Bolfarine, Heleno

Gómez, H. W.

Martínez, E. H.

Varela, H.

Revistes Catalanes amb Accés Obert

Statistics & Operations Research Transactions
SORT 32 (1) January-June 2008, 57-66
Statistics &
Operations Research
Transactions
A note on the likelihood and moments of the
skew-normal distribution
c© Institut d’Estadı´stica de Catalunya
sort@idescat.esISSN: 1696-2281
www.idescat.net/sort
Eliseo H. Martı´nez1, He´ctor Varela2,
He´ctor W. Go´mez3 and Heleno Bolfarine
Abstract
In this paper an alternative approach to the one in Henze (1986) is proposed for deriving the odd
moments of the skew-normal distribution considered in Azzalini (1985). The approach is based on
a Pascal type triangle, which seems to greatly simplify moments computation. Moreover, it is shown
that the likelihood equation for estimating the asymmetry parameter in such model is generated as
orthogonal functions to the sample vector. As a consequence, conditions for a unique solution of the
likelihood equation are established, which seem to hold in more general setting.
MSC: 62F10, 65C20
Keywords: Skew-normal distribution, maximum likelihood equations, moments
1 Introduction
The density function of the skew-normal distribution with parameters ε, ω > 0 and λ
(location, scale and asymmetry parameters, respectively) introduced by Azzalini (1985),
is given by
1 Departamento de Matema´ticas, Facultad de Ciencias Ba´sicas, Universidad de Antofagasta, Chile. E-mail:
emartinez@uantof.cl. Partially supported by Grant DIRINV 1323-06.
2 Departamento de Matema´ticas, Facultad de Ciencias Ba´sicas, Universidad de Antofagasta, Chile. E-mail:
hvarela@uantof.cl. Partially supported by Grant DIRINV 1323-06.
3 Departamento de Matema´tica, Facultad de Ingenierı´a, Universidad de Atacama, Chile. E-mail: hgomez@
mat.uda.cl. Partially supported by Grant FONDECYT 1060727.
4 Departamento de Estatistica, IME, Universidad de Sao Paulo, Brasil. E-mail: hbolfar@ime.usp.br. Partially
supported by CNPq-Brasil.
Received: December 2006
Accepted: November 2007
58 A note on the likelihood and moments of the skew-normal distribution
φ(x; ε, ω, λ) = 2
ω
φ
(
x − ε
ω
)
Φ
(
x − ε
ω
· λ
)
,
where φ and Φ are, respectively, the density and cumulative distribution functions of
the standard normal distribution. We use the notation X ∼ S N(ε, ω, λ) to denote this
distribution, and in the special case of the standard version, X ∼ S N(0, 1, λ). As is well
known, the density function of the ordinary normal distribution follows as a special
case (i.e., λ = 0). Moreover, the k-th moment of a random variable X with S N(ε, ω, λ)
distribution can be computed directly from the integral
E
[
Xk
]
=
∞∫
−∞
xk
2
ω
φ
(
x − ε
ω
)
Φ
(
x − ε
ω
· λ
)
dx,
which is quite complicated to deal with. Using this general expectation, moments can
be computed as, for example, if X ∼ S N(0, 1, λ), then
μ1 = E [X] =
√
2 λ√
π
√(λ2 + 1) ; μ2 = E
[
X2
]
= 1.
It is also well known that if X ∼ S N(0, 1, λ) then X2 ∼ χ2(1) (Azzalini, 1985), so that
the even moments are equal to the moments of the ordinary normal distribution. Henze
(1986) employs a stochastic representation for this model, so that if X ∼ S N(0, 1, λ),
then
X = δ|Z0| + (1 − δ2)1/2Z1,
where δ = λ/(1 + λ2)1/2 and Z0 and Z1 are independent N(0, 1) random variates. Odd
moments can then be computed in a more straightforward way.
Azzalini and Capitanio (1999) report on some applications of the multivariate skew-
normal distribution while Genton et al. (2001) derive the moments for the multivariate
version. Pewsey (2000) discusses inference problems for the univariate skew-normal
distribution such as unbounded likelihood and singular information matrix. A recent
review encompassing most of the recent advances on the topic is found in Azzalini
(2005).
The main object of this note is to present alternative approaches for deriving the
odd moments of the skew-normal distribution and study the behaviour of the likelihood
equation for the asymmetry parameter. In particular, conditions for a unique finite root
are discussed. We propose using a Pascal type triangle, which seems to greatly simplify
computing the odd moments. This approach seems to be new and completely overlooked
by the literature.
The paper is organized as follows. Section 2 deals with odd moments of the skew-
normal distribution, which are derived using a Pascal type triangle, diﬀerent from the
Eliseo H. Martı´nez, He´ctor Varela, He´ctor W. Go´mez and Heleno Bolfarine 59
approach considered in Henze (1986). Section 3 is devoted to the study of the maximum
likelihood estimator for the S N(0, 1, λ) distribution. In Section 4 final comments are
presented.
2 Odd moments of the skew-normal distributions
In this section we discuss alternative derivations for the odd moments of the skew-
normal distribution. We recall that the even moments are the same as the ones for
the symmetric normal model. We start by presenting a recursive relation for moments
computation.
Proposition 1 Let X ∼ S N(0, 1, λ). Then
μ2n+1 = E
[
X2n+1
]
= 2nμ2n−1 +
√
2
π
(2n)!
2n(n)!
λ
(1 + λ2)n+1/2 , (1)
for n = 1, 2, . . . .
Proof. Solving μ2n−1 =
∞∫
−∞
x2n−12φ(x)Φ(λx) dx using integration by parts, we obtain
μ2n−1 =
[
x2n
n
φ(x)Φ(λx)
]∞
−∞
+
μ2n+1
2n
− 1
2n
∞∫
−∞
2λφ(x)φ(λx) x2n dx.
Solving for the integral on the right hand side, we obtain
μ2n−1 =
μ2n+1
2n
− 1
2n
√
2
π
(2n)!
2n(n)!
λ
(1 + λ2)n+1/2 (2)
and the result follows by re-expressing the moment μ2n+1 = μ2n+1 in the last expression.

Equation (1) allows computing moments for the skew normal distribution in a
recursive fashion, which can be troublesome. We can, however, use a more direct
approach, as described next. Recall the double factorial function for a positive integer:
n!! =
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
n(n − 2) · · · 3 · 1 n > 0, odd
n(n − 2) · · · 4 · 2 n > 0, even
1 n = 0, 1.
60 A note on the likelihood and moments of the skew-normal distribution
Of its many properties, the one we may use is
(2n − 1)!! = (2n)!
2nn! ,
so that recursive equation (2) can be written as
μ2n+1 = 2nμ2n−1 +
√
2
π
(2n − 1)!! λ(1 + λ2)n+1/2 ,
for n = 1, 2, . . . .
The first few iterations of this equation lead to the following moments:
μ1 =
√
2 λ√
π
√(λ2 + 1) ,
μ3 =
√
2/π
[
2
λ
(1 + λ2)1/2 + 1!!
λ
(1 + λ2)3/2
]
,
μ5 =
√
2/π
[
2 · 4 λ(1 + λ2)1/2 + 4 · 1!!
λ
(1 + λ2)3/2 + 3!!
λ
(1 + λ2)5/2 ,
]
and
μ7 =
√
2/π
[
2 · 4 · 6 λ(1 + λ2)1/2 + 6 · 4 · 1!!
λ
(1 + λ2)3/2 + 6 · 3!!
λ
(1 + λ2)5/2 + 5!!
λ
(1 + λ2)7/2
]
,
leading to the following compact equation:
μ2n+1 =
√
2
π
n∑
k=0
(2n)!!
(2k)!!
(2k − 1)!!λ
(1 + λ2)(2k+1)/2 ,
which can be rewritten after a straightforward factorization as
μ2n+1 =
√
2/π
λ
(1 + λ2)(2n+1)/2
n∑
k=0
(2n)!!
(2k)!! (2k − 1)!!(1 + λ
2)n−k.
Expanding (1 + λ2)n−k leads to
μ2n+1 =
√
2/π
λ
(1 + λ2)(2n+1)/2
n∑
k=0
n−k∑
j=0
(2n)!!
(2k)!! (2k − 1)!!
(
n − k
j
)
λ2 j.
Eliseo H. Martı´nez, He´ctor Varela, He´ctor W. Go´mez and Heleno Bolfarine 61
It can be shown after extensive but straightforward algebraic manipulations that the
above expressions and the expression given in Hence (1986), Corollary 4, are equivalent.
Hence, using this last expression, the first four odd moments of the skew normal
distribution can be written as
μ1 =
√
2/π
1
(1 + λ2)1/2λ · 1!!
μ3 =
√
2/π
1
(1 + λ2)3/2 (2!!λ
3 + 3!!λ)
μ5 =
√
2/π
1
(1 + λ2)5/2 (4!!λ
5 + 4(2!! + 3!!)λ3 + 5!!λ)
μ7 =
√
2/π
1
(1 + λ2)7/2 (6!!λ
7 + 6[4!! + 4(2!! + 3!!)]λ5 + 6[4(2!! + 3!!) + 5!!]λ3 + 7!!λ)
and by fixing attention to the coeﬃcients of λ2 j+1, a Pascal type triangle formation can
be depicted. The results are then formalized in the next proposition.
Proposition 2 If X ∼ S N(0, 1, λ) and n is a positive integer then the odd moments are
given by
E
[
X 2 n−1
]
=
√
2√
π (λ2 + 1)(2 n − 1) / 2 Qn(λ)
where
Qn(λ) =
n∑
k=1
an(k) λ2 k−1
and the coeﬃcients an(k) are computed iteratively by using
a1(1) = 1;
an(1) = (2n − 1) an−1(1) ; n ≥ 2;
an(k) = (2n − 2) (an−1(k) + an−1(k − 1)) ; n ≥ 2, 1 < k < n;
an(n) = (2n − 2) an−1(n − 1).
62 A note on the likelihood and moments of the skew-normal distribution
Hence, the odd moments of X can be computed iteratively by computing the
coeﬃcients an(k) which can be cumbersome. So, it would be interesting to have a more
convenient and direct way, without having to use recursion.
We show in the sequel that the coeﬃcients an(k) can be computed by using a Pascal
type triangle, constructed according to the following steps. It starts with 1 as the ordinary
Pascal triangle, and the left hand side of the triangle has even levels greater than 1
(2,4,6,. . . ), for example, level 2 has two numbers, 2 and 3 and the right hand side of the
triangle has odd levels greater than 1 (3,5,7,. . . ). For example, the leftmost number in
level 3 is number 8, which results from multiplying basis number (2) by the following
even number which is 4. On the other hand, the rightmost number in that level follows
by multiplying the basis number (3) by the following odd number which is 5, so that 15
results.
Moreover, all the other numbers are generated similarly as the Pascal Triangle except
that each resulting number is multiplied by the next corresponding even level. For
example, 20 is obtained by adding 2 and 3 (equals 5), which multiplied by 4 makes
20. 168 is obtained by adding 8 and 20 and multiplying by the next even level which is
6, leading to 168. Similarly, 210 is obtained by adding 20 and 15 and also multiplying
by the line even level, 6, and so on.
The polynomial Qn(λ) can then be computed for odd degrees. It also presents a form
of regularity which we believe has not been noticed so far in the literature. It is expressed
by the fact that the odd moments of the polynomial Qn(λ), of which the first and last
coeﬃcients, namely an(1) and an(n), are odd and even double factorials, respectively, and
the remaining coeﬃcients follow a Pascal type triangle regularity and are, consequently,
easily computable.
As an example, we compute in the sequel the coeﬃcients for the moments of order
1, 3, 5, 7, 9 for the skew normal distribution. As described above, we obtain:
1
2 3
8 20 15
48 168 210 105
384 1728 3024 2520 945
As an illustration, we compute the seventh moment of the skew-normal distribution,
which follows by considering the fourth line (n = 4) in the triangle above, leading to
Q4(λ) = 48λ7 + 168λ5 + 210λ3 + 105λ,
from which the seventh moment can be computed as
E
[
X7
]
=
√
2λ (48λ6 + 168λ4 + 210λ2 + 105)√
π(λ2 + 1)7/2 .
Eliseo H. Martı´nez, He´ctor Varela, He´ctor W. Go´mez and Heleno Bolfarine 63
Note that the right hand side of the triangle reproduces the even moments for the skew
normal distribution.
Using the above results, moments for the location-scale situation, that is, for Z ∼
S N(ε, ω, λ), can be easily computed by using the fact that
E[Zr] =
r∑
k=0
(
r
k
)
εr−kωkμk.
which follows directly by using the binomial expansion.
3 Maximum likelihood equation for the asymmetry parameter
Suppose that X ∼ S N(0, 1, λ) with λ unknown, and let x1, · · · , xn a random sample from
this distribution. The likelihood function for the parameter λ based on the observed
sample is then given by
l(λ) = n log 2 +
n∑
i=1
logφ(xi) +
n∑
i=1
logΦ (λxi)
Diﬀerentiating the log of the likelihood with respect to parameter λ, the following
equation is obtained:
∂l(λ)
∂ λ
=
n∑
i=1
R(i)xi = 0, (3)
where
R(i) = φ(λxi)
Φ(λxi) . (4)
The estimator of λ can be found easily. In fact, we only have to solve the equation:
n∑
i=1
R(i)xi = 0.
Hence, in this case, to find the maximum likelihood estimator of λ, we have to
find the R(i) (or λ) defined according to (4), so that the vector R = (R(1), · · · ,R(n))
is orthogonal to x = (x1, · · · , xn). Note that, Rx is a function of λ, that is, we can write
f (λ) = R · x, (5)
64 A note on the likelihood and moments of the skew-normal distribution
and the maximum likelihood estimator of λ makes R and x orthogonal. This function
clearly is continuous for all values of λ. Conditions for a unique root in (5) are
established next.
Proposition 3 If x = (x1, · · · , xn)′, and if there exists k and j such that xk < 0 and xj > 0
then f (λ) = 0 has a unique real root.
Proof. We can separate in the likelihood function the positive and negative sample
elements, so that f (λ) = 0 if and only if
f (λ) =
∑
xi>0
R(i) xi +
∑
x j<0
R( j) xj = 0
Moreover, defining
h(λ) =
∑
xi>0
R(i)xi ; g(λ) =
∑
x j<0
R( j) xj,
it clearly follows that h(λ) > 0 and g(λ) < 0 for any λ. Notice that R(x) > 0 and it is
a diﬀerentiable monotonically decreasing function from +∞ to 0 as x ranges from −∞
to +∞. Hence, h(−∞) = ∞ and h(+∞) = 0 so that the proof can proceed as follows.
Notice that f (λ) is the sum of a positive decreasing function h and a negative increasing
function g, such that
f (−∞) = ∞ + 0 = ∞, f (∞) = 0 −∞ = −∞
and therefore f (λ) has a root; monotonicity of h and g imply uniqueness of this root. 
Liseo (1990) has noticed the inverse statement to Proposition 3, namely that if all
sample elements have the same sign, then the MLE does not exist. Pewsey (2006)
considers the general case, that is, the function Φ(.) is replaced by a general G.
We call attention to the fact that the above results can be extended to any density of
the form
h(x) = 2g(x)Φ(λx),
where g(.) is a symmetric density function and Φ(.) is as above. Such more general
families of densities are considered in Arellano-Valle and Genton (2005) and Go´mez et
al. (2007).
Eliseo H. Martı´nez, He´ctor Varela, He´ctor W. Go´mez and Heleno Bolfarine 65
4 Discussion
In this note we have shown new approaches for computing odd moments of the skew
normal distribution. The first method uses a Pascal type triangle making computing the
moments more accessible, while the second method uses a recursive approach. Further,
it is also shown that the maximum likelihood equation for estimating the asymmetry
parameter λ is a product of an orthogonal function to the sample vector, leading to
conditions for a unique root of the likelihood equation. Clearly, Proposition 3 holds in
more general settings.
Acknowledgments
The authors acknowledge helpful comments and suggestions from two referees which
substantially improved the presentation.
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A note on the likelihood and moments of the skew-normal distribution

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