The skew-normal distribution was introduced by Azzalini [1], typically denoted as {SN(λ),λ∈R}, with asymmetry parameter λ, so that SN(0) becomes the standard normal distribution. Hence, Y∼SN(λ) has a density function given byf(y|λ)=2ϕ(y)Φ(λx),y∈R(1)with λ∈R and where ϕ and Φ denote, respectively, the density and distribution functions of the N(0,1) distribution. Some well known facts about this distribution areE(Y)=2π−−√λ1+λ2−−−−−√;Var(Y)=1−2λ2π(1+λ2)(2)β1−−√=12(4−π)(E2(X)Var(Y))3/2;β2=2(π−3)(E2(Y)Var(Y))2−3(3)where β1−−√ and β2 are asymmetry and kurtosis coefficients, respectively. From (3) it is well known that−0.9953≤β1−−√≤0.9953(4)0≤β2≤0.8692.(5)Henze [2] developed the stochastic representation for the skew-normal density and computed its odd moments. The stochastic representation and some more general representations for skew models are also discussed in Azzalini [3]. Arnold et al. [4] make use of the above results to develop truncations of the normal model. Pewsey [5] studied inference problems faced by the skew-normal model with some general results, especially consequences of the singularity of the Fisher information matrix (FIM) in the vicinity of symmetry. Gupta and Chen [6] developed a goodness of fit test. Reliability studies for the skew-normal model were developed by Gupta and Brown [7]. Univariate extensions to the skew-normal model are studied in Arellano-Valle et al. [8], Azzalini [9], Gómez et al. ([10,11]), [12], etc.When it is necessary to model data with more than one mode, mixtures of distributions are always used. The importance of such studies rests on the fact that these models present computational difficulties due to identifiability problems. One major motivation of this paper is to develop models than can be seen as alternative parametric models to replace the use of mixtures of distributions, as these present estimation problems from either classical or Bayesian points of view ([13,14]). Bimodal distributions generated from skew distributions can be found in Azzalini and Capitanio [15], Ma and Genton [16], Arellano-Valle et al. [17], Kim [18], Lin et al. ([19,20]), Elal-Olivero et al. [21], Arnold et al. [22], Arellano-Valle et al. [23], Elal-Olivero [24], Gómez et al. [25], Arnold et al. [26], Braga et al. [27], Venegas et al. [28], Shah et al. [29], Gómez-Déniz et al. [30], Esmaeili et al. [31], Imani and Ghoreishi [32], Maleki et al. [33], etc. The main object of this paper is to study the properties of the bimodal skew-normal model introduced by Elal-Olivero et al. [21]. In particular, we derive results related to stochastic representation of the distribution and density function; this makes it simple to derive distributional moments and inferences by maximum likelihood (ML) estimation, among other quantities. The paper is organized as follows. Section 2 develops a bimodal normal distribution, its basic properties, representation, moments and moment generating function. Section 3 develops a s
kew-normal bimodal distribution, its basic properties, stochastic representation, moments and moment generating function. In Section 4, we perform a small scale simulation study of the ML estimators for parameters. A real data application is discussed in Section 5, which illustrates the usefulness of the proposed model. Conclusions and future work are presented in Section 6.
https://www.mdpi.com/2227-7390/8/5/703/htm
David Elal-Olivero, Héctor W. Gómez, Heleno Bolfarine, Juan F. Olivares-Pacheco, Osvaldo Venegas
