50
Research Article
Received: 17 July 2008, Accepted: 19 July 2009, Published online in Wiley Online Library: 11 September 2009
(wileyonlinelibrary.com) DOI: 10.1002/env.1026
Bimodal extension based on the skew-normal
distribution with application to pollen data
H´ ector W. G´ omez
a∗
, David Elal-Olivero
b
, Hugo S. Salinas
b
and Heleno
Bolfarine
c
This paper considers an extension to the skew-normal model through the inclusion of an additional parameter which can
lead to both uni- and bi-modal distributions. The paper presents various basic properties of this family of distributions
and provides a stochastic representation which is useful for obtaining theoretical properties and to simulate from the
distribution. Moreover, the singularity of the Fisher information matrix is investigated and maximum likelihood estimation
for a random sample with no covariates is considered. The main motivation is thus to avoid using mixtures in fitting bimodal
data as these are well known to be complicated to deal with, particularly because of identifiability problems. Data-based
illustrations show that such model can be useful. Copyright © 2009 John Wiley & Sons, Ltd.
Keywords: asymmetry; kurtosis; uni-bimodality; maximum likelihood estimation; singular information matrix
1. INTRODUCTION
In an important paper, Azzalini (1985) studies properties of the {SN(α),α ∈ R} family, namely the skew-normal family of distributions with
asymmetry parameter α, so that SN(0) indicates the standard normal distribution. This family has been extensively studied in the recent
statistical literature, with extensions in different directions. However, a limitation of this family of densities is the fact that the distributions
are unimodal. Hence, we say that Z ∼ SN(α) if its density function is given by
f (z | α) = 2φ(z)(αz), z, α ∈ R (1)
with φ(·) and (·) the density and distribution function of the standard normal distribution, respectively. Some well-known properties of the
distribution SN(α) are
E(Z) =
2/πλ (2)
β
1
=
1
2
(4 − π)
E
2
(Z)
Var(Z)
3/2
,β
2
= 2(π − 3)
E
2
(Z)
Var(Z)
2
(3)
M
Z
(t ) = 2 exp
t
2
2
(λt ) (4)
where λ = α/
√
1 + α
2
,
√
β
1
, and β
2
are the asymmetry and kurtosis coefficients, respectively. M
Z
(t ) denotes the moment generating function
of the random variable Z.
Henze (1986) develops a stochastic representation for this density and derives its odd moments. Azzalini (1986) also discusses the stochastic
representation and studies more general models than the skew-normal model. Pewsey (2000) reports on some inference problems for this
model. A goodness of fit test is discussed by Gupta and Chen (2001). Arellano-Valle et al. (2004) introduce a skew symmetric model which
∗
Correspondence to: H. W. G´ omez, Facultad de Ciencias B´ asicas, Departamento de Matem ´ aticas, Universidad de Antofagasta, Chile. E-mail: hgomez@uantof.cl
a H´ ector W. G´ omez
Facultad de Ciencias B´ asicas, Departamento de Matem ´ aticas, Universidad de Antofagasta, Chile
b David Elal-Olivero
Facultad de Ingenier
´
ia, Departamento de Matem ´ atica, Universidad de Atacama, Chile
c Heleno Bolfarine
Departamento de Estatistica, IME, Universidad de Sao Paulo, Brasil
Environmetrics 2011; 22: 50–62 Copyright © 2009 John Wiley & Sons, Ltd.