and M =

λ/{1 − exp(−λ)};

3. accept X = x as a TESS(λ) variate if Y < fY (x). If Y ≥ fY (x) return to step 2.

If λ < 0 then apply the above scheme with M = −λ/{1 − exp(λ)} and take the negatives of the

simulated variates.

10

4 Some particular cases

In this section we study some particular cases of truncated–exponential skew–symmetric distribu-

tions. Here, we consider normal, t and Cauchy cases (Cauchy is a particular case of t and normal is

a limiting case of t). As Nadarajah and Kotz (2006) did, some other distributions such as Laplace,

logistic and uniform can also be studied.

If fX(·) = φ(·) and FX(·) = Φ(·), in Definition 1, then (3) yields the pdf:

fY (y) =λ

1 − exp(−λ)φ(y) exp {−λΦ(y)} , y ∈ R, λ ∈ R. (17)

The corresponding Azzalini’s distribution has the pdf

fY (y) = 2φ(y)Φ(λy), y ∈ R, λ ∈ R. (18)

We shall refer to (17) as the truncated–exponential skew–normal distribution with parameter λ,

ES–normal(λ). It follows by Theorem 1 that E|Y |r exists for all r > 0. So, by Theorem 2 and the

results in Nadarajah (2007a),

E (Y r) =2r/2λ√

π {1 − exp(−λ)}

∞∑

k=0

(−λ)k

2kk!

k∑

p = 0

p + r even

(k

p

)π−p/22pΓ

(p + r + 1

2

)

×F(p)A

(p + r + 1

2;1

2, . . . ,

1

2;3

2, . . . ,

3

2;−1, . . . ,−1

),

where F(n)A (· · · ) denotes the Lauricella function of type A (Exton, 1978) defined by

F(n)A (a, b1, . . . , bn; c1, . . . , cn;x1, . . . , xn)

=∞∑

m1=0

· · ·∞∑

mn=0

(a)m1+···+mn(b1)m1

· · · (bn)mn

(c1)m1· · · (cn)mn

xm1

1 · · · xmn

n

m1! · · ·mn!,

where (f)k = f(f + 1) · · · (f + k − 1) denotes the ascending factorial with the convention (f)0 = 1.

One can show using equations (6)–(9) that (17) has a unique mode corresponding to a maximum

at y0, the unique root of λφ(y) + y = 0.

If fX(·) and FX(·) in Definition 1 are the Student’s t pdf and the Student’s t cdf with ν degrees

of freedom, respectively, then (3) yields the pdf:

fY (y) =λ

1 − exp(−λ)

1√νB(ν/2, 1/2)

(1 +

y2

ν

)−(1+ν)/2

exp {−λFX(y)} , y ∈ R, λ ∈ R. (19)

where B(a, b) = Γ(a)Γ(b)/Γ(a + b) is the beta function. The corresponding Azzalini’s distribution

has the pdf

fY (y) =2√

νB(ν/2, 1/2)

(1 +

y2

ν

)−(1+ν)/2

FX(λy), y ∈ R, λ ∈ R. (20)

11

For general ν, the cdf term, FX(y), takes the form

FX(y) =1

2+

yΓ ((ν + 1)/2)√πνΓ (ν/2)

2F1

(1

2,ν + 1

2;3

2;−y2

ν

), (21)

where 2F1(· · · ) denotes the Gauss hypergeometric function defined by

2F1 (a, b; c;x) =∞∑

k=0

(a)k (b)k(c)k

xk

k!.

If ν is an integer then one can simplify (21) to:

FX(y) =

1

2+

1

πarctan

(y√ν

)+

1

2π

(ν−1)/2∑

i=1

B

(i,

1

2

)νi−1/2y

(ν + y2)i, for ν odd,

1

2+

1

2π

ν/2∑

i=1

B

(i − 1

2,1

2

)νi−1y

(ν + y2)i−1/2, for ν even,

see Nadarajah and Kotz (2003). We shall refer to (19) as the truncated–exponential skew–t(ν)

distribution with ν degrees of freedom and parameter λ, ES–t(λ, ν). It follows from Theorem 1

that E|Y |r exists for r < ν. So, if r < ν then by Theorem 2 and the results in Nadarajah (2007b),

E (Y r) =λ

1 − exp(−λ)

∞∑

k=0

(−λ)k

(k + 1)!{A(r, k + 1, k + 1) + (−1)rA(r, k + 1, 1)} ,

where

A(k, n, r) =n!νk/2

2n(r − 1)!(n − r)!

r−1∑

p=0

n−r∑

q=0

(r − 1

p

)(n − r

q

)(−1)q2p+q

×B−1−p−q(1/2, ν/2)B

(ν − k

2,k + p + q + 1

2

)

×F 1:21:1

((k + p + q + 1

2

):

(1 − ν

2,1

2

); . . . ;

(1 − ν

2,1

2

);

(ν + p + q + 1

2

):

(3

2

); . . . ;

(3

2

); 1, . . . , 1

),

where FA:BC:D (· · · ) denotes the generalized Kampe de Feriet function (Exton, 1978) defined by

FA:BC:D ((a) : (b1); . . . ; (bn); (c) : (d1); . . . ; (dn);x1, . . . , xn)

=∞∑

m1=0

· · ·∞∑

mn=0

((a))m1+···+mn((b1))m1

· · · ((bn))mn

((c))m1+···+mn((d1))m1

· · · ((dn))mn

xm1

1 · · · xmn

n

m1! · · ·mn!,

where a = (a1, a2, . . . , aA), bi = (bi,1, bi,2, . . . , bi,B) for i = 1, 2, . . . , n, c = (c1, c2, . . . , cC), di =

(di,1, di,2, . . . , di,D) for i = 1, 2, . . . , n, and ((f))k = ((f1, f2, . . . , fp))k = (f1)k(f2)k · · · (fp)k denotes

the product of ascending factorials. The mode of (19) is the root of the equation y(1+y2/ν)(ν−1)/2 =

−λ√

ν/{(1 + ν)B(ν/2, 1/2)}. It is well known that the asymptotic distribution of Student’s t as

12

ν → ∞ is standard normal, see e.g. Johnson et al. (1995, page 363). A similar result can be stated

for ES–t(λ, ν): if Z ∼ ES–normal(λ), Y ∼ ES–t(λ, ν) and ν → ∞ then Z and Y are identically

distributed.

A particular case of (19) with interesting properties is:

fY (y) =λ

1 − exp(−λ)

1

π(1 + y2)exp

{−λ

(1

2+

1

πarctan(y)

)}, y ∈ R, λ ∈ R. (22)

We shall refer to (22) as the truncated–exponential skew–Cauchy distribution, ES–Cauchy. Note

that (22) is a particular case of the Pearson type IV distribution, so the ES–Cauchy is a Pearson

type IV distribution. Equations (4) and (5) for the ES–Cauchy have closed forms. ES–Cauchy has

the heaviest tails within the class of distributions ES–t(λ, ν) if ν is limited to be an integer. It

follows by Theorem 1 that E|Y |r does not exist for all r ≥ 1. Furthermore, one can show using

equations (6)–(9) that (22) has a unique mode corresponding to a maximum at y0 = −λ/(2π).

[Figures 2 to 5 about here.]

Figure 2 to 5 show how the skewness and kurtosis measures of (19) compare to those of (20)

for ν = 5, 10, 20,∞ and λ = −100,−99, . . . , 99, 100 (note that (19) and (20) reduce to (17) and

(18), respectively, in the case ν = ∞). The formulas used to compute the skewness and kurtosis

measures are:

Skewness(Y ) =E(Y 3)− 3E(Y )E

(Y 2)

+ 2E3(Y ){E(Y 2)− E2 (Y )

}3/2

and

Kurtosis(Y ) =E(Y 4)− 4E(Y )E

(Y 3)

+ 6E(Y 2)E2(Y ) − 3E4(Y )

{E(Y 2)− E2 (Y )

}2 .

It is evident from the figures that the truncated–exponential skew distributions take a wider range

of values for both skewness and kurtosis. The two curves for skewness in Figures 4 and 5 appear to

approach the same limit as λ → ±∞: this was verified by redrawing the figures over a wider range

of λ values. The same comment applies with respect to the two curves for kurtosis in Figure 5. The

gain in terms of skewness and kurtosis appears to be greater for the distributions with heavier tails.

In the figures, the truncated–exponential skew–t(5) distribution achieves the greatest gain. This is

interesting because heavy tail distributions are becoming increasing popular models for real–world

data because their tails are more realistic than the normal tails.

5 Simulation study

In this section, we compare the performances of the methods of moments and maximum likelihood

presented in Section 3. For this purpose, we generated samples of size n = 100 from (13) for

13

λ = −1, 1, 2, 5, κ = −1, 0, 1, 5 and δ = 1, 2, 5, 10 when X is standard normal and Student’s t with

ν = 5, 10, 20. For each sample, we computed the moments and maximum likelihood estimates

following the procedures described in Section 3. We repeated this process 100 times and computed

the average of the estimates (AE) and the mean squared error (MSE). The results are reported in

Tables 1 to 4.

[Table 1 to 4 about here.]

It is clear that maximum likelihood performs consistently better than the moments methods for all

values of λ, κ, δ, for all four distributions and with respect to the AE and MSE. This is expected

of course. The observations were similar for other values of λ, κ, δ.

6 Application

The aim of this section is to illustrate the usefulness of (3) over (2) using some real data sets. We

use the annual maximum daily rainfall data for the years from 1907 to 2000 for fourteen locations

in west central Florida: Clermont, Brooksville, Orlando, Bartow, Avon Park, Arcadia, Kissimmee,

Inverness, Plant City, Tarpon Springs, Tampa International Airport, St Leo, Gainesville, and Ocala.

The data were obtained from the Department of Meteorology in Tallahassee, Florida. By definition

annual maximum daily rainfall is non–negative: to have (2) and (3) as possible models and to avoid

computational difficulties, we define standardized annual maximum rainfall = (annual maximum

rainfall - m)/s, where m and s are the observed mean and standard deviation, respectively.

We would like to emphasize that the aim here is not to provide a complete statistical modeling or

inferences for the data sets involved. We refer the readers to Nadarajah (2005) for a comprehensive

analysis of the data sets used.

[Figures 6 to 11 about here.]

We fitted location–scale variations of Azzalini skew–normal and the truncated–exponential

skew–normal distributions to the standardized annual maximum rainfall from each of the four-

teen locations. The maximum likelihood procedure described by equations (14)–(16) was used.

The results for four of the locations are:

• Clermont with sample size n = 94: (17) yielded − ln L = 99.0 with λ = −709.783 (42.488),

κ = −6.603 (0.533) and δ = 2.094 (0.178); (18) yielded − ln L = 102.6 with λ = 15.892

(13.484), κ = −1.005 (0.072) and δ = 1.414 (0.118).

• Avon Park with sample size n = 94: (17) yielded − ln L = 93.0 with λ = −709.781 (42.082),


(8.083), κ = −1.021 (0.054) and δ = 1.425 (0.117).

14

• Gainesville with sample size n = 94: (17) yielded − ln L = 110.0 with λ = −709.781 (24.686),


(12.449), κ = −1.164 (0.045) and δ = 1.531 (0.117).

• Ocala with sample size n = 94: (17) yielded − lnL = 97.6 with λ = −709.782 (33.526),


(82.923), κ = −1.136 (0.039) and δ = 1.509 (0.119).

The numbers within brackets are the standard errors computed by inverting the expected informa-

tion matrix, see Section 3.

The two fitted models given by (17) and (18) are not nested. So, their comparison should be

based on criteria such as the Akaike’s information criterion or the Bayesian information criterion.

However, the two models have the same number of parameters. In this case, these criteria reduce

to the usual likelihood ratio test.

Comparing the likelihood values, we see that (17) provides a significantly better fit than (18)

for each of the four locations. The results were the same for other locations. The locations

– Clermont, Avon Park, Gainesville and Ocala – are illustrated because they showed the most

significant improvements.

The conclusion based on the likelihood values can be verified by means of probability–probability

plots and density plots. A probability–probability plot consists of plots of the observed probabilities

against the probabilities predicted by the fitted model. For example, for the model given by (17),

[1−exp{−λΦ((x(j)− κ)/δ)}]/[1−exp(−λ)] was plotted versus (j−0.375)/(n+0.25), j = 1, 2, . . . , n

(as recommended by Blom (1958) and Chambers et al. (1983)), where x(j) are the sorted values of

the observed standardized annual maximum rainfall. The probability–probability plots for the two

fitted models and for the four locations are shown in Figures 6 to 9. We can see that the model

given by (17) has the points much closer to the diagonal line for each location.

A density plot compares the fitted pdfs of the models with the empirical histogram of the

observed data. The density plots for the four locations are shown in Figures 10 and 11. Again the

fitted pdfs for (17) appear to capture the general pattern of the empirical histograms much better.

7 Conclusions

We have introduced a new family of skew distributions as a competitor to the well–known Azzalini

skew distributions. We have studied various mathematical properties and developed procedures for

estimation, hypotheses testing and simulation. We have assessed the performances of the estimation

procedures by simulation. We have also studied three particular members of the family and their

flexibility and illustrated a real data application. The new family of distributions has several

15

advantages over Azzalini skew distributions. Some of these are: 1) it belongs to the exponential

family; 2) has closed form expressions for pdf, cdf and quantiles; 3) moments, mgf and the chf can

be expressed as a series expansion of those of the original symmetric distribution (see Theorem 2);

4) moments, mgf and the chf of order statistics follow directly from those of the original sample; 5)

exhibits the same tail behaviors as those of the original symmetric distribution; 6) the maximum

likelihood estimator for the shape parameter always exists and is unique (see Theorem 3); 7) the

standard error for the shape parameter has a closed form; 8) admits a uniformly most powerful

test for hypotheses about the shape parameter; 9) admits wider range of values for skewness and

kurtosis and so more flexibility especially when the tails are heavier; 10) provides better fits to real

data sets.

The work of this paper can be extended in several directions. One is to compare the truncated–

exponential skew–t distributions with other skew–t distributions proposed in the literature (for

example, those by Azzalini and Capitanio (2003), Jones and Faddy (2003), Ma and Genton (2004)

and Azzalini and Genton (2008)). Another is to perform a simulation study to explore the properties

of the maximum likelihood and moments estimates for small, medium and large sample sizes.

A third direction is to construct multivariate generalizations of the truncated–exponential skew–

symmetric distributions. These issues are beyond the scope of the present investigation, but we

may consider some of these in a future paper.

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18

Table 1. Comparison of maximum likelihood versus moments estimation

for the truncated–exponential skew–t(5) distribution.

λ κ δ Maximum likelihood Moments

AE(λ) AE(κ) AE(δ) MSE(λ) MSE(κ) MSE(δ) AE(λ) AE(κ) AE(δ) MSE(λ) MSE(κ) MSE(δ)

-1 -1 1 -0.922 -0.965 1.007 0.856 0.087 0.001 -0.908 -0.963 1.008 1.032 0.090 0.001

-1 -1 2 -0.777 -0.852 1.995 0.990 0.426 0.005 -0.738 -0.844 1.995 1.099 0.462 0.006

-1 -1 5 -1.135 -1.117 5.042 1.210 3.081 0.041 -1.155 -1.123 5.049 1.233 3.357 0.045

-1 -1 10 -0.521 0.604 9.939 0.996 10.771 0.123 -0.406 0.705 9.932 1.121 11.268 0.134

-1 0 1 -0.801 0.074 1.009 1.036 0.105 0.001 -0.800 0.090 1.009 1.270 0.126 0.001

-1 0 2 -0.543 0.305 2.013 1.415 0.542 0.004 -0.488 0.356 2.014 1.491 0.591 0.004

-1 0 5 -0.980 0.037 5.018 0.963 2.505 0.030 -0.976 0.038 5.021 1.130 3.050 0.036

-1 0 10 -0.760 0.906 9.969 1.220 12.265 0.117 -0.753 1.121 9.966 1.272 13.458 0.130

-1 1 1 -0.898 1.044 1.011 1.482 0.157 0.001 -0.890 1.048 1.012 1.808 0.181 0.001

-1 1 2 -0.534 1.308 2.016 1.410 0.591 0.004 -0.490 1.381 2.018 1.431 0.633 0.005

-1 1 5 -1.056 0.978 5.039 1.533 3.776 0.037 -1.061 0.974 5.040 1.843 4.384 0.040

-1 1 10 -0.964 1.150 10.075 0.783 7.716 0.115 -0.956 1.185 10.084 0.913 8.995 0.124

-1 5 1 -0.986 5.013 1.009 1.365 0.141 0.001 -0.985 5.014 1.009 1.531 0.155 0.002

-1 5 2 -0.879 5.080 2.008 1.369 0.549 0.005 -0.875 5.083 2.009 1.546 0.589 0.007

-1 5 5 -0.841 5.303 4.984 1.033 2.639 0.023 -0.806 5.348 4.984 1.091 3.047 0.028

-1 5 10 -0.707 5.966 10.181 1.587 16.988 0.237 -0.684 6.040 10.209 1.741 20.618 0.286

1 -1 1 0.531 -1.146 1.015 1.638 0.177 0.001 0.462 -1.150 1.016 1.914 0.199 0.001

1 -1 2 0.739 -1.164 2.002 1.064 0.441 0.003 0.691 -1.196 2.002 1.106 0.507 0.004

1 -1 5 0.446 -1.897 5.005 1.432 3.768 0.031 0.420 -2.013 5.006 1.636 4.281 0.036

1 -1 10 0.939 -1.159 10.169 1.270 13.230 0.221 0.938 -1.175 10.207 1.410 14.324 0.247

1 0 1 0.952 -0.029 1.008 1.163 0.123 0.001 0.947 -0.032 1.009 1.414 0.149 0.002

1 0 2 1.097 0.036 1.990 0.673 0.256 0.004 1.100 0.038 1.990 0.839 0.317 0.004

1 0 5 0.857 -0.271 5.022 0.923 2.444 0.034 0.848 -0.313 5.023 0.963 2.899 0.037

1 0 10 0.624 -1.289 10.060 2.099 22.436 0.156 0.603 -1.322 10.063 2.411 25.141 0.173

1 1 1 0.899 0.971 1.004 1.485 0.151 0.001 0.895 0.966 1.004 1.652 0.174 0.001

1 1 2 0.716 0.797 2.002 1.479 0.584 0.004 0.654 0.790 2.003 1.825 0.676 0.004

1 1 5 1.153 1.225 5.028 0.916 2.452 0.031 1.174 1.249 5.035 1.099 2.600 0.032

1 1 10 0.859 0.326 10.055 0.835 8.858 0.075 0.853 0.268 10.063 0.869 9.189 0.077

1 5 1 0.946 4.971 1.002 0.806 0.090 0.002 0.942 4.967 1.002 0.952 0.096 0.002

1 5 2 0.571 4.732 2.017 1.319 0.587 0.004 0.541 4.712 2.020 1.623 0.705 0.004

1 5 5 1.012 5.017 5.040 1.733 4.302 0.048 1.013 5.021 5.043 1.889 4.392 0.050

1 5 10 0.820 4.395 9.952 0.895 8.958 0.098 0.804 4.277 9.952 1.117 10.857 0.122

2 -1 1 2.014 -1.011 1.000 0.710 0.058 0.002 2.014 -1.011 1.000 0.852 0.060 0.002

2 -1 2 1.837 -1.091 1.991 0.635 0.227 0.002 1.803 -1.094 1.989 0.770 0.261 0.003

2 -1 5 2.156 -0.740 5.053 0.448 1.036 0.036 2.172 -0.711 5.059 0.464 1.096 0.041

2 -1 10 1.646 -2.126 10.060 1.308 13.029 0.154 1.606 -2.216 10.062 1.457 14.325 0.154

19

2 0 1 1.955 -0.013 1.003 0.573 0.053 0.002 1.951 -0.014 1.003 0.606 0.059 0.002

2 0 2 1.580 -0.239 1.989 0.927 0.363 0.005 1.489 -0.285 1.988 0.941 0.402 0.007

2 0 5 2.050 0.077 5.003 0.451 1.040 0.044 2.053 0.086 5.003 0.514 1.235 0.052

2 0 10 2.053 0.095 10.150 0.975 9.184 0.191 2.064 0.095 10.167 1.136 10.562 0.196

2 1 1 1.738 0.919 0.995 0.910 0.088 0.001 1.687 0.906 0.995 0.953 0.104 0.001

2 1 2 1.651 0.780 1.991 1.477 0.528 0.007 1.580 0.777 1.991 1.762 0.531 0.008

2 1 5 2.068 1.085 5.086 0.610 1.431 0.039 2.074 1.088 5.100 0.682 1.558 0.043

2 1 10 1.658 -0.069 9.918 0.567 5.897 0.110 1.582 -0.179 9.915 0.602 6.627 0.128

2 5 1 1.729 4.917 1.008 1.325 0.132 0.003 1.701 4.913 1.009 1.420 0.134 0.003

2 5 2 2.077 5.046 2.022 0.608 0.232 0.008 2.089 5.052 2.023 0.622 0.281 0.009

2 5 5 2.147 5.217 5.058 0.406 0.994 0.038 2.176 5.237 5.063 0.485 1.122 0.043

2 5 10 1.890 4.581 9.897 0.458 5.186 0.178 1.883 4.523 9.897 0.507 6.366 0.221

5 -1 1 4.824 -1.040 0.986 0.584 0.025 0.001 4.784 -1.042 0.983 0.619 0.026 0.002

5 -1 2 5.158 -0.970 2.004 0.672 0.112 0.008 5.188 -0.966 2.004 0.695 0.123 0.009

5 -1 5 5.098 -0.960 5.007 0.393 0.413 0.031 5.108 -0.951 5.007 0.425 0.463 0.036

5 -1 10 4.995 -1.068 9.990 0.300 1.657 0.175 4.994 -1.074 9.990 0.323 1.754 0.178

5 0 1 5.126 0.018 1.001 0.432 0.017 0.001 5.138 0.022 1.001 0.537 0.019 0.002

5 0 2 5.000 -0.009 2.006 0.512 0.103 0.008 5.000 -0.010 2.006 0.531 0.123 0.008

5 0 5 5.348 0.269 5.021 0.798 0.624 0.030 5.428 0.307 5.023 0.957 0.687 0.033

5 0 10 5.057 0.007 9.902 0.276 0.842 0.081 5.058 0.009 9.896 0.280 0.931 0.087

5 1 1 5.035 0.998 0.999 0.532 0.022 0.002 5.036 0.997 0.999 0.661 0.024 0.002

5 1 2 4.892 0.955 2.000 0.300 0.063 0.005 4.884 0.949 2.000 0.338 0.074 0.006

5 1 5 5.155 1.145 5.036 0.502 0.356 0.023 5.167 1.166 5.040 0.566 0.419 0.027

5 1 10 4.729 0.377 9.879 0.929 4.201 0.224 4.720 0.345 9.850 1.116 4.960 0.235

5 5 1 5.100 5.014 1.002 0.354 0.014 0.001 5.122 5.016 1.002 0.392 0.016 0.001

5 5 2 5.236 5.078 2.032 0.380 0.057 0.005 5.243 5.086 2.039 0.464 0.067 0.005

5 5 5 5.089 5.033 4.979 0.444 0.538 0.037 5.094 5.034 4.977 0.480 0.596 0.039

5 5 10 5.128 5.271 10.010 0.501 1.954 0.193 5.148 5.273 10.011 0.575 2.190 0.210

20





-1 -1 1 -0.717 -0.908 1.005 1.479 0.135 0.001 -0.658 -0.886 1.005 1.567 0.143 0.002

-1 -1 2 -0.923 -0.940 2.022 1.822 0.695 0.013 -0.910 -0.930 2.024 2.095 0.786 0.014

-1 -1 5 -0.752 -0.646 5.103 1.329 3.161 0.056 -0.708 -0.643 5.103 1.542 3.539 0.069

-1 -1 10 -1.011 -0.982 10.250 2.019 19.314 0.332 -1.012 -0.978 10.268 2.362 20.935 0.383

-1 0 1 -0.893 0.050 1.015 1.756 0.161 0.002 -0.883 0.055 1.018 2.138 0.181 0.003

-1 0 2 -0.925 0.044 2.019 0.634 0.245 0.003 -0.918 0.052 2.023 0.689 0.271 0.003

-1 0 5 -0.615 0.604 5.051 2.287 5.121 0.041 -0.596 0.625 5.055 2.360 6.326 0.047

-1 0 10 -0.576 1.212 10.150 1.736 16.607 0.268 -0.489 1.229 10.170 1.763 19.537 0.316

-1 1 1 -1.119 0.960 1.016 1.328 0.126 0.002 -1.126 0.955 1.017 1.626 0.154 0.002

-1 1 2 -0.793 1.130 2.014 0.928 0.344 0.004 -0.758 1.157 2.016 1.152 0.424 0.005

-1 1 5 -1.077 0.982 5.110 1.672 3.868 0.069 -1.090 0.978 5.116 1.852 4.067 0.080

-1 1 10 -0.967 1.196 10.125 1.700 15.623 0.272 -0.963 1.245 10.156 1.741 16.429 0.310

-1 5 1 -1.195 4.942 1.040 2.863 0.246 0.005 -1.196 4.942 1.043 3.468 0.293 0.005

-1 5 2 -0.666 5.204 2.022 1.862 0.642 0.008 -0.594 5.215 2.026 1.912 0.714 0.009

-1 5 5 -0.807 5.283 5.021 1.074 2.450 0.044 -0.773 5.304 5.026 1.229 2.470 0.046

-1 5 10 -0.892 5.258 10.138 1.322 12.745 0.194 -0.870 5.281 10.153 1.358 13.566 0.203

1 -1 1 1.309 -0.900 1.020 1.062 0.106 0.002 1.376 -0.875 1.020 1.189 0.108 0.003

1 -1 2 0.926 -1.027 2.032 1.712 0.591 0.010 0.923 -1.033 2.040 2.054 0.738 0.013

1 -1 5 1.233 -0.644 5.120 1.291 2.940 0.056 1.280 -0.644 5.143 1.456 2.985 0.058

1 -1 10 0.857 -1.382 10.194 1.632 16.100 0.209 0.848 -1.418 10.195 1.703 17.343 0.248

1 0 1 0.928 -0.036 1.014 1.952 0.175 0.002 0.923 -0.044 1.017 2.410 0.191 0.002

1 0 2 1.089 0.040 2.034 1.271 0.481 0.007 1.109 0.049 2.038 1.531 0.563 0.008

1 0 5 1.454 0.646 5.168 1.725 3.972 0.096 1.493 0.798 5.171 2.046 4.237 0.099

1 0 10 0.835 -0.460 10.086 1.503 13.874 0.182 0.833 -0.518 10.100 1.869 14.665 0.225

1 1 1 1.015 0.999 1.019 1.604 0.148 0.002 1.015 0.999 1.020 1.914 0.178 0.003

1 1 2 0.892 0.930 2.032 1.780 0.685 0.009 0.889 0.917 2.037 1.817 0.708 0.010

1 1 5 1.108 1.127 5.109 1.740 4.104 0.091 1.125 1.130 5.118 1.830 4.563 0.093

1 1 10 0.626 -0.204 10.172 2.522 23.778 0.319 0.583 -0.359 10.179 3.113 27.381 0.367

1 5 1 0.661 4.893 1.015 1.770 0.166 0.002 0.659 4.883 1.017 2.054 0.168 0.002

1 5 2 0.795 4.856 2.025 1.482 0.550 0.009 0.758 4.831 2.026 1.744 0.608 0.011

1 5 5 0.897 4.805 5.105 1.891 4.277 0.057 0.896 4.803 5.120 2.151 4.327 0.068

1 5 10 1.326 5.936 10.187 0.969 8.816 0.182 1.335 6.131 10.193 1.006 9.894 0.193

2 -1 1 1.990 -1.015 1.016 1.562 0.134 0.004 1.990 -1.015 1.018 1.595 0.156 0.005

2 -1 2 1.932 -1.076 2.006 1.347 0.418 0.011 1.922 -1.087 2.007 1.498 0.516 0.013

2 -1 5 2.043 -1.000 5.076 1.422 3.170 0.073 2.047 -0.999 5.083 1.727 3.184 0.077

2 -1 10 1.606 -2.190 9.991 1.919 16.498 0.306 1.595 -2.463 9.991 2.013 17.455 0.364

21

2 0 1 2.263 0.070 1.019 1.168 0.096 0.004 2.284 0.072 1.020 1.405 0.111 0.005

2 0 2 1.580 -0.253 1.972 1.258 0.421 0.008 1.546 -0.268 1.969 1.392 0.502 0.009

2 0 5 2.213 0.243 5.076 1.043 2.149 0.065 2.223 0.249 5.090 1.304 2.266 0.069

2 0 10 2.011 -0.017 10.080 1.087 8.546 0.213 2.012 -0.017 10.087 1.229 10.163 0.233

2 1 1 2.218 1.056 1.015 1.162 0.090 0.003 2.267 1.058 1.018 1.212 0.098 0.003

2 1 2 2.200 1.076 2.019 1.397 0.462 0.016 2.203 1.087 2.021 1.653 0.562 0.019

2 1 5 1.713 0.523 4.999 2.101 4.334 0.061 1.705 0.507 4.999 2.586 5.100 0.070

2 1 10 1.789 0.346 10.013 0.942 7.975 0.227 1.784 0.233 10.016 0.955 8.933 0.231

2 5 1 1.863 4.956 1.009 2.333 0.194 0.006 1.841 4.948 1.011 2.401 0.230 0.006

2 5 2 1.947 4.957 2.009 1.071 0.359 0.007 1.942 4.949 2.010 1.274 0.367 0.008

2 5 5 1.747 4.624 5.002 1.002 2.141 0.060 1.740 4.535 5.003 1.027 2.391 0.073

2 5 10 1.945 4.803 9.989 1.031 8.568 0.230 1.944 4.779 9.986 1.165 9.336 0.231

5 -1 1 4.706 -1.057 0.985 0.590 0.024 0.002 4.697 -1.063 0.982 0.639 0.024 0.003

5 -1 2 4.948 -1.076 1.993 1.946 0.375 0.012 4.937 -1.080 1.993 2.420 0.440 0.013

5 -1 5 4.871 -1.146 4.972 0.570 0.556 0.047 4.866 -1.162 4.970 0.655 0.566 0.054

5 -1 10 4.918 -1.282 9.881 0.613 2.509 0.241 4.900 -1.327 9.867 0.660 2.533 0.297

5 0 1 4.804 -0.047 0.995 0.674 0.032 0.001 4.757 -0.049 0.995 0.737 0.038 0.001

5 0 2 4.611 -0.148 1.970 0.750 0.121 0.008 4.530 -0.164 1.965 0.857 0.135 0.008

5 0 5 4.648 -0.394 4.907 1.268 1.591 0.062 4.588 -0.451 4.887 1.379 1.906 0.064

5 0 10 4.708 -0.670 9.869 1.914 8.658 0.411 4.654 -0.749 9.843 2.231 10.382 0.501

5 1 1 5.167 1.026 1.005 0.743 0.031 0.003 5.172 1.031 1.006 0.796 0.033 0.004

5 1 2 5.070 1.026 2.005 0.703 0.103 0.006 5.082 1.031 2.005 0.757 0.110 0.006

5 1 5 5.204 1.147 5.038 0.584 0.566 0.046 5.219 1.148 5.042 0.605 0.677 0.054

5 1 10 5.184 1.352 10.067 0.527 1.653 0.121 5.212 1.405 10.076 0.548 1.806 0.125

5 5 1 5.164 5.013 0.999 0.482 0.019 0.002 5.169 5.016 0.999 0.537 0.021 0.002

5 5 2 5.146 5.044 2.016 1.048 0.155 0.010 5.171 5.046 2.019 1.237 0.157 0.012

5 5 5 5.209 5.190 5.042 0.713 0.688 0.047 5.258 5.204 5.047 0.725 0.743 0.058

5 5 10 5.162 5.234 9.995 0.532 1.679 0.140 5.167 5.236 9.995 0.610 1.849 0.172

22





-1 -1 1 -0.593 -0.883 1.010 1.688 0.141 0.002 -0.508 -0.877 1.011 2.015 0.160 0.002

-1 -1 2 -0.789 -0.887 2.005 0.853 0.302 0.003 -0.776 -0.880 2.007 1.022 0.372 0.004

-1 -1 5 -1.053 -1.078 5.053 0.627 1.295 0.036 -1.063 -1.087 5.054 0.675 1.374 0.044

-1 -1 10 -0.976 -0.978 10.046 0.831 7.204 0.100 -0.975 -0.976 10.050 0.907 8.473 0.112

-1 0 1 -1.025 -0.009 1.023 1.725 0.145 0.004 -1.030 -0.009 1.024 1.831 0.148 0.004

-1 0 2 -1.223 -0.133 2.053 1.747 0.619 0.016 -1.243 -0.149 2.054 2.149 0.692 0.017

-1 0 5 -0.841 0.218 5.014 1.135 2.516 0.092 -0.835 0.246 5.014 1.205 2.813 0.108

-1 0 10 -1.263 -0.722 10.217 1.621 13.308 0.475 -1.293 -0.864 10.225 1.783 16.536 0.519

-1 1 1 -1.146 0.957 1.011 0.775 0.066 0.001 -1.179 0.949 1.012 0.861 0.074 0.002

-1 1 2 -1.173 0.886 2.033 0.773 0.255 0.008 -1.205 0.870 2.034 0.885 0.299 0.008

-1 1 5 -1.327 0.507 5.144 1.328 2.962 0.105 -1.353 0.408 5.173 1.361 3.523 0.109

-1 1 10 -0.757 1.734 10.132 1.614 14.266 0.348 -0.742 1.777 10.135 1.897 14.679 0.392

-1 5 1 -0.768 5.063 1.013 1.398 0.125 0.002 -0.749 5.068 1.013 1.568 0.156 0.002

-1 5 2 -1.217 4.878 2.043 1.331 0.485 0.017 -1.219 4.875 2.045 1.487 0.486 0.021

-1 5 5 -0.884 5.176 5.046 0.827 1.786 0.024 -0.882 5.185 5.057 0.966 1.955 0.028

-1 5 10 -1.240 4.382 10.114 0.811 6.921 0.240 -1.268 4.266 10.139 0.852 7.119 0.254

1 -1 1 0.727 -1.096 1.011 1.540 0.137 0.001 0.663 -1.096 1.012 1.873 0.166 0.001

1 -1 2 0.890 -1.089 2.034 1.805 0.590 0.011 0.883 -1.109 2.037 2.071 0.625 0.011

1 -1 5 0.823 -1.297 5.000 0.954 2.064 0.040 0.821 -1.365 5.000 1.018 2.578 0.041

1 -1 10 1.408 0.189 10.391 1.755 15.497 0.524 1.474 0.338 10.420 2.121 15.944 0.550

1 0 1 1.032 0.003 0.998 0.848 0.074 0.002 1.039 0.003 0.998 0.990 0.085 0.002

1 0 2 1.122 0.076 2.018 0.805 0.289 0.006 1.138 0.091 2.018 0.959 0.321 0.007

1 0 5 0.949 -0.122 5.131 2.035 4.528 0.149 0.939 -0.126 5.147 2.232 5.502 0.171

1 0 10 1.290 0.858 10.147 0.942 7.766 0.285 1.330 1.070 10.156 0.987 8.204 0.351

1 1 1 1.111 1.042 1.021 1.223 0.110 0.004 1.134 1.044 1.022 1.410 0.117 0.004

1 1 2 0.838 0.915 2.034 1.762 0.599 0.015 0.834 0.913 2.035 2.076 0.643 0.017

1 1 5 1.161 1.166 5.075 0.686 1.486 0.056 1.196 1.175 5.080 0.811 1.789 0.056

1 1 10 1.395 2.229 10.317 1.685 15.258 0.403 1.440 2.254 10.345 2.006 18.727 0.481

1 5 1 1.297 5.083 1.026 1.319 0.112 0.004 1.319 5.084 1.029 1.463 0.127 0.005

1 5 2 0.742 4.849 2.006 1.049 0.363 0.009 0.679 4.836 2.007 1.053 0.411 0.010

1 5 5 1.020 4.994 5.071 0.783 1.786 0.054 1.023 4.994 5.072 0.893 2.081 0.056

1 5 10 0.877 4.568 10.091 1.330 11.272 0.216 0.876 4.522 10.094 1.646 13.625 0.232

2 -1 1 2.120 -0.965 1.015 0.950 0.074 0.003 2.123 -0.961 1.017 1.035 0.074 0.003

2 -1 2 1.468 -1.312 1.984 2.216 0.735 0.025 1.425 -1.316 1.981 2.270 0.757 0.031

2 -1 5 2.075 -0.957 5.004 0.993 1.942 0.099 2.091 -0.949 5.005 1.013 1.984 0.105

2 -1 10 1.497 -2.461 9.914 1.554 13.000 0.318 1.439 -2.469 9.910 1.741 14.369 0.319

23

2 0 1 1.929 -0.029 1.003 1.226 0.095 0.004 1.913 -0.033 1.004 1.272 0.103 0.004

2 0 2 1.924 -0.051 1.996 0.858 0.281 0.009 1.917 -0.056 1.996 1.011 0.328 0.012

2 0 5 2.039 0.019 5.107 1.899 3.857 0.168 2.041 0.023 5.112 1.973 4.230 0.201

2 0 10 2.040 0.062 10.136 1.607 12.715 0.669 2.044 0.077 10.159 1.775 15.471 0.764

2 1 1 1.938 0.981 1.014 1.662 0.134 0.004 1.930 0.978 1.016 1.935 0.144 0.004

2 1 2 1.887 0.908 1.996 1.127 0.333 0.012 1.865 0.886 1.996 1.189 0.369 0.013

2 1 5 1.714 0.631 4.981 1.153 2.128 0.066 1.713 0.545 4.979 1.378 2.520 0.083

2 1 10 2.007 0.970 10.053 1.337 10.000 0.506 2.009 0.970 10.054 1.544 11.294 0.599

2 5 1 1.973 4.990 0.999 1.123 0.093 0.004 1.973 4.988 0.999 1.390 0.101 0.004

2 5 2 1.984 4.953 1.999 1.852 0.511 0.025 1.984 4.946 1.999 2.272 0.566 0.027

2 5 5 1.801 4.676 4.935 0.575 0.985 0.042 1.766 4.634 4.929 0.695 1.166 0.043

2 5 10 1.470 3.525 9.923 1.158 9.301 0.264 1.414 3.414 9.915 1.189 10.985 0.327

5 -1 1 4.789 -1.052 0.986 1.083 0.050 0.004 4.738 -1.055 0.985 1.144 0.055 0.004

5 -1 2 5.260 -0.927 2.023 0.704 0.115 0.013 5.293 -0.911 2.024 0.844 0.135 0.014

5 -1 5 5.338 -0.723 5.069 1.049 0.865 0.068 5.394 -0.661 5.086 1.198 1.068 0.070

5 -1 10 4.851 -1.367 9.913 1.099 4.221 0.317 4.845 -1.446 9.892 1.187 4.274 0.339

5 0 1 4.922 -0.022 0.990 0.837 0.034 0.003 4.905 -0.022 0.988 0.881 0.041 0.003

5 0 2 5.221 0.054 2.018 1.074 0.158 0.014 5.237 0.062 2.020 1.167 0.184 0.014

5 0 5 5.265 0.201 5.049 1.241 1.002 0.081 5.316 0.245 5.058 1.466 1.127 0.089

5 0 10 4.808 -0.476 9.894 1.617 7.465 0.336 4.795 -0.526 9.891 1.666 8.396 0.373

5 1 1 4.787 0.955 0.990 0.658 0.029 0.003 4.759 0.945 0.990 0.796 0.030 0.003

5 1 2 4.723 0.857 1.976 2.080 0.411 0.028 4.689 0.849 1.973 2.434 0.476 0.031

5 1 5 4.663 0.550 4.890 2.945 3.294 0.193 4.626 0.522 4.886 2.960 3.745 0.230

5 1 10 4.732 0.420 9.888 1.777 8.065 0.461 4.676 0.355 9.884 1.914 8.120 0.568

5 5 1 5.248 5.035 1.005 1.107 0.037 0.003 5.275 5.042 1.005 1.168 0.038 0.003

5 5 2 4.805 4.881 1.976 1.557 0.276 0.019 4.759 4.867 1.975 1.703 0.314 0.023

5 5 5 5.170 5.162 5.064 1.090 0.979 0.091 5.188 5.164 5.067 1.092 1.072 0.093

5 5 10 4.900 4.819 9.950 0.620 2.431 0.233 4.879 4.807 9.946 0.722 2.733 0.276

24


for the truncated–exponential skew–normal distribution.



-1 -1 1 -1.176 -1.049 1.014 0.765 0.064 0.003 -1.197 -1.055 1.015 0.803 0.068 0.003

-1 -1 2 -1.206 -1.113 2.034 0.935 0.274 0.016 -1.217 -1.120 2.042 1.089 0.286 0.016

-1 -1 5 -0.882 -0.834 5.036 0.608 1.246 0.038 -0.863 -0.794 5.039 0.671 1.387 0.047

-1 -1 10 -1.121 -1.425 10.191 0.870 7.149 0.385 -1.135 -1.458 10.231 1.008 8.455 0.388

-1 0 1 -1.060 -0.009 1.012 0.443 0.035 0.001 -1.072 -0.010 1.013 0.491 0.040 0.001

-1 0 2 -0.989 -0.002 2.020 0.431 0.150 0.006 -0.986 -0.003 2.020 0.510 0.162 0.006

-1 0 5 -0.939 0.127 4.999 0.744 1.403 0.066 -0.931 0.128 4.999 0.928 1.408 0.072

-1 0 10 -1.076 -0.219 10.085 0.402 3.239 0.102 -1.092 -0.230 10.105 0.407 3.463 0.113

-1 1 1 -1.216 0.942 1.003 0.529 0.047 0.002 -1.223 0.930 1.003 0.529 0.054 0.002

-1 1 2 -1.228 0.869 2.031 0.676 0.217 0.007 -1.275 0.844 2.035 0.725 0.230 0.008

-1 1 5 -0.979 1.066 5.032 0.160 0.351 0.014 -0.978 1.068 5.039 0.191 0.414 0.015

-1 1 10 -0.936 1.106 10.136 0.600 5.101 0.300 -0.927 1.125 10.140 0.651 5.412 0.304

-1 5 1 -0.982 5.003 1.002 0.243 0.021 0.001 -0.980 5.004 1.003 0.291 0.021 0.001

-1 5 2 -1.126 4.950 1.999 0.701 0.230 0.015 -1.142 4.948 1.999 0.835 0.264 0.018

-1 5 5 -0.924 5.102 4.998 0.261 0.571 0.026 -0.917 5.111 4.998 0.287 0.571 0.030

-1 5 10 -0.805 5.569 9.942 0.323 2.951 0.122 -0.760 5.612 9.935 0.372 3.396 0.150

1 -1 1 1.076 -0.982 1.008 0.938 0.063 0.002 1.080 -0.982 1.008 0.970 0.063 0.003

1 -1 2 1.040 -0.981 2.008 0.705 0.243 0.012 1.048 -0.978 2.010 0.879 0.262 0.015

1 -1 5 1.042 -1.000 5.018 0.526 1.054 0.045 1.046 -1.000 5.019 0.589 1.309 0.050

1 -1 10 0.908 -1.262 10.014 0.456 3.929 0.072 0.898 -1.278 10.015 0.551 4.401 0.079

1 0 1 0.925 -0.031 1.001 0.367 0.030 0.001 0.923 -0.036 1.002 0.447 0.033 0.001

1 0 2 1.030 0.012 2.012 0.250 0.081 0.004 1.030 0.013 2.012 0.296 0.095 0.004

1 0 5 1.255 0.328 5.097 1.012 1.945 0.123 1.282 0.356 5.105 1.013 1.949 0.140

1 0 10 0.963 -0.045 9.984 0.337 2.679 0.084 0.958 -0.054 9.983 0.352 2.753 0.085

1 1 1 1.116 1.038 1.012 0.910 0.069 0.003 1.133 1.038 1.015 1.098 0.075 0.003

1 1 2 0.897 0.936 2.017 0.340 0.123 0.004 0.888 0.929 2.019 0.406 0.129 0.005

1 1 5 0.986 1.036 5.065 0.581 1.164 0.032 0.983 1.037 5.078 0.691 1.358 0.035

1 1 10 1.078 1.305 10.094 0.884 6.883 0.378 1.084 1.381 10.103 1.076 7.995 0.420

1 5 1 1.053 5.020 1.007 0.778 0.062 0.003 1.062 5.023 1.008 0.922 0.068 0.003

1 5 2 0.884 4.937 2.014 0.336 0.100 0.003 0.856 4.932 2.015 0.389 0.104 0.003

1 5 5 0.892 4.800 5.002 0.271 0.567 0.018 0.876 4.780 5.003 0.319 0.574 0.020

1 5 10 1.231 5.663 10.152 0.811 6.545 0.327 1.250 5.706 10.161 0.912 6.899 0.372

2 -1 1 1.835 -1.057 0.993 0.476 0.040 0.003 1.812 -1.060 0.992 0.544 0.041 0.003

2 -1 2 2.084 -0.973 2.020 1.311 0.359 0.023 2.085 -0.970 2.021 1.401 0.391 0.025

2 -1 5 2.065 -0.920 5.074 0.872 1.608 0.093 2.068 -0.919 5.090 1.017 1.749 0.116

2 -1 10 1.782 -1.516 9.874 0.469 3.534 0.207 1.741 -1.635 9.850 0.554 3.897 0.231

25

2 0 1 2.178 0.048 1.021 0.815 0.060 0.004 2.179 0.048 1.026 0.972 0.069 0.005

2 0 2 2.041 0.036 2.023 1.188 0.350 0.020 2.045 0.039 2.027 1.214 0.415 0.022

2 0 5 2.187 0.230 5.085 0.714 1.289 0.085 2.215 0.240 5.102 0.743 1.494 0.089

2 0 10 1.893 -0.334 10.028 0.958 6.887 0.381 1.888 -0.379 10.029 1.007 7.000 0.382

2 1 1 1.908 0.977 1.007 1.064 0.077 0.006 1.906 0.975 1.007 1.327 0.087 0.007

2 1 2 2.144 1.071 2.031 0.856 0.229 0.012 2.148 1.087 2.032 1.010 0.279 0.015

2 1 5 2.008 0.986 5.040 0.733 1.405 0.062 2.008 0.984 5.047 0.772 1.441 0.074

2 1 10 2.045 1.136 10.024 0.591 4.458 0.289 2.051 1.139 10.028 0.596 5.049 0.318

2 5 1 2.479 5.127 1.044 1.295 0.087 0.007 2.562 5.142 1.050 1.391 0.089 0.008

2 5 2 2.091 5.039 2.015 0.620 0.171 0.009 2.097 5.045 2.016 0.718 0.182 0.010

2 5 5 2.251 5.353 5.086 0.866 1.484 0.107 2.312 5.391 5.090 1.041 1.644 0.108

2 5 10 1.956 4.871 10.033 0.574 3.910 0.200 1.947 4.864 10.039 0.586 4.252 0.221

5 -1 1 4.790 -1.058 0.983 2.259 0.087 0.009 4.783 -1.065 0.981 2.568 0.106 0.009

5 -1 2 5.112 -0.970 2.009 0.908 0.132 0.015 5.120 -0.963 2.010 1.000 0.133 0.017

5 -1 5 5.181 -0.876 5.030 1.580 1.434 0.137 5.202 -0.853 5.034 1.855 1.719 0.170

5 -1 10 4.518 -2.090 9.658 2.512 10.479 0.861 4.468 -2.360 9.608 2.655 12.546 0.937

5 0 1 4.574 -0.099 0.971 1.865 0.071 0.007 4.496 -0.101 0.967 2.004 0.081 0.008

5 0 2 5.056 -0.003 2.008 2.424 0.335 0.028 5.059 -0.004 2.009 2.529 0.378 0.034

5 0 5 4.789 -0.251 4.920 1.488 1.647 0.144 4.759 -0.252 4.918 1.768 1.928 0.170

5 0 10 5.287 0.375 10.141 1.386 4.296 0.365 5.297 0.444 10.144 1.547 4.514 0.450

5 1 1 4.692 0.932 0.983 1.541 0.068 0.007 4.648 0.919 0.979 1.693 0.076 0.007

5 1 2 4.796 0.927 1.985 1.591 0.257 0.022 4.755 0.925 1.983 1.688 0.275 0.024

5 1 5 5.575 1.445 5.129 1.678 1.292 0.127 5.603 1.463 5.132 1.785 1.328 0.136

5 1 10 4.648 0.290 9.811 1.604 6.425 0.534 4.614 0.117 9.797 1.819 7.308 0.566

5 5 1 4.775 4.953 0.986 1.574 0.063 0.006 4.740 4.943 0.985 1.705 0.068 0.008

5 5 2 4.639 4.831 1.952 1.947 0.315 0.030 4.563 4.814 1.947 2.128 0.380 0.032

5 5 5 4.435 4.366 4.806 2.924 3.221 0.271 4.371 4.227 4.767 3.605 3.255 0.298

5 5 10 4.541 3.964 9.721 2.146 8.929 0.908 4.466 3.714 9.678 2.554 9.391 1.048

26

0.0 0.2 0.4 0.6 0.8 1.0

1.2

1.3

1.4

1.5

u

f 1(F

1−1(u

))f 2

(F2−1(u

))

Figure 1. Plot of f1(F−11 (u))/f2(F

−12 (u)) versus u when X is a standard normal random variable,

λ1 = −10 and λ2 = 5.

27

−100 −50 0 50 100

−3

−1

12

3

λ

Ske

wn

ess

−100 −50 0 50 100

−3

−1

12

3

−100 −50 0 50 100

10

20

30

λ

Ku

rto

sis

−100 −50 0 50 100

10

20

30

Figure 2. The skewness and kurtosis of the truncated–exponential skew–t(5) and Azzalini skew–

t(5) distributions in (19) and (20), respectively, for λ = −100,−99, . . . , 99, 100. The solid and

broken curves correspond to Azzalini skew–t(5) and the truncated–exponential skew–t(5) distribu-

tions, respectively.

28

−100 −50 0 50 100

−1

.5−

0.5

0.5

1.5

λ

Ske

wn

ess

−100 −50 0 50 100

−1

.5−

0.5

0.5

1.5

−100 −50 0 50 100

45

67

λ

Ku

rto

sis

−100 −50 0 50 100

45

67



broken curves correspond to Azzalini skew–t(10) and the truncated–exponential skew–t(10) distri-

butions, respectively.

29

−100 −50 0 50 100

−1

.00

.01

.0

λ

Ske

wn

ess

−100 −50 0 50 100

−1

.00

.01

.0

−100 −50 0 50 100

3.5

4.0

4.5

5.0

λ

Ku

rto

sis

−100 −50 0 50 100

3.5

4.0

4.5

5.0



broken curves correspond to Azzalini skew–t(20) and the truncated–exponential skew–t(20) distri-

butions, respectively.

30

−100 −50 0 50 100

−1

.00

.00

.51

.0

λ

Ske

wn

ess

−100 −50 0 50 100

−1

.00

.00

.51

.0

−100 −50 0 50 100

3.0

3.2

3.4

3.6

3.8

λ

Ku

rto

sis

−100 −50 0 50 100

3.0

3.2

3.4

3.6

3.8

Figure 5. The skewness and kurtosis of the truncated–exponential skew–normal and Azzalini

skew–normal distributions in (17) and (18), respectively, for λ = −100,−99, . . . , 99, 100. The solid

and broken curves correspond to Azzalini skew–normal and the truncated–exponential skew–normal

distributions, respectively.

31

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Expected Probability

Obse

rved P

roba

bility

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0


Obse

rved P

roba

bility

Figure 6. Probability–probability plots of the fits of (17) (top) and (18) (bottom) for the annual

maximum rainfall data from Clermont with sample size n = 94.

32

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0


Obse

rved P

roba

bility

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0


Obse

rved P

roba

bility


maximum rainfall data from Avon Park with sample size n = 94.

33

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0


Obse

rved P

roba

bility

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0


Obse

rved P

roba

bility


maximum rainfall data from Gainesville with sample size n = 94.

34

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0


Obse

rved P

roba

bility

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0


Obse

rved P

roba

bility


maximum rainfall data from Ocala with sample size n = 94.

35

Standardized Annual Maximum Rainfall

Fitted

PDF

s

−1 0 1 2 3 4 5

0.00.1

0.20.3

0.40.5

0.60.7

−1 0 1 2 3 4 5

0.00.1

0.20.3

0.40.5

0.60.7

−1 0 1 2 3 4 5

0.00.1

0.20.3

0.40.5

0.60.7


Fitted

PDF

s

−1 0 1 2 3 4

0.00.2

0.40.6

0.81.0

−1 0 1 2 3 4

0.00.2

0.40.6

0.81.0

−1 0 1 2 3 4

0.00.2

0.40.6

0.81.0

Figure 10. Fitted pdfs of (17) and (18) for the annual maximum rainfall data from Clermont (top)

and Avon Park (bottom). The solid and broken curves correspond to (18) and (17), respectively.

The sample sizes for both locations are n = 94.

36


Fitted

PDF

s

−1 0 1 2 3 4 5

0.00.1

0.20.3

0.40.5

0.6

−1 0 1 2 3 4 5

0.00.1

0.20.3

0.40.5

0.6

−1 0 1 2 3 4 5

0.00.1

0.20.3

0.40.5

0.6


Fitted

PDF

s

−1 0 1 2 3 4

0.00.2

0.40.6

0.8

−1 0 1 2 3 4

0.00.2

0.40.6

0.8

−1 0 1 2 3 4

0.00.2

0.40.6

0.8

Figure 11. Fitted pdfs of (17) and (18) for the annual maximum rainfall data from Gainesville

(top) and Ocala (bottom). The solid and broken curves correspond to (18) and (17), respectively.

The sample sizes for both locations are n = 94.

37

truncated-exponential skew-symmetric distributions · vahid nassiri and adel mohammadpour...

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