Download Truncated-Exponential Skew-Symmetric Distributions

Truncated-Exponential Skew-Symmetric
Distributions
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) FX−1 (x))/fX (FX−1 (x)), 0 < λ < ∞ then (1) yields the family of distributions
studied by Fernández and Steel (1998). The most popular version of (1) are the skew distributions
introduced by Azzalini (1985). Take ω(x) = 2FX (λFX−1 (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 simulation. 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, T ESS(λ), if its pdf has the following form:
fY (y; λ) =
λ
fX (y) exp {−λFX (y)} ,
1 − exp(−λ)
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:
1 − exp {−λFX (y)}
FY (y; λ) =
, y ∈ R, λ ∈ R.
(4)
1 − exp(−λ)
The inverse cdf is:
FY−1 (y; λ) = FX−1 (−(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 ∼ T ESS(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 ∼ T ESS(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 Fi−1 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 F2−1 (F1 (y)) is convex in y or equivalently f1 (F1−1 (u))/f2 (F2−1 (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 (F1−1 (u))/f2 (F2−1 (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
′
fX (y)
= λfX (y).
fX (y)
3
(6)
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
′′
fX (y0 )
′
− λfX (y0 ) <
fX (y0 )
′
fX (y0 )
fX (y0 )
!2
.
(7)
!2
.
(8)
!2
.
(9)
The root corresponds to a minimum if
′′
fX (y0 )
′
− λfX (y0 ) >
fX (y0 )
′
fX (y0 )
fX (y0 )
The root corresponds to a point of inflexion if
′′
fX (y0 )
′
− λfX (y0 ) =
fX (y0 )
′
fX (y0 )
fX (y0 )
′ (y) > λf 2 (y) for all
The mode corresponding to a maximum is unique if the y0 is such that fX
X
′
2
y < y0 and fX (y) < λfX (y) for all y > y0 . The mode corresponding to a minimum is unique if
′ (y) < λf 2 (y) for all y < y and f ′ (y) > λf 2 (y) for all y > y . The mode
the y0 is such that fX
0
0
X
X
X
′
2 (y) for all
corresponding to a point of inflexion is unique if the y0 is such that either fX (y) < λfX
′ (y) > λf 2 (y) for all y 6= y . Note that (3) can be multi–modal if, for example, f (·)
y 6= y0 or fX
0
X
X
is multi–modal.
Note that the tails of Y ∼ T ESS(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 ∼ T ESS(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 1 − exp(s − λ)
, λ ∈ R,
λ − s 1 − exp(−λ)
where MT (s) is the moment generating function of T . In particular,
E(T ) =
1 − exp(−λ)(λ + 1)
, λ ∈ R.
λ {1 − exp(−λ)}
Note that the pdf of T is the same as the ω(·) chosen to construct (3).
4
(10)
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):
1 − exp(−λ) 1 − exp(−λ)(λ + 1)
λ
+
−
I,
λ
1 − exp(−λ)
1 − exp(−λ)
sY = ln
where
I=
Z
(11)
∞
−∞
ln fX (y)fX (y) exp {−λFX (y)} dy.
By the series expansion for exponential, one can express
I = −sX +
where
Ik =
Z
∞
X
(−1)k λk
k=1
k!
Ik ,
∞
ln fX (y)fX (y)FXk (y)dy.
−∞
Let rY (γ) and rX (γ) denote Rényi entropies (Rényi, 1961) corresponding to fY (·) and fX (·),
respectively. Similar calculations using the series expansion for exponential show that
Z ∞
1
λγ
γ
rY (γ) =
ln
γ fX (y) exp {−γλFX (y)} dy
1−γ
−∞ {1 − exp(−λ)}
"
#
Z
∞
X
1
λγ
(−λγ)k ∞ γ
=
ln
fX (y)FXk (y)dy
1−γ
{1 − exp(−λ)}γ
k!
−∞
k=0
(Z
"
#)
∞
γ
X
(−λγ)k
1
λ
exp {(1 − γ)rX (γ)} +
,
(12)
=
ln
Jk
1−γ
{1 − exp(−λ)}γ
k!
k=1
where
Jk =
Z
∞
γ
fX
(y)fX (y)FXk (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 ∼ T ESS(λ).
Theorem 1. Let Y ∼ T ESS(λ). If E|X|r , r > 0, exists, then E|Y |r exists.
Proof. We have
r
E|Y |
=
=
=
Z
+∞
λ
fX (y) exp {−λFX (y)} dy
1 − exp(−λ)
−∞
Z +∞
λ
|y|r exp {−λFX (y)} fX (y)dy
1 − exp(−λ) −∞
λ
E [|X|r exp {−λFX (X)}] .
1 − exp(−λ)
|y|r
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 ∼ T ESS(λ). If the conditions of Theorem 1 hold then
∞
E (Y r ) =
X (−λ)k
λ
r
E Xk+1:k+1
.
1 − exp(−λ)
(k + 1)!
k=0
Proof. By the series expansion for exponential, one can express
Z ∞
λ
r
E (Y ) =
y r fX (y) exp {−λFX (y)} dy
1 − exp(−λ) −∞
Z
∞
X
λ
(−λ)k ∞ r
=
y fX (y)FXk (y)dy
1 − exp(−λ)
k!
−∞
=
λ
1 − exp(−λ)
k=0
∞
X
(−λ)k
1
r
E Xk+1:k+1
k! (k + 1)
k=0
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 ∼ T ESS(λ) are
∞
X (−λ)k
λ
E[exp(tY )] =
Mk+1:k+1 (t)
1 − exp(−λ)
(k + 1)!
k=0
and
∞
X (−λ)k
λ
E[exp(itY )] =
φk+1:k+1 (t),
1 − exp(−λ)
(k + 1)!
k=0
respectively.
Let Yk:n denote the kth order statistic for a random sample of size n from fY (·). Write Y = Y (λ)
when Y ∼ T ESS(λ). Then the pdf and the cdf of Yk:n can be expressed as
fYk:n (y) =
k−1 n−k n! {1 − exp(−λ)}−n X X k − 1 n − k (−1)l+n−k−m
(k − 1)!(n − k)!
l
m
l+m+1
l=0 m=0
× exp {−(n − k − m)λ} [1 − exp {−(l + m + 1)λ}] fY ((l+m+1)λ) (y)
and
FYk:n (y) =
k−1 n−k n! {1 − exp(−λ)}−n X X k − 1 n − k (−1)l+n−k−m
(k − 1)!(n − k)!
l
m
l+m+1
l=0 m=0
× 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
r
E (Yk:n
)
=
k−1 n−k n! {1 − exp(−λ)}−n X X k − 1 n − k (−1)l+n−k−m
l
m
(k − 1)!(n − k)!
l+m+1
l=0 m=0
× exp {−(n − k − m)λ} [1 − exp {−(l + m + 1)λ}] E (Y r ((l + m + 1)λ)) ,
E [exp (tYk:n )] =
k−1 n−k n! {1 − exp(−λ)}−n X X k − 1 n − k (−1)l+n−k−m
l
m
(k − 1)!(n − k)!
l+m+1
m=0
l=0
× exp {−(n − k − m)λ} [1 − exp {−(l + m + 1)λ}] MY ((l+m+1)λ) (t)
and
E [exp (itYk:n )] =
k−1 n−k n! {1 − exp(−λ)}−n X X k − 1 n − k (−1)l+n−k−m
l
m
(k − 1)!(n − k)!
l+m+1
m=0
l=0
× 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 ∼ T ESS(λ) can be
expressed as
r−1
X
r−1+j
r−1−j r − 1
λr =
βj ,
(−1)
j
j
j=0
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 variable with additional location and scale parameters.
Definition 2. A random variable Y has the truncated–exponential skew–symmetric distribution
with parameters (λ, κ, δ), T ESS(λ, κ, δ), if its pdf has the following form:
λ
y−κ
y−κ
fY (y; λ, κ, δ) =
fX
exp −λFX
(13)
δ {1 − exp(−λ)}
δ
δ
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),
P
page 23) and ni=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
P
y1 , y2 , . . . , yn is a random sample from (13). Let mk = (1/n) nj=1 yjk for k = 1, 2, 3. By equating
the theoretical moments of (13) with the sample moments, we obtain the equations:
∞
k X
λ
k k−l l X (−λ)m
l
E Xm+1:m+1
κ δ
= mk
1 − exp(−λ)
l
(m + 1)!
m=0
l=0
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
)
(
)
n ( Y
n
n
X
λ
L(λ, κ, δ) =
fX (zi ) exp −λ
FX (zi ) ,
δ(1 − exp(−λ))
i=1
i=1
where zi = (yi − κ)/δ. So, the maximum likelihood estimators are the simultaneous solutions of
the equations
X
n
1
1
n
−
=
FX (zi ) ,
(14)
λ exp(λ) − 1
i=1
′
n
n
X
X
fX (zi )
=λ
fX (zi )
fX (zi )
i=1
(15)
i=1
and
n
X
i=1
n
X
fX (zi )
=λ
zi fX (zi ) − n,
fX (zi )
′
zi
(16)
i=1
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.
P
Proof. Note that (14) can be re–expressed as ng(λ) − r(z) = 0, where r(z) = ni=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:
2
n
∂ ln L
n exp(λ)
= 2−
E −
,
2
∂λ
λ
{exp(λ) − 1}2
∂ 2 ln L
E −
∂λ∂κ
∂ 2 ln L
E −
∂λ∂δ
E −
E −
∂ 2 ln L
∂κ∂δ
∂ 2 ln L
∂κ2
= −
and
∂ 2 ln L
E −
∂δ2
n
= − E [fX (Z)] ,
δ
n
= − E [ZfX (Z)] ,
δ
!2 
i
f (Z)
f (Z)  nλ h ′
n
n
= − 2E X
+ 2E  X
+ 2 E fX (Z) ,
δ
fX (Z)
δ
fX (Z)
δ
"
"
′′
#
′

#
"
′′
′
#

f (Z)
f (Z)
n
n
n
E X
− 2E Z X
+ 2 E Z
2
δ
fX (Z)
δ
fX (Z)
δ
i
nλ h ′
+ 2 E ZfX (Z)
δ
!2 
fX (Z)  nλ
+ 2 E [fX (Z)]
fX (Z)
δ
′

#
#
" ′
"
′′
fX (Z)
f
(Z)
n
2n
n
n
= − 2 − 2E Z
− 2 E Z2 X
+ 2 E Z 2
δ
δ
fX (Z)
δ
fX (Z)
δ
i
2nλ
′
nλ h
+ 2 E [ZfX (Z)] + 2 E Z 2 fX (Z) ,
δ
δ
′
fX (Z)
fX (Z)
!2 

where Z = (Y − κ)/δ and Y ∼ T ESS(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

" ′′
#
!2 
2
′
f (Z)
f (Z) 
∂ ln L
n
n
+ 2E X
E −
= − 2E X
,
∂κ2
δ
fX (Z)
δ
fX (Z)
E −
∂ 2 ln L
∂κ∂δ
and
E −
∂ 2 ln L
∂δ2

#
" ′′
#
′
f
(Z)
f
(Z)
n
n
n
= − 2E X
− 2E Z X
+ 2 E Z
δ
fX (Z)
δ
fX (Z)
δ
"
′
fX (Z)
fX (Z)

#
"
#
′
′′
f
(Z)
f
(Z)
n
2n
n
n
=− 2 − 2E Z X
− 2 E Z2 X
+ 2 E Z 2
δ
δ
fX (Z)
δ
fX (Z)
δ
"
9
′
!2 

fX (Z)
fX (Z)
!2 
,
where Z = (Y − κ)/δ and Y ∼ T ESS(0, κ, δ). If in addition X is standard normal then
2
n
∂ ln L
E −
= 2,
2
∂κ
δ
∂ 2 ln L
E −
∂κ∂δ
=
2n
E [Z]
δ2
and
∂ 2 ln L
E −
∂δ2
=
where Z = (Y − κ)/δ and Y ∼ T ESS(0, κ, δ).
n 2
3E Z − 1 ,
2
δ
P
We noted earlier that T = ni=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
(
1, if T ≥ t0 ,
ϕ(X) =
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 T ESS(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 = U M fX (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 T ESS(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 distributions. 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:
λ
φ(y) exp {−λΦ(y)} ,
1 − exp(−λ)
fY (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 ) =
√
∞
X
2r/2 λ
(−λ)k
π {1 − exp(−λ)} k=0 2k k!
(p)
×FA
k
X
p=0
p + r even
k −p/2 p
p+r+1
π
2 Γ
2
p
p+r+1 1
1 3
3
; , . . . , ; , . . . , ; −1, . . . , −1 ,
2
2
2 2
2
(n)
where FA (· · · ) denotes the Lauricella function of type A (Exton, 1978) defined by
(n)
FA (a, b1 , . . . , bn ; c1 , . . . , cn ; x1 , . . . , xn )
∞
∞
mn
1
X
X
(a)m1 +···+mn (b1 )m1 · · · (bn )mn xm
1 · · · xn
=
···
,
(c1 )m1 · · · (cn )mn
m1 ! · · · mn !
m =0
m =0
1
n
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:
−(1+ν)/2
λ
1
y2
√
fY (y) =
1+
exp {−λFX (y)} , y ∈ R, λ ∈ R. (19)
1 − exp(−λ) νB(ν/2, 1/2)
ν
where B(a, b) = Γ(a)Γ(b)/Γ(a + b) is the beta function. The corresponding Azzalini’s distribution
has the pdf
−(1+ν)/2
2
y2
fY (y) = √
1+
FX (λy), y ∈ R, λ ∈ R.
(20)
ν
νB(ν/2, 1/2)
11
For general ν, the cdf term, FX (y), takes the form
1 yΓ ((ν + 1)/2)
FX (y) = + √
2 F1
2
πνΓ (ν/2)
1 ν + 1 3 y2
,
; ;−
2
2
2
ν
,
(21)
where 2 F1 (· · · ) denotes the Gauss hypergeometric function defined by
2 F1 (a, b; c; x) =
∞
X
(a)k (b)k xk
.
(c)k k!
k=0
If ν is an integer then one can simplify (21) to:

(ν−1)/2

 1 1
y
1 X
1 ν i−1/2 y


B i,
, for ν odd,

 2 + π arctan √ν + 2π
2 (ν + y 2 )i
i=1
FX (y) =
ν/2


1
1 X
1 1
ν i−1 y


+
B
i
−
,
,
for ν even,

 2 2π
2 2 (ν + y 2 )i−1/2
i=1
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 ) =
X (−λ)k
λ
{A(r, k + 1, k + 1) + (−1)r A(r, k + 1, 1)} ,
1 − exp(−λ)
(k + 1)!
k=0
where
A(k, n, r) =
r−1 n−r
X
X r − 1n − r n!ν k/2
(−1)q 2p+q
2n (r − 1)!(n − r)!
p
q
p=0 q=0
ν −k k+p+q+1
,
×B −1−p−q (1/2, ν/2)B
2
2
k+p+q+1
ν 1
ν 1
1:2
×F1:1
: 1− ,
;...; 1 − ,
;
2
2 2
2 2
!
ν+p+q+1
3
3
:
;...;
; 1, . . . , 1 ,
2
2
2
A:B (· · · ) denotes the generalized Kampé de Fériet function (Exton, 1978) defined by
where FC:D
A:B
FC:D
((a) : (b1 ); . . . ; (bn ); (c) : (d1 ); . . . ; (dn ); x1 , . . . , xn )
∞
∞
mn
1
X
X
((a))m1 +···+mn ((b1 ))m1 · · · ((bn ))mn xm
1 · · · xn
=
···
,
((c))
((d
))
·
·
·
((d
))
m
!
·
·
·
m
!
m
+···+m
1
n
1
n
n
1
m
m
n
1
m =0
m =0
1
n
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+y 2 /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:
λ
1
1 1
exp −λ
+ arctan(y)
,
fY (y) =
1 − exp(−λ) π(1 + y 2 )
2 π
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:
E Y 3 − 3E(Y )E Y 2 + 2E 3 (Y )
Skewness(Y ) =
3/2
E Y 2 − E 2 (Y )
and
E Y 4 − 4E(Y )E Y 3 + 6E Y 2 E 2 (Y ) − 3E 4 (Y )
.
Kurtosis(Y ) =
2
E Y 2 − E 2 (Y )
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 fourteen locations. The maximum likelihood procedure described by equations (14)–(16) was used.
The results for four of the locations are:
b = −709.783 (42.488),
• Clermont with sample size n = 94: (17) yielded − ln L = 99.0 with λ
b = 15.892
κ
b = −6.603 (0.533) and δb = 2.094 (0.178); (18) yielded − ln L = 102.6 with λ
(13.484), κ
b = −1.005 (0.072) and δb = 1.414 (0.118).
b = −709.781 (42.082),
• Avon Park with sample size n = 94: (17) yielded − ln L = 93.0 with λ
b = 16.196
κ
b = −6.989 (0.586) and δb = 2.214 (0.196); (18) yielded − ln L = 100.3 with λ
(8.083), κ
b = −1.021 (0.054) and δb = 1.425 (0.117).
14
b = −709.781 (24.686),
• Gainesville with sample size n = 94: (17) yielded − ln L = 110.0 with λ
b = 22.212
κ
b = −7.436 (0.567) and δb = 2.359 (0.192); (18) yielded − ln L = 116.5 with λ
(12.449), κ
b = −1.164 (0.045) and δb = 1.531 (0.117).
b = −709.782 (33.526),
• Ocala with sample size n = 94: (17) yielded − ln L = 97.6 with λ
b = 58.873
κ
b = −7.171 (0.582) and δb = 2.272 (0.197); (18) yielded − ln L = 104.2 with λ
(82.923), κ
b = −1.136 (0.039) and δb = 1.509 (0.119).
The numbers within brackets are the standard errors computed by inverting the expected information 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),
b
b
b was plotted versus (j − 0.375)/(n + 0.25), j = 1, 2, . . . , n
[1 − exp{−λΦ((x
b)/δ)}]/[1
− exp(−λ)]
(j) − κ
(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
b AE(b
b MSE(λ)
b MSE(b
b AE(λ)
b AE(b
b MSE(λ)
b MSE(b
b
AE(λ)
κ) AE(δ)
κ) MSE(δ)
κ) AE(δ)
κ) MSE(δ)
-1 -1 1 -0.922
-1 -1 2 -0.777
-1 -1 5 -1.135
-1 -1 10 -0.521
-1 0 1 -0.801
-1 0 2 -0.543
-1 0 5 -0.980
-1 0 10 -0.760
-1 1 1 -0.898
-1 1 2 -0.534
-1 1 5 -1.056
-1 1 10 -0.964
-1 5 1 -0.986
-1 5 2 -0.879
-1 5 5 -0.841
-1 5 10 -0.707
1 -1 1 0.531
1 -1 2 0.739
1 -1 5 0.446
1 -1 10 0.939
1 0 1 0.952
1 0 2 1.097
1 0 5 0.857
1 0 10 0.624
1 1 1 0.899
1 1 2 0.716
1 1 5 1.153
1 1 10 0.859
1 5 1 0.946
1 5 2 0.571
1 5 5 1.012
1 5 10 0.820
2 -1 1 2.014
2 -1 2 1.837
2 -1 5 2.156
2 -1 10 1.646
-0.965
-0.852
-1.117
0.604
0.074
0.305
0.037
0.906
1.044
1.308
0.978
1.150
5.013
5.080
5.303
5.966
-1.146
-1.164
-1.897
-1.159
-0.029
0.036
-0.271
-1.289
0.971
0.797
1.225
0.326
4.971
4.732
5.017
4.395
-1.011
-1.091
-0.740
-2.126
1.007 0.856
1.995 0.990
5.042 1.210
9.939 0.996
1.009 1.036
2.013 1.415
5.018 0.963
9.969 1.220
1.011 1.482
2.016 1.410
5.039 1.533
10.075 0.783
1.009 1.365
2.008 1.369
4.984 1.033
10.181 1.587
1.015 1.638
2.002 1.064
5.005 1.432
10.169 1.270
1.008 1.163
1.990 0.673
5.022 0.923
10.060 2.099
1.004 1.485
2.002 1.479
5.028 0.916
10.055 0.835
1.002 0.806
2.017 1.319
5.040 1.733
9.952 0.895
1.000 0.710
1.991 0.635
5.053 0.448
10.060 1.308
0.087
0.426
3.081
10.771
0.105
0.542
2.505
12.265
0.157
0.591
3.776
7.716
0.141
0.549
2.639
16.988
0.177
0.441
3.768
13.230
0.123
0.256
2.444
22.436
0.151
0.584
2.452
8.858
0.090
0.587
4.302
8.958
0.058
0.227
1.036
13.029
0.001
0.005
0.041
0.123
0.001
0.004
0.030
0.117
0.001
0.004
0.037
0.115
0.001
0.005
0.023
0.237
0.001
0.003
0.031
0.221
0.001
0.004
0.034
0.156
0.001
0.004
0.031
0.075
0.002
0.004
0.048
0.098
0.002
0.002
0.036
0.154
19
-0.908
-0.738
-1.155
-0.406
-0.800
-0.488
-0.976
-0.753
-0.890
-0.490
-1.061
-0.956
-0.985
-0.875
-0.806
-0.684
0.462
0.691
0.420
0.938
0.947
1.100
0.848
0.603
0.895
0.654
1.174
0.853
0.942
0.541
1.013
0.804
2.014
1.803
2.172
1.606
-0.963
-0.844
-1.123
0.705
0.090
0.356
0.038
1.121
1.048
1.381
0.974
1.185
5.014
5.083
5.348
6.040
-1.150
-1.196
-2.013
-1.175
-0.032
0.038
-0.313
-1.322
0.966
0.790
1.249
0.268
4.967
4.712
5.021
4.277
-1.011
-1.094
-0.711
-2.216
1.008 1.032
1.995 1.099
5.049 1.233
9.932 1.121
1.009 1.270
2.014 1.491
5.021 1.130
9.966 1.272
1.012 1.808
2.018 1.431
5.040 1.843
10.084 0.913
1.009 1.531
2.009 1.546
4.984 1.091
10.209 1.741
1.016 1.914
2.002 1.106
5.006 1.636
10.207 1.410
1.009 1.414
1.990 0.839
5.023 0.963
10.063 2.411
1.004 1.652
2.003 1.825
5.035 1.099
10.063 0.869
1.002 0.952
2.020 1.623
5.043 1.889
9.952 1.117
1.000 0.852
1.989 0.770
5.059 0.464
10.062 1.457
0.090
0.462
3.357
11.268
0.126
0.591
3.050
13.458
0.181
0.633
4.384
8.995
0.155
0.589
3.047
20.618
0.199
0.507
4.281
14.324
0.149
0.317
2.899
25.141
0.174
0.676
2.600
9.189
0.096
0.705
4.392
10.857
0.060
0.261
1.096
14.325
0.001
0.006
0.045
0.134
0.001
0.004
0.036
0.130
0.001
0.005
0.040
0.124
0.002
0.007
0.028
0.286
0.001
0.004
0.036
0.247
0.002
0.004
0.037
0.173
0.001
0.004
0.032
0.077
0.002
0.004
0.050
0.122
0.002
0.003
0.041
0.154
2
2
2
2
2
2
2
2
2
2
2
2
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
0 1 1.955
0 2 1.580
0 5 2.050
0 10 2.053
1 1 1.738
1 2 1.651
1 5 2.068
1 10 1.658
5 1 1.729
5 2 2.077
5 5 2.147
5 10 1.890
-1 1 4.824
-1 2 5.158
-1 5 5.098
-1 10 4.995
0 1 5.126
0 2 5.000
0 5 5.348
0 10 5.057
1 1 5.035
1 2 4.892
1 5 5.155
1 10 4.729
5 1 5.100
5 2 5.236
5 5 5.089
5 10 5.128
-0.013
-0.239
0.077
0.095
0.919
0.780
1.085
-0.069
4.917
5.046
5.217
4.581
-1.040
-0.970
-0.960
-1.068
0.018
-0.009
0.269
0.007
0.998
0.955
1.145
0.377
5.014
5.078
5.033
5.271
1.003 0.573
1.989 0.927
5.003 0.451
10.150 0.975
0.995 0.910
1.991 1.477
5.086 0.610
9.918 0.567
1.008 1.325
2.022 0.608
5.058 0.406
9.897 0.458
0.986 0.584
2.004 0.672
5.007 0.393
9.990 0.300
1.001 0.432
2.006 0.512
5.021 0.798
9.902 0.276
0.999 0.532
2.000 0.300
5.036 0.502
9.879 0.929
1.002 0.354
2.032 0.380
4.979 0.444
10.010 0.501
0.053
0.363
1.040
9.184
0.088
0.528
1.431
5.897
0.132
0.232
0.994
5.186
0.025
0.112
0.413
1.657
0.017
0.103
0.624
0.842
0.022
0.063
0.356
4.201
0.014
0.057
0.538
1.954
0.002
0.005
0.044
0.191
0.001
0.007
0.039
0.110
0.003
0.008
0.038
0.178
0.001
0.008
0.031
0.175
0.001
0.008
0.030
0.081
0.002
0.005
0.023
0.224
0.001
0.005
0.037
0.193
20
1.951
1.489
2.053
2.064
1.687
1.580
2.074
1.582
1.701
2.089
2.176
1.883
4.784
5.188
5.108
4.994
5.138
5.000
5.428
5.058
5.036
4.884
5.167
4.720
5.122
5.243
5.094
5.148
-0.014
-0.285
0.086
0.095
0.906
0.777
1.088
-0.179
4.913
5.052
5.237
4.523
-1.042
-0.966
-0.951
-1.074
0.022
-0.010
0.307
0.009
0.997
0.949
1.166
0.345
5.016
5.086
5.034
5.273
1.003 0.606
1.988 0.941
5.003 0.514
10.167 1.136
0.995 0.953
1.991 1.762
5.100 0.682
9.915 0.602
1.009 1.420
2.023 0.622
5.063 0.485
9.897 0.507
0.983 0.619
2.004 0.695
5.007 0.425
9.990 0.323
1.001 0.537
2.006 0.531
5.023 0.957
9.896 0.280
0.999 0.661
2.000 0.338
5.040 0.566
9.850 1.116
1.002 0.392
2.039 0.464
4.977 0.480
10.011 0.575
0.059
0.402
1.235
10.562
0.104
0.531
1.558
6.627
0.134
0.281
1.122
6.366
0.026
0.123
0.463
1.754
0.019
0.123
0.687
0.931
0.024
0.074
0.419
4.960
0.016
0.067
0.596
2.190
0.002
0.007
0.052
0.196
0.001
0.008
0.043
0.128
0.003
0.009
0.043
0.221
0.002
0.009
0.036
0.178
0.002
0.008
0.033
0.087
0.002
0.006
0.027
0.235
0.001
0.005
0.039
0.210
Table 2. Comparison of maximum likelihood versus moments estimation
for the truncated–exponential skew–t(10) distribution.
λ κ δ
Maximum likelihood
Moments
b AE(b
b MSE(λ)
b MSE(b
b AE(λ)
b AE(b
b MSE(λ)
b MSE(b
b
AE(λ)
κ) AE(δ)
κ) MSE(δ)
κ) AE(δ)
κ) MSE(δ)
-1 -1 1 -0.717
-1 -1 2 -0.923
-1 -1 5 -0.752
-1 -1 10 -1.011
-1 0 1 -0.893
-1 0 2 -0.925
-1 0 5 -0.615
-1 0 10 -0.576
-1 1 1 -1.119
-1 1 2 -0.793
-1 1 5 -1.077
-1 1 10 -0.967
-1 5 1 -1.195
-1 5 2 -0.666
-1 5 5 -0.807
-1 5 10 -0.892
1 -1 1 1.309
1 -1 2 0.926
1 -1 5 1.233
1 -1 10 0.857
1 0 1 0.928
1 0 2 1.089
1 0 5 1.454
1 0 10 0.835
1 1 1 1.015
1 1 2 0.892
1 1 5 1.108
1 1 10 0.626
1 5 1 0.661
1 5 2 0.795
1 5 5 0.897
1 5 10 1.326
2 -1 1 1.990
2 -1 2 1.932
2 -1 5 2.043
2 -1 10 1.606
-0.908
-0.940
-0.646
-0.982
0.050
0.044
0.604
1.212
0.960
1.130
0.982
1.196
4.942
5.204
5.283
5.258
-0.900
-1.027
-0.644
-1.382
-0.036
0.040
0.646
-0.460
0.999
0.930
1.127
-0.204
4.893
4.856
4.805
5.936
-1.015
-1.076
-1.000
-2.190
1.005 1.479
2.022 1.822
5.103 1.329
10.250 2.019
1.015 1.756
2.019 0.634
5.051 2.287
10.150 1.736
1.016 1.328
2.014 0.928
5.110 1.672
10.125 1.700
1.040 2.863
2.022 1.862
5.021 1.074
10.138 1.322
1.020 1.062
2.032 1.712
5.120 1.291
10.194 1.632
1.014 1.952
2.034 1.271
5.168 1.725
10.086 1.503
1.019 1.604
2.032 1.780
5.109 1.740
10.172 2.522
1.015 1.770
2.025 1.482
5.105 1.891
10.187 0.969
1.016 1.562
2.006 1.347
5.076 1.422
9.991 1.919
0.135
0.695
3.161
19.314
0.161
0.245
5.121
16.607
0.126
0.344
3.868
15.623
0.246
0.642
2.450
12.745
0.106
0.591
2.940
16.100
0.175
0.481
3.972
13.874
0.148
0.685
4.104
23.778
0.166
0.550
4.277
8.816
0.134
0.418
3.170
16.498
0.001
0.013
0.056
0.332
0.002
0.003
0.041
0.268
0.002
0.004
0.069
0.272
0.005
0.008
0.044
0.194
0.002
0.010
0.056
0.209
0.002
0.007
0.096
0.182
0.002
0.009
0.091
0.319
0.002
0.009
0.057
0.182
0.004
0.011
0.073
0.306
21
-0.658
-0.910
-0.708
-1.012
-0.883
-0.918
-0.596
-0.489
-1.126
-0.758
-1.090
-0.963
-1.196
-0.594
-0.773
-0.870
1.376
0.923
1.280
0.848
0.923
1.109
1.493
0.833
1.015
0.889
1.125
0.583
0.659
0.758
0.896
1.335
1.990
1.922
2.047
1.595
-0.886
-0.930
-0.643
-0.978
0.055
0.052
0.625
1.229
0.955
1.157
0.978
1.245
4.942
5.215
5.304
5.281
-0.875
-1.033
-0.644
-1.418
-0.044
0.049
0.798
-0.518
0.999
0.917
1.130
-0.359
4.883
4.831
4.803
6.131
-1.015
-1.087
-0.999
-2.463
1.005 1.567
2.024 2.095
5.103 1.542
10.268 2.362
1.018 2.138
2.023 0.689
5.055 2.360
10.170 1.763
1.017 1.626
2.016 1.152
5.116 1.852
10.156 1.741
1.043 3.468
2.026 1.912
5.026 1.229
10.153 1.358
1.020 1.189
2.040 2.054
5.143 1.456
10.195 1.703
1.017 2.410
2.038 1.531
5.171 2.046
10.100 1.869
1.020 1.914
2.037 1.817
5.118 1.830
10.179 3.113
1.017 2.054
2.026 1.744
5.120 2.151
10.193 1.006
1.018 1.595
2.007 1.498
5.083 1.727
9.991 2.013
0.143
0.786
3.539
20.935
0.181
0.271
6.326
19.537
0.154
0.424
4.067
16.429
0.293
0.714
2.470
13.566
0.108
0.738
2.985
17.343
0.191
0.563
4.237
14.665
0.178
0.708
4.563
27.381
0.168
0.608
4.327
9.894
0.156
0.516
3.184
17.455
0.002
0.014
0.069
0.383
0.003
0.003
0.047
0.316
0.002
0.005
0.080
0.310
0.005
0.009
0.046
0.203
0.003
0.013
0.058
0.248
0.002
0.008
0.099
0.225
0.003
0.010
0.093
0.367
0.002
0.011
0.068
0.193
0.005
0.013
0.077
0.364
2
2
2
2
2
2
2
2
2
2
2
2
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
0 1 2.263
0 2 1.580
0 5 2.213
0 10 2.011
1 1 2.218
1 2 2.200
1 5 1.713
1 10 1.789
5 1 1.863
5 2 1.947
5 5 1.747
5 10 1.945
-1 1 4.706
-1 2 4.948
-1 5 4.871
-1 10 4.918
0 1 4.804
0 2 4.611
0 5 4.648
0 10 4.708
1 1 5.167
1 2 5.070
1 5 5.204
1 10 5.184
5 1 5.164
5 2 5.146
5 5 5.209
5 10 5.162
0.070
-0.253
0.243
-0.017
1.056
1.076
0.523
0.346
4.956
4.957
4.624
4.803
-1.057
-1.076
-1.146
-1.282
-0.047
-0.148
-0.394
-0.670
1.026
1.026
1.147
1.352
5.013
5.044
5.190
5.234
1.019 1.168
1.972 1.258
5.076 1.043
10.080 1.087
1.015 1.162
2.019 1.397
4.999 2.101
10.013 0.942
1.009 2.333
2.009 1.071
5.002 1.002
9.989 1.031
0.985 0.590
1.993 1.946
4.972 0.570
9.881 0.613
0.995 0.674
1.970 0.750
4.907 1.268
9.869 1.914
1.005 0.743
2.005 0.703
5.038 0.584
10.067 0.527
0.999 0.482
2.016 1.048
5.042 0.713
9.995 0.532
0.096
0.421
2.149
8.546
0.090
0.462
4.334
7.975
0.194
0.359
2.141
8.568
0.024
0.375
0.556
2.509
0.032
0.121
1.591
8.658
0.031
0.103
0.566
1.653
0.019
0.155
0.688
1.679
0.004
0.008
0.065
0.213
0.003
0.016
0.061
0.227
0.006
0.007
0.060
0.230
0.002
0.012
0.047
0.241
0.001
0.008
0.062
0.411
0.003
0.006
0.046
0.121
0.002
0.010
0.047
0.140
22
2.284
1.546
2.223
2.012
2.267
2.203
1.705
1.784
1.841
1.942
1.740
1.944
4.697
4.937
4.866
4.900
4.757
4.530
4.588
4.654
5.172
5.082
5.219
5.212
5.169
5.171
5.258
5.167
0.072
-0.268
0.249
-0.017
1.058
1.087
0.507
0.233
4.948
4.949
4.535
4.779
-1.063
-1.080
-1.162
-1.327
-0.049
-0.164
-0.451
-0.749
1.031
1.031
1.148
1.405
5.016
5.046
5.204
5.236
1.020 1.405
1.969 1.392
5.090 1.304
10.087 1.229
1.018 1.212
2.021 1.653
4.999 2.586
10.016 0.955
1.011 2.401
2.010 1.274
5.003 1.027
9.986 1.165
0.982 0.639
1.993 2.420
4.970 0.655
9.867 0.660
0.995 0.737
1.965 0.857
4.887 1.379
9.843 2.231
1.006 0.796
2.005 0.757
5.042 0.605
10.076 0.548
0.999 0.537
2.019 1.237
5.047 0.725
9.995 0.610
0.111
0.502
2.266
10.163
0.098
0.562
5.100
8.933
0.230
0.367
2.391
9.336
0.024
0.440
0.566
2.533
0.038
0.135
1.906
10.382
0.033
0.110
0.677
1.806
0.021
0.157
0.743
1.849
0.005
0.009
0.069
0.233
0.003
0.019
0.070
0.231
0.006
0.008
0.073
0.231
0.003
0.013
0.054
0.297
0.001
0.008
0.064
0.501
0.004
0.006
0.054
0.125
0.002
0.012
0.058
0.172
Table 3. Comparison of maximum likelihood versus moments estimation
for the truncated–exponential skew–t(20) distribution.
λ κ δ
Maximum likelihood
Moments
b AE(b
b MSE(λ)
b MSE(b
b AE(λ)
b AE(b
b MSE(λ)
b MSE(b
b
AE(λ)
κ) AE(δ)
κ) MSE(δ)
κ) AE(δ)
κ) MSE(δ)
-1 -1 1 -0.593
-1 -1 2 -0.789
-1 -1 5 -1.053
-1 -1 10 -0.976
-1 0 1 -1.025
-1 0 2 -1.223
-1 0 5 -0.841
-1 0 10 -1.263
-1 1 1 -1.146
-1 1 2 -1.173
-1 1 5 -1.327
-1 1 10 -0.757
-1 5 1 -0.768
-1 5 2 -1.217
-1 5 5 -0.884
-1 5 10 -1.240
1 -1 1 0.727
1 -1 2 0.890
1 -1 5 0.823
1 -1 10 1.408
1 0 1 1.032
1 0 2 1.122
1 0 5 0.949
1 0 10 1.290
1 1 1 1.111
1 1 2 0.838
1 1 5 1.161
1 1 10 1.395
1 5 1 1.297
1 5 2 0.742
1 5 5 1.020
1 5 10 0.877
2 -1 1 2.120
2 -1 2 1.468
2 -1 5 2.075
2 -1 10 1.497
-0.883
-0.887
-1.078
-0.978
-0.009
-0.133
0.218
-0.722
0.957
0.886
0.507
1.734
5.063
4.878
5.176
4.382
-1.096
-1.089
-1.297
0.189
0.003
0.076
-0.122
0.858
1.042
0.915
1.166
2.229
5.083
4.849
4.994
4.568
-0.965
-1.312
-0.957
-2.461
1.010 1.688
2.005 0.853
5.053 0.627
10.046 0.831
1.023 1.725
2.053 1.747
5.014 1.135
10.217 1.621
1.011 0.775
2.033 0.773
5.144 1.328
10.132 1.614
1.013 1.398
2.043 1.331
5.046 0.827
10.114 0.811
1.011 1.540
2.034 1.805
5.000 0.954
10.391 1.755
0.998 0.848
2.018 0.805
5.131 2.035
10.147 0.942
1.021 1.223
2.034 1.762
5.075 0.686
10.317 1.685
1.026 1.319
2.006 1.049
5.071 0.783
10.091 1.330
1.015 0.950
1.984 2.216
5.004 0.993
9.914 1.554
0.141
0.302
1.295
7.204
0.145
0.619
2.516
13.308
0.066
0.255
2.962
14.266
0.125
0.485
1.786
6.921
0.137
0.590
2.064
15.497
0.074
0.289
4.528
7.766
0.110
0.599
1.486
15.258
0.112
0.363
1.786
11.272
0.074
0.735
1.942
13.000
0.002
0.003
0.036
0.100
0.004
0.016
0.092
0.475
0.001
0.008
0.105
0.348
0.002
0.017
0.024
0.240
0.001
0.011
0.040
0.524
0.002
0.006
0.149
0.285
0.004
0.015
0.056
0.403
0.004
0.009
0.054
0.216
0.003
0.025
0.099
0.318
23
-0.508
-0.776
-1.063
-0.975
-1.030
-1.243
-0.835
-1.293
-1.179
-1.205
-1.353
-0.742
-0.749
-1.219
-0.882
-1.268
0.663
0.883
0.821
1.474
1.039
1.138
0.939
1.330
1.134
0.834
1.196
1.440
1.319
0.679
1.023
0.876
2.123
1.425
2.091
1.439
-0.877
-0.880
-1.087
-0.976
-0.009
-0.149
0.246
-0.864
0.949
0.870
0.408
1.777
5.068
4.875
5.185
4.266
-1.096
-1.109
-1.365
0.338
0.003
0.091
-0.126
1.070
1.044
0.913
1.175
2.254
5.084
4.836
4.994
4.522
-0.961
-1.316
-0.949
-2.469
1.011 2.015
2.007 1.022
5.054 0.675
10.050 0.907
1.024 1.831
2.054 2.149
5.014 1.205
10.225 1.783
1.012 0.861
2.034 0.885
5.173 1.361
10.135 1.897
1.013 1.568
2.045 1.487
5.057 0.966
10.139 0.852
1.012 1.873
2.037 2.071
5.000 1.018
10.420 2.121
0.998 0.990
2.018 0.959
5.147 2.232
10.156 0.987
1.022 1.410
2.035 2.076
5.080 0.811
10.345 2.006
1.029 1.463
2.007 1.053
5.072 0.893
10.094 1.646
1.017 1.035
1.981 2.270
5.005 1.013
9.910 1.741
0.160
0.372
1.374
8.473
0.148
0.692
2.813
16.536
0.074
0.299
3.523
14.679
0.156
0.486
1.955
7.119
0.166
0.625
2.578
15.944
0.085
0.321
5.502
8.204
0.117
0.643
1.789
18.727
0.127
0.411
2.081
13.625
0.074
0.757
1.984
14.369
0.002
0.004
0.044
0.112
0.004
0.017
0.108
0.519
0.002
0.008
0.109
0.392
0.002
0.021
0.028
0.254
0.001
0.011
0.041
0.550
0.002
0.007
0.171
0.351
0.004
0.017
0.056
0.481
0.005
0.010
0.056
0.232
0.003
0.031
0.105
0.319
2
2
2
2
2
2
2
2
2
2
2
2
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
0 1 1.929
0 2 1.924
0 5 2.039
0 10 2.040
1 1 1.938
1 2 1.887
1 5 1.714
1 10 2.007
5 1 1.973
5 2 1.984
5 5 1.801
5 10 1.470
-1 1 4.789
-1 2 5.260
-1 5 5.338
-1 10 4.851
0 1 4.922
0 2 5.221
0 5 5.265
0 10 4.808
1 1 4.787
1 2 4.723
1 5 4.663
1 10 4.732
5 1 5.248
5 2 4.805
5 5 5.170
5 10 4.900
-0.029
-0.051
0.019
0.062
0.981
0.908
0.631
0.970
4.990
4.953
4.676
3.525
-1.052
-0.927
-0.723
-1.367
-0.022
0.054
0.201
-0.476
0.955
0.857
0.550
0.420
5.035
4.881
5.162
4.819
1.003 1.226
1.996 0.858
5.107 1.899
10.136 1.607
1.014 1.662
1.996 1.127
4.981 1.153
10.053 1.337
0.999 1.123
1.999 1.852
4.935 0.575
9.923 1.158
0.986 1.083
2.023 0.704
5.069 1.049
9.913 1.099
0.990 0.837
2.018 1.074
5.049 1.241
9.894 1.617
0.990 0.658
1.976 2.080
4.890 2.945
9.888 1.777
1.005 1.107
1.976 1.557
5.064 1.090
9.950 0.620
0.095
0.281
3.857
12.715
0.134
0.333
2.128
10.000
0.093
0.511
0.985
9.301
0.050
0.115
0.865
4.221
0.034
0.158
1.002
7.465
0.029
0.411
3.294
8.065
0.037
0.276
0.979
2.431
0.004
0.009
0.168
0.669
0.004
0.012
0.066
0.506
0.004
0.025
0.042
0.264
0.004
0.013
0.068
0.317
0.003
0.014
0.081
0.336
0.003
0.028
0.193
0.461
0.003
0.019
0.091
0.233
24
1.913
1.917
2.041
2.044
1.930
1.865
1.713
2.009
1.973
1.984
1.766
1.414
4.738
5.293
5.394
4.845
4.905
5.237
5.316
4.795
4.759
4.689
4.626
4.676
5.275
4.759
5.188
4.879
-0.033
-0.056
0.023
0.077
0.978
0.886
0.545
0.970
4.988
4.946
4.634
3.414
-1.055
-0.911
-0.661
-1.446
-0.022
0.062
0.245
-0.526
0.945
0.849
0.522
0.355
5.042
4.867
5.164
4.807
1.004 1.272
1.996 1.011
5.112 1.973
10.159 1.775
1.016 1.935
1.996 1.189
4.979 1.378
10.054 1.544
0.999 1.390
1.999 2.272
4.929 0.695
9.915 1.189
0.985 1.144
2.024 0.844
5.086 1.198
9.892 1.187
0.988 0.881
2.020 1.167
5.058 1.466
9.891 1.666
0.990 0.796
1.973 2.434
4.886 2.960
9.884 1.914
1.005 1.168
1.975 1.703
5.067 1.092
9.946 0.722
0.103
0.328
4.230
15.471
0.144
0.369
2.520
11.294
0.101
0.566
1.166
10.985
0.055
0.135
1.068
4.274
0.041
0.184
1.127
8.396
0.030
0.476
3.745
8.120
0.038
0.314
1.072
2.733
0.004
0.012
0.201
0.764
0.004
0.013
0.083
0.599
0.004
0.027
0.043
0.327
0.004
0.014
0.070
0.339
0.003
0.014
0.089
0.373
0.003
0.031
0.230
0.568
0.003
0.023
0.093
0.276
Table 4. Comparison of maximum likelihood versus moments estimation
for the truncated–exponential skew–normal distribution.
λ κ δ
Maximum likelihood
Moments
b AE(b
b MSE(λ)
b MSE(b
b AE(λ)
b AE(b
b MSE(λ)
b MSE(b
b
AE(λ)
κ) AE(δ)
κ) MSE(δ)
κ) AE(δ)
κ) MSE(δ)
-1 -1 1 -1.176
-1 -1 2 -1.206
-1 -1 5 -0.882
-1 -1 10 -1.121
-1 0 1 -1.060
-1 0 2 -0.989
-1 0 5 -0.939
-1 0 10 -1.076
-1 1 1 -1.216
-1 1 2 -1.228
-1 1 5 -0.979
-1 1 10 -0.936
-1 5 1 -0.982
-1 5 2 -1.126
-1 5 5 -0.924
-1 5 10 -0.805
1 -1 1 1.076
1 -1 2 1.040
1 -1 5 1.042
1 -1 10 0.908
1 0 1 0.925
1 0 2 1.030
1 0 5 1.255
1 0 10 0.963
1 1 1 1.116
1 1 2 0.897
1 1 5 0.986
1 1 10 1.078
1 5 1 1.053
1 5 2 0.884
1 5 5 0.892
1 5 10 1.231
2 -1 1 1.835
2 -1 2 2.084
2 -1 5 2.065
2 -1 10 1.782
-1.049
-1.113
-0.834
-1.425
-0.009
-0.002
0.127
-0.219
0.942
0.869
1.066
1.106
5.003
4.950
5.102
5.569
-0.982
-0.981
-1.000
-1.262
-0.031
0.012
0.328
-0.045
1.038
0.936
1.036
1.305
5.020
4.937
4.800
5.663
-1.057
-0.973
-0.920
-1.516
1.014 0.765
2.034 0.935
5.036 0.608
10.191 0.870
1.012 0.443
2.020 0.431
4.999 0.744
10.085 0.402
1.003 0.529
2.031 0.676
5.032 0.160
10.136 0.600
1.002 0.243
1.999 0.701
4.998 0.261
9.942 0.323
1.008 0.938
2.008 0.705
5.018 0.526
10.014 0.456
1.001 0.367
2.012 0.250
5.097 1.012
9.984 0.337
1.012 0.910
2.017 0.340
5.065 0.581
10.094 0.884
1.007 0.778
2.014 0.336
5.002 0.271
10.152 0.811
0.993 0.476
2.020 1.311
5.074 0.872
9.874 0.469
0.064
0.274
1.246
7.149
0.035
0.150
1.403
3.239
0.047
0.217
0.351
5.101
0.021
0.230
0.571
2.951
0.063
0.243
1.054
3.929
0.030
0.081
1.945
2.679
0.069
0.123
1.164
6.883
0.062
0.100
0.567
6.545
0.040
0.359
1.608
3.534
0.003
0.016
0.038
0.385
0.001
0.006
0.066
0.102
0.002
0.007
0.014
0.300
0.001
0.015
0.026
0.122
0.002
0.012
0.045
0.072
0.001
0.004
0.123
0.084
0.003
0.004
0.032
0.378
0.003
0.003
0.018
0.327
0.003
0.023
0.093
0.207
25
-1.197
-1.217
-0.863
-1.135
-1.072
-0.986
-0.931
-1.092
-1.223
-1.275
-0.978
-0.927
-0.980
-1.142
-0.917
-0.760
1.080
1.048
1.046
0.898
0.923
1.030
1.282
0.958
1.133
0.888
0.983
1.084
1.062
0.856
0.876
1.250
1.812
2.085
2.068
1.741
-1.055
-1.120
-0.794
-1.458
-0.010
-0.003
0.128
-0.230
0.930
0.844
1.068
1.125
5.004
4.948
5.111
5.612
-0.982
-0.978
-1.000
-1.278
-0.036
0.013
0.356
-0.054
1.038
0.929
1.037
1.381
5.023
4.932
4.780
5.706
-1.060
-0.970
-0.919
-1.635
1.015 0.803
2.042 1.089
5.039 0.671
10.231 1.008
1.013 0.491
2.020 0.510
4.999 0.928
10.105 0.407
1.003 0.529
2.035 0.725
5.039 0.191
10.140 0.651
1.003 0.291
1.999 0.835
4.998 0.287
9.935 0.372
1.008 0.970
2.010 0.879
5.019 0.589
10.015 0.551
1.002 0.447
2.012 0.296
5.105 1.013
9.983 0.352
1.015 1.098
2.019 0.406
5.078 0.691
10.103 1.076
1.008 0.922
2.015 0.389
5.003 0.319
10.161 0.912
0.992 0.544
2.021 1.401
5.090 1.017
9.850 0.554
0.068
0.286
1.387
8.455
0.040
0.162
1.408
3.463
0.054
0.230
0.414
5.412
0.021
0.264
0.571
3.396
0.063
0.262
1.309
4.401
0.033
0.095
1.949
2.753
0.075
0.129
1.358
7.995
0.068
0.104
0.574
6.899
0.041
0.391
1.749
3.897
0.003
0.016
0.047
0.388
0.001
0.006
0.072
0.113
0.002
0.008
0.015
0.304
0.001
0.018
0.030
0.150
0.003
0.015
0.050
0.079
0.001
0.004
0.140
0.085
0.003
0.005
0.035
0.420
0.003
0.003
0.020
0.372
0.003
0.025
0.116
0.231
2
2
2
2
2
2
2
2
2
2
2
2
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
0 1 2.178
0 2 2.041
0 5 2.187
0 10 1.893
1 1 1.908
1 2 2.144
1 5 2.008
1 10 2.045
5 1 2.479
5 2 2.091
5 5 2.251
5 10 1.956
-1 1 4.790
-1 2 5.112
-1 5 5.181
-1 10 4.518
0 1 4.574
0 2 5.056
0 5 4.789
0 10 5.287
1 1 4.692
1 2 4.796
1 5 5.575
1 10 4.648
5 1 4.775
5 2 4.639
5 5 4.435
5 10 4.541
0.048
0.036
0.230
-0.334
0.977
1.071
0.986
1.136
5.127
5.039
5.353
4.871
-1.058
-0.970
-0.876
-2.090
-0.099
-0.003
-0.251
0.375
0.932
0.927
1.445
0.290
4.953
4.831
4.366
3.964
1.021 0.815
2.023 1.188
5.085 0.714
10.028 0.958
1.007 1.064
2.031 0.856
5.040 0.733
10.024 0.591
1.044 1.295
2.015 0.620
5.086 0.866
10.033 0.574
0.983 2.259
2.009 0.908
5.030 1.580
9.658 2.512
0.971 1.865
2.008 2.424
4.920 1.488
10.141 1.386
0.983 1.541
1.985 1.591
5.129 1.678
9.811 1.604
0.986 1.574
1.952 1.947
4.806 2.924
9.721 2.146
0.060
0.350
1.289
6.887
0.077
0.229
1.405
4.458
0.087
0.171
1.484
3.910
0.087
0.132
1.434
10.479
0.071
0.335
1.647
4.296
0.068
0.257
1.292
6.425
0.063
0.315
3.221
8.929
0.004
0.020
0.085
0.381
0.006
0.012
0.062
0.289
0.007
0.009
0.107
0.200
0.009
0.015
0.137
0.861
0.007
0.028
0.144
0.365
0.007
0.022
0.127
0.534
0.006
0.030
0.271
0.908
26
2.179
2.045
2.215
1.888
1.906
2.148
2.008
2.051
2.562
2.097
2.312
1.947
4.783
5.120
5.202
4.468
4.496
5.059
4.759
5.297
4.648
4.755
5.603
4.614
4.740
4.563
4.371
4.466
0.048
0.039
0.240
-0.379
0.975
1.087
0.984
1.139
5.142
5.045
5.391
4.864
-1.065
-0.963
-0.853
-2.360
-0.101
-0.004
-0.252
0.444
0.919
0.925
1.463
0.117
4.943
4.814
4.227
3.714
1.026 0.972
2.027 1.214
5.102 0.743
10.029 1.007
1.007 1.327
2.032 1.010
5.047 0.772
10.028 0.596
1.050 1.391
2.016 0.718
5.090 1.041
10.039 0.586
0.981 2.568
2.010 1.000
5.034 1.855
9.608 2.655
0.967 2.004
2.009 2.529
4.918 1.768
10.144 1.547
0.979 1.693
1.983 1.688
5.132 1.785
9.797 1.819
0.985 1.705
1.947 2.128
4.767 3.605
9.678 2.554
0.069
0.415
1.494
7.000
0.087
0.279
1.441
5.049
0.089
0.182
1.644
4.252
0.106
0.133
1.719
12.546
0.081
0.378
1.928
4.514
0.076
0.275
1.328
7.308
0.068
0.380
3.255
9.391
0.005
0.022
0.089
0.382
0.007
0.015
0.074
0.318
0.008
0.010
0.108
0.221
0.009
0.017
0.170
0.937
0.008
0.034
0.170
0.450
0.007
0.024
0.136
0.566
0.008
0.032
0.298
1.048
1.5
1.4
1.3
1.2
−1
f1(F−1
1 (u)) f2(F2 (u))
0.0
0.2
0.4
0.6
0.8
1.0
u
Figure 1. Plot of f1 (F1−1 (u))/f2 (F2−1 (u)) versus u when X is a standard normal random variable,
λ1 = −10 and λ2 = 5.
27
3
2
1
−1
−3
Skewness
−100
−50
0
50
100
50
100
30
20
10
Kurtosis
λ
−100
−50
0
λ
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) distributions, respectively.
28
1.5
0.5
−0.5
−1.5
Skewness
−100
−50
0
50
100
50
100
6
4
5
Kurtosis
7
λ
−100
−50
0
λ
Figure 3. The skewness and kurtosis of the truncated–exponential skew–t(10) and Azzalini skew–
t(10) distributions in (19) and (20), respectively, for λ = −100, −99, . . . , 99, 100. The solid and
broken curves correspond to Azzalini skew–t(10) and the truncated–exponential skew–t(10) distributions, respectively.
29
1.0
0.0
−1.0
Skewness
−100
−50
0
50
100
50
100
4.5
4.0
3.5
Kurtosis
5.0
λ
−100
−50
0
λ
Figure 4. The skewness and kurtosis of the truncated–exponential skew–t(20) and Azzalini skew–
t(20) distributions in (19) and (20), respectively, for λ = −100, −99, . . . , 99, 100. The solid and
broken curves correspond to Azzalini skew–t(20) and the truncated–exponential skew–t(20) distributions, respectively.
30
1.0
0.5
0.0
−1.0
Skewness
−100
−50
0
50
100
50
100
3.6
3.4
3.0
3.2
Kurtosis
3.8
λ
−100
−50
0
λ
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
1.0
0.8
0.6
0.4
0.0
0.2
Observed Probability
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.6
0.4
0.0
0.2
Observed Probability
0.8
1.0
Expected Probability
0.0
0.2
0.4
0.6
Expected Probability
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
1.0
0.8
0.6
0.4
0.0
0.2
Observed Probability
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.6
0.4
0.0
0.2
Observed Probability
0.8
1.0
Expected Probability
0.0
0.2
0.4
0.6
Expected Probability
Figure 7. Probability–probability plots of the fits of (17) (top) and (18) (bottom) for the annual
maximum rainfall data from Avon Park with sample size n = 94.
33
1.0
0.8
0.6
0.4
0.0
0.2
Observed Probability
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.6
0.4
0.0
0.2
Observed Probability
0.8
1.0
Expected Probability
0.0
0.2
0.4
0.6
Expected Probability
Figure 8. Probability–probability plots of the fits of (17) (top) and (18) (bottom) for the annual
maximum rainfall data from Gainesville with sample size n = 94.
34
1.0
0.8
0.6
0.4
0.0
0.2
Observed Probability
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
0.6
0.4
0.0
0.2
Observed Probability
0.8
1.0
Expected Probability
0.0
0.2
0.4
0.6
Expected Probability
Figure 9. Probability–probability plots of the fits of (17) (top) and (18) (bottom) for the annual
maximum rainfall data from Ocala with sample size n = 94.
35
0.7
0.6
0.5
0.4
0.3
0.0
0.1
0.2
Fitted PDFs
−1
0
1
2
3
4
5
0.6
0.4
0.0
0.2
Fitted PDFs
0.8
1.0
Standardized Annual Maximum Rainfall
−1
0
1
2
3
4
Standardized Annual Maximum Rainfall
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
0.6
0.5
0.4
0.3
0.0
0.1
0.2
Fitted PDFs
−1
0
1
2
3
4
5
0.4
0.0
0.2
Fitted PDFs
0.6
0.8
Standardized Annual Maximum Rainfall
−1
0
1
2
3
4
Standardized Annual Maximum Rainfall
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