The Extended Half-Skew Normal Distribution

Citation: Santoro, K.I.; Gómez, H.J.; Gallardo, D.I.; Barranco-Chamorro, I.; Gómez, H.W. The Extended Half-Skew Normal Distribution. Mathematics 2022,10, 3740. https:// doi.org/10.3390/math10203740 Academic Editor: Jiancang Zhuang Received: 1 September 2022 Accepted: 30 September 2022 Published: 12 October 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article The Extended Half-Skew Normal Distribution Karol I. Santoro 1, Héctor J. Gómez 2,*, Diego I. Gallardo 3, Inmaculada Barranco-Chamorro 4,* and Héctor W. Gómez 5 1Department of Mathematics, Faculty of Sciences, North Catholic University, Antofagasta 1240000, Chile 2Department of Physics and Mathematics Sciences, Faculty of Engineering, Catholic University of Temuco, Temuco 4780000, Chile 3Department of Mathematics, Faculty of Engineering, University of Atacama, Copiapó 1530000, Chile 4Department of Statistics and Operations Research, Faculty of Mathematics, University of Seville, 41012 Seville, Spain 5 Department of Mathematics, Faculty of Basic Sciences, University of Antofagasta, Antofagasta 1240000, Chile *Correspondence: [email protected] (H.J.G.); [email protected] (I.B.-C.) Abstract: A new class of densities for modelling non-negative data, which is based on the skewsymmetric family of distributions proposed by Azzalini is introduced.We focus on the model generated by the skew-normal distribution, called Extended Half Skew-Normal distribution. Its relevant properties are studied. These are pdf, cdf, moments, mgf, and stochastic representation. The parameters are estimated by moment and maximum likelihood methods. A simulation study to assess the performance of the maximum likelihood estimators in finite samples was carried out. Two real applications are included, in which the EHSN provides a better fit than other proposals in the literature. Keywords: lifetime distributions; skew-symmetric distributions; maximum likelihood MSC: 62E10; 62E15; 62F10 1. Introduction In the last few years, a considerable amount of research activity has been carried out in the field of skew-symmetric distributions theory and applications. Skew-symmetric distributions are of interest since they can be applied to datasets with an asymmetric structure without the need of applying arbitrary transformations to original data in order to reach symmetry. Quite often these transformations cause a loss of interpretability with respect to the original data. Nowadays, the focus of a number of statisticians is the development of non-symmetric parametric extensions of symmetric distributions. That is, to consider distributions with support in R , to which parameters are incorporated to deal with skewness and kurtosis. In this sense, we can cite the pioneering paper by Azzalini [ 1 ] in which the skew-normal model was introduced. Other univariate parametric extensions of this model can be seen in the works by Azzalini [ 2 ], Azzalini and Capitano [ 3 ], Gupta et al. [ 4 ], Arellano-Valle et al. [ 5 ], DiCiccio and Monti [ 6 ], Gómez et al. [ 7 ], Adcock and Azzalini [ 8 ], and Gómez-Déniz et al. [ 9 ], among others. In all these papers, results and applications of great interest can be found. The model studied in this paper is based on the univariate skew-normal distribution introduced by Azzalini [ 1 ]. There, a general method to build skew distributions on the basis of symmetric distributions was proposed. This key result is pointed out in next lemma, and will be one of the starting points to develop our proposal. Lemma 1. Let f be a symmetric at zero probability density function (pdf) and G an absolutely continuous cumulative distribution function (cdf), such that G0(x) = G0(−x). Then h(x;λ) = 2f(x)G(λx),x∈R,λ∈R, (1) Mathematics 2022,10, 3740. https://doi.org/10.3390/math10203740 https://www.mdpi.com/journal/mathematics Mathematics 2022,10, 3740 2 of 19 is the pdf of a random variable (rv) X . In (1), λ is a shape parameter related to the skewness of the distribution. In literature, it is said that hλ=h(x ; λ) introduced in (1) is the skew version of f with skewing function G. Gómez et al. [ 7 ] showed that cases of great interest are obtained if the N( 0, 1 ) cdf, Φ , is considered as skewing function. Specifically they proposed the following family of skew densities g(x;λ) = 2f0(x)Φ(λx),λ,x∈R, (2) where f0 is any symmetric around zero pdf and λ is a skewness parameter. Recall that, if f0=φ and G=Φ then gλ reduces to the pdf of the skew normal SN(λ) introduced by Azzalini [ 1 ]. As merit of (2), it can be cited that the models obtained are more flexible as for its skewness parameter than the distributions proposed by other authors, such as those proposed in Nadarajah and Kotz [10]. Flexible models can be of interest in real applications where is quite common to find non-negative data, which need for their modelling distributions with positive support. In this sense, we consider the model introduced in Elal-Olivero et al. [11], whose pdf is fY(y;α) = 2α+y2 α+kf0(y),y≥0, α≥0, (3) where f0 is the pdf of a symmetric around zero rv Y with E[Y2] = k<∞ . If f0(·) = φ , the pdf of the N( 0, 1 ) distribution, then k= 1 and (3) is called the Extended Half-Normal (EHN) density, which was studied in [ 11 ]. Other choices of f0 are possible, for instance, taken as f0(·) the Power Exponential pdf, [ 12 ], the Extended Half-Power Exponential model was recently proposed in [13]. The aim of this paper is to obtain a new model of distributions, based on (2), which will be flexible enough to modelling non-negative datasets. The outline of this paper is the following one. In Section 2, the new general family of distributions is introduced along with basic properties. In Section 3, the particular case based on the skew-normal pdf is considered. This model is called the Extended Half Skew-Normal (EHSN) distribution. Its cdf, moments and stochastic representation are studied in detail. Section 4is devoted to inference in the EHSN model. The method of moments and maximum likelihood (ML) are discussed. In Section 5, a simulation study is carried out to assess the consistence of ML estimators in the EHSN model. Finally, two real applications are given in Section 6, which illustrate the usefulness of our proposal. 2. A New General Family of Distributions In this section, based on (1), a new class of distributions for modelling non-negative data is introduced. Some basic properties of this family are also given. Lemma 2. Let gλ=g(x;λ)introduced in (2), and h a non-negative scalar function such that Z∞ 0h(x)gλ(x)dx <∞. Then, f(x) = cgλ(x)h(x),x>0 , is a pdf in R+with c−1=Z∞ 0h(x)gλ(x)dx <∞. Mathematics 2022,10, 3740 3 of 19 Note that, varying f0 or h in Lemma 2, a diversity of distributions with non-negative support can be obtained. In Proposition 1, we apply the idea given in (3) with k= 1 to the skew density gλdefined in (2). Proposition 1. Let gλ(·) defined in (2) and c−1 α,λ=Z∞ 0α+x2 α+1gλ(x)dx . Provided that c−1 α,λ< ∞, and by applying Lemma 2, the following family of densities is obtained fX(x;α,λ)=cα,λα+x2 α+1gλ(x),x≥0, (4) where α>0and λ∈R. Proof. Straightforward by using Lemma 2. Remark 1. Particular cases of interest of (4) are: P1. Taking limit when λ→∞ , then gλ tends to the half (or folded at zero) density of f0 ,2 f0(x) with x ≥0, and therefore fX(·)tends to the pdf introduced in (3). P2. If α→∞ , then fX tends to the density gλ truncated to ( 0, +∞) , which can be denoted as fX(x;α,λ)→cgλ(x), x ≥0. In a general setting, it is of interest to introduce a scale parameter δ> 0. Then, the pdf of our family will be δ−1fX(x/δ), that is fX(x;α,λ,δ) = cα,λ δα+ (x/δ)2 α+1gλx δ, (5) with cα,λgiven in Proposition 1. The notation X∼fX(x;α,λ,δ)will be used to refer to the density defined in (5). 3. The Extended Half Skew-Normal In this section, we focus on the particular case in which gλ(·) given in (2), is the pdf of the skew normal model, that is , f0=φ . The model, which results of applying (4), will be called the Extended Half Skew-Normal (EHSN). Results in this new model will be obtained by applying next lemmas, whose proofs can be seen in Nadarajah and Kotz [ 10 ] and Huang et al. [14], respectively. Lemma 3 ((Nadarajah and Kotz [ 10 ])) . Let Ftr be the cdf of a t-Student’s distribution with r> 0 degrees of freedom. Then, for every positive integer r, Ftris given by Ftr(t) =            1 2+1 πarctant √r+1 2√π (r−1)/2 ∑ i=1 Γ(i)ri−1/2 Γ(i+1/2) t (r+t2)i, if ris odd 1 2+1 2√π r/2 ∑ i=1 Γ(i−1/2)ri−1 Γ(i) t (r+t2)i−1/2 , if ris even (6) where t ∈Rand 0 ∑ i=1 is defined as 0. Lemma 4 (Huang et al. [14]).For s ≥ −1and λ∈R, the following result holds Z∞ 0vsφ(v)Φ(λv)dv =2s/2−1Γ((s+1)/2) √πFts+1λ√s+1. (7) Mathematics 2022,10, 3740 4 of 19 Lemma 5. Let X have pdf fX(x ; α , λ , δ) as introduced in (5). Then, the normalizing constant, cα,λ , in (5) is cα,λ=2π π+2 arctan(λ) + 2λ (1+λ2)(α+1) . (8) Proof. Note that, making the change of variable v=x δ, c−1 α,λ=Z∞ 0 2 δ3αδ2+x2 α+1φx δΦλx δd=Z∞ 02α+v2 α+1φ(v)Φ(λv)dv . By applying the results given in Lemma 3and Lemma 4[10] Z∞ 0φ(v)Φ(λv)dv =Γ(1/2) 2√πFt1(λ) = 1 4+1 2πarctan(λ), (9) Z∞ 0v2φ(v)Φ(λv)dv =Γ(3/2) √πFt3λ√3=1 4+1 2πλ 1+λ2+arctan(λ). (10) From (8)–(10) is obtained. Proposition 2. Let X ∼EHSN(α,δ,λ). Then, the pdf of X is given by fX(x;α,δ,λ) = cα,λ δ3αδ2+x2 α+12φx δΦλx δ,x≥0, (11) with α> 0shape parameter, δ> 0scale parameter, λ≥ 0skewness parameter and cα,λ given in (8). Proof. It follows from Lemma 5. Remark 2. In the model introduced in Proposition 2, we restrict the skewness parameter to λ≥ 0 since negative values of λ skew the distribution to negative values of x , and the resulting pdf’s are not of interest for the purpose of modelling non-negative data. Corollary 1. The following models are particular cases of the EHSN distribution: 1. If λ→ 0or λ→+∞ then EHSN (α , δ , λ) reduces to the Extended Half-Normal distribution, EHN(α,δ), introduced in [11]. 2. If λ= 0, α= 0, and δ= 1, then EHSN (α= 0, δ= 1, λ= 0 ) reduces to the Right Half Bimodal Normal model proposed in [15], RHBN(2). 3. If λ= 0and α→∞ then EHPE (α→∞ , δ , λ= 0 ) reduces to the Half-Normal distribution, HN(δ). Figure 1summarizes the relationships among the EHSN and the particular cases previously cited. In Figure 2, plots for the pdf of EHSN model are given. Without loss of generality, the scale parameter δ= 1 is taken. Four values of λ are fixed ( λ= 0.1, 1, 2, 5), and several values of αare considered. Next, it is proven that the cdf of the EHSN model can be expressed in terms of the cdf of a skew-normal, cdf and pdf of a Generalized Gamma, and pdf of a N(0, 1)distribution. Details about the Generalized Gamma introduced by Stacy [16] are given in Appendix A. Mathematics 2022,10, 3740 5 of 19 EHSN(α,δ,λ) α=0, δ=1, λ=0 !! λ=0 || λ=0, α→∞  EHN(α,δ)α=0, δ=1// α→∞ "" RHBN(2) HN(δ) Figure 1. Particular cases for the EHSN distribution. (a) (b) (c) (d) Figure 2. ( a ) Pdf of EHSN ( α , δ= 1, λ= 0.1) for different values of α . ( b ) Pdf of EHSN ( α , δ= 1, λ= 1) for different values of α . ( c ) Pdf of EHSN ( α , δ= 1, λ= 2) model for different values of α . ( d ) Pdf of EHSN (α,δ=1, λ=5) model for different values of α. Proposition 3. Let X ∼EHSN(α,δ,λ),λ>0. Then, the cdf X, FX, can be obtained as: FX(x) = cα,λ (α+1)αΦx δ−2Ox δ,λ−1 2+1 πarctan(λ)+1 2FGGx δ+Zx/δ 0fGG(t)Zλt 0φ(u)dudt, (12) where φ(·) and Φ(·) denote the pdf and cdf of a N( 0, 1 ) , FGG and fGG denotes the cdf and pdf of a Generalized Gamma distribution, GG(√2, 3, 2), and O(·,λ)is the Owen [17]function. Proof. Given X∼EHSN(α,δ,λ), let us consider Z=X δ∼EHSN(α, 1, λ). We have FX(x) = FZx δ. (13) Mathematics 2022,10, 3740 6 of 19 Next, we are going to obtain the cdf of Zfor z>0 FZ(z)=Zz 0fZ(t)dt =cα,λ α (α+1)Zz 02φ(t)Φ(λt)dt +cα,λ 1 (α+1)Zz 02t2φ(t)Φ(λt)dt . Note that 2 φ(t)Φ(λt) for t∈R is the pdf of a skew-normal distribution, SN(λ) , as it can be seen in [1]. Therefore, we can write Zz 02φ(t)Φ(λt)dt =Φx δ−2Ox δ,λ−1 2+1 πarctan(λ), (14) On the other hand, let us next consider Zz 02t2φ(t)Φ(λt)dt . Note that 2 t2φ(t) for t> 0 is the pdf of the Generalized Gamma distribution introduced by Stacy [ 16 ], GG(a=√2 , d= 3, p= 2 ) . By proceeding similarly to [ 18 ], we have that Zz 02t2φ(t)Φ(λt)dt =Zz 02t2φ(t)Zλt −∞φ(u)dudt , can be obtained in terms of the joint distribution of a random vector (T , U) with T∼ GG(√2, 3, 2),fGG(t) = 2t2φ(t)with t>0, and U∼N(0, 1)independent. Specifically, for λ>0 Zz 02t2φ(t)Zλt −∞φ(u)dudt =FGG(z)Φ(0) + Zz 02t2φ(t)Zλt 0φ(u)dudt . Taking into account that Φ( 0 ) = 1 / 2, fGG(t) = 2 t2φ(t) with t> 0, and (12) and (13) follows. Since the EHSN distribution can be used to model lifetime data is of interest to study its survival and hazard rate function, see [ 19 , 20 ]. These functions are next obtained. Some plots are given in Figure 3. (a) (b) Figure 3. ( a ) Plots of the survival function for the EHSN (α , 1, 0.1 ) . ( b ) Plots of the hazard rate function for the EHSN(α, 1, 0.1). Corollary 2. Let X ∼EHSN(α,δ,λ)with λ>0. Then 1. The survival function, SX, is SX(t) = 1−FX(t), t >0, and FXwas given in (12). 2. The hazard rate function, hX, is hX(t) = fX(t) SX(t), t >0, and fXgiven in (11). Mathematics 2022,10, 3740 7 of 19 Plots of the survival and hazard rate function of X∼EHSN(α , δ , λ) are given in Figure 3for fixed λ=0.1, δ=1, and α∈ {0.5, 1, 1.5, 2, 5}. Remark 3. Similar plots were obtained for the values of parameters considered in Figure 2. The plot obtained in Figure 3for the hazard rate function suggests that for λand δfixed, the EHSN model is stochastically ordered with respect to α. 3.1. Moments From now on, the notation Y∼fY(y) will be used to denote that Y has pdf fY . In this section, first the moments of EHSN model are obtained. Second it is proven that the moment generating function (mgf) of EHSN model can be obtained from the mgf of Y∼c1φ(y)Φ(λy),y≥0. Proposition 4. Let X ∼EHSN(α,δ,λ). Then, the moment of order n can be obtained as E[Xn] = 2cα,λ α+1(α2n/2−1Γ((n+1)/2) √πFtn+1(λ√n+1) + 2n/2Γ((n+3)/2) √πFtn+3(λ√n+3)), where Ftr(·)was given in (6). Proof. From (6) and (7), the proposed result follows. Next, the moment generating function (mgf) is obtained. Proposition 5. Let X ∼EHSN(α,δ,λ). Then the mgf of X, MX, can be obtained as MX(t) = MZ(δt)with Z ∼EHSN(α, 1, λ) and where the mgf of Z is given by MZ(t) = cα,λ (α+1)(1+α+t2)MY(t) +cα,λ (α+1)"t √2π+t√2 √π λ √1+λ2MHN(t) + λ π(1+λ2)MGG(t)#, where MY(t) is the mgf of Y∼c1 2 φ(y)Φ(λy) , y> 0; MHN(t) is the mgf of a half normal distribution, HN1 1+λ2; and MGG is the mgf of a Generalized Gamma, GG√2 √1+λ2, 2, 2. Proof. MZ(t) = E[exp(tZ)]=Z∞ 0etz fz(z)dz, with fzthe pdf of Z∼EHSN(α, 1, λ). Note that Z∞ 0etz2φ(z)Φ(λz)dz =MY(t)(15) with Y∼c1 2 φ(y)Φ(λy) , y> 0. That is, the mgf of a skew normal distribution, SN(λ) , truncated at zero. Let us consider now Z∞ 0etz2z2φ(z)Φ(λz)dz =I. (16) This integral is solved by applying integration by parts twice, taking in both cases dv =zφ(z), (v=−φ(z)). First, we have I=My(t) + 2tZ∞ 0etzzφ(z)Φ(λz)dz +2λZ∞ 0etzzφ(z)φ(λz)dz. Mathematics 2022,10, 3740 8 of 19 Second, we obtain 2tZ∞ 0etzzφ(z)Φ(λz)dz =t √2π+t2My(t) + λtZ∞ 0etz2φ(z)φ(λz)dz =t √2π+t2My(t) + λt√2 √π 1 √1+λ2MHN(t), where MHN(t)is the mgf of a half normal distribution, HN1 1+λ2. Finally, we consider 2λZ∞ 0etzzφ(z)φ(λz)dz =λ πZ∞ 0etz z exp−z2(1+λ2) 2dz =λ π(1+λ2)MGG(t) where MGG is the mgf of a Generalized Gamma, GGa=√2 √1+λ2,d=2, p=2. As a direct consequence of Proposition 4, the first four moments of EHSN(α , δ , λ) model are given, µk=E[Xk], with k=1, . . . , 4. Corollary 3. Let X ∼EHSN(α,λ,δ). Then, 1. µ1=2cα,λδ √2π(α+1)nαFt2λ√2+2Ft4λ√4o. 2. µ2=cα,λδ2 α+1nαFt3λ√3+3Ft5λ√5o. 3. µ3=2√2cα,λδ3 √π(α+1)nαFt4λ√4+4Ft6λ√6o. 4. µ4=cα,λδ4 α+1n3αFt5λ√5+15Ft7λ√7o. Expressions of variance, skewness and kurtosis can be obtained from Corollary 3. Corollary 4. Let X ∼EHSN(α,λ,δ). Then, 1. The variance of X, Var[X] = E[X2]−(E[X])2, is Var[X] = cα,λδ2 α+1αFt3(λ√3) + 3Ft5(λ√5)−4cα,λ 2π(α+1)hαFt2(λ√2+2Ft4(λ√4))i2. 2. The skewness, pβ1, and kurtosis, β2, coefficients can be obtained by using pβ1=µ3−3µ1µ2+2µ3 1 (µ2−µ2 1)3 2 ,and β2=µ4−4µ1µ3+6µ2µ2 1−3µ4 1 (µ2−µ2 1)2. Without loss of generality, the scale parameter can be taken as one, δ= 1. Plots for pβ1and β2, as functions of αand λare given in Figure 4. Mathematics 2022,10, 3740 9 of 19 (a) (b) (c) (d) Figure 4. ( a ) pβ1 as function of λ(α=δ= 1 ) ; ( b ) β2 as function of λ(α=δ= 1 ) ; ( c ) pβ1 as function of αand λ(δ=1); (d)β2as function of αand λ(δ=1). 3.2. Stochastic Representation In this section, the stochastic representation of the EHSN model is given. This result will be used to generate random values of this model. Next propositions allow us to obtain the stochastic representation of the EHSN model. Proposition 6. Let Y and U be independent rv’s, with Y2∼χ2 3 and P(U=− 1 ) = P(U= 1 ) = 1 2. Then, the rv W =U|Y|is distributed as W ∼w2φ(w)with w ∈R. Proof. It can be seen in Elal-Olivero et al. [11]. Proposition 7. Let Z and W be independent rv’s, with Z∼N( 0, 1 ) and W∼w2φ(w) , w∈R . Let us define H =qα α+1Z+q1 α+1W. Then H ∼α+h2 α+1φ(h), with h ∈Rand α>0. Proof. Let us denote a=qα α+1and b=q1 α+1. We have that the cdf of H,FH, is FH(h) = P(aZ +bW ≤h) = PZ≤h−bW a=Z∞ −∞FZh−bw afW(w)dw . Mathematics 2022,10, 3740 16 of 19 Figure 7. Fitted EHSN and EHN model by maximum likelihood for speed of wind dataset. 6.2. Application 2 This dataset corresponds to heights (100 × feet) of 219 volcanoes studied in [ 27 ]. In Table 4, the descriptive summaries are provided. These are: the sample mean ( x ), the sample variance ( s2 ), the sample skewness coefficient ( √b1 ), and the sample kurtosis coefficient (b2). In this application, the EHSN model is compared to LSN and EHN. Moments estimates in the EHSN model are obtained by applying results in Section 4.1, these are bαMM = 1.579, b δMM = 59.668, and b λMM = 2.788. These estimates are used as initial values to obtain MLEs by using numerical methods. ML estimates for the LSN, EHN, and EHSN models, along with their standard errors are provided in Table 5. For purposes of comparison, log-likelihood, AIC and BIC are also included in this table. These summaries support the fact that the EHSN model provides a better fit to this dataset. As plots, the histogram, along with the estimated pdfs, are given in Figure 8. The QQ-plots comparing EHN and EHSN models are given in Figure 9. In Figure 10, the empirical cdf is plotted along with the cdf estimated for the EHSN model. All these plots support our conclusions. Table 4. Descriptive summaries for volcano heights (in 100 ×feet). n x s2√b1b2 219 70.247 1850.548 0.8344 3.4439 Table 5. Estimates of parameters in EHN and EHSN models for volcanoes dataset. Estimates of Parameters LSN EHN EHSN α4.015 (1.051)0.783 (0.286)1.579 (0.923) δ0.770 (0.037)56.514 (2.894)59.668 (4.187) λ0.007 (1.709)2.788 (1.616) Log-likelihood −1133.636 −1117.914 −1115.100 AIC 2273.272 2241.828 2236.200 BIC 2283.440 2251.996 2246.367 Mathematics 2022,10, 3740 17 of 19 Figure 8. Histogram for volcano heights dataset and estimated models. EHSN (solid line), EHN (dashed line), LSN (dotted line). (a) (b) Figure 9. qqplots for volcano heights dataset: (a) EHN model, (b) EHSN model. Figure 10. Empirical and fitted EHSN cdf for volcano heights in dataset 2. 7. Final Discussion and Conclusions This study presents a new model with positive support based on the skew-normal distribution, which has been called the extended half skew-normal distribution. This distribution is useful as a more general model compared to the EHN model proposed by Elal-Olivero [ 11 ], pursuing to increase kurtosis and improve the modeling of positive datasets with high kurtosis. Relevant properties of the model are given. Closed expressions Mathematics 2022,10, 3740 18 of 19 are given for the pdf, cdf, mgf, and moments. Additionally, a stochastic representation is proposed, which will be the basis to generate random values in this model. Estimation of parameters is carried out via maximum likelihood methods by using numerical techniques. The simulation study illustrates the good performance of estimators. Two applications with real datasets were also carried out, verifying that the new model performs better than the competing models. One of the referees points out that the pdf’s introduced throughout this paper, see, for instance, the general family defined in (4) or the pdf for the EHN introduced in (3), can be considered as weighted distributions, [ 28 ]. For the pdf introduced in (4), this would be to write fX(x) = w(x;α) EgλI[x≥0][w(X;α)]gλ(x)I[x≥0], with weight function w(x ; α) = α+x2 . This idea may be studied in future works along with its implications. Author Contributions: Conceptualization, K.I.S., H.J.G., D.I.G. and I.B.-C.; Formal analysis, I.B.-C. and D.I.G.; Investigation, K.I.S., H.J.G. and I.B.-C.; Software, D.I.G.; Supervision, H.W.G. All of the authors contributed significantly to this research article. All authors have read and agreed to the published version of the manuscript. Funding: The research of I. Barranco-Chamorro was supported by IOAP of University of Seville, Spain. The research of H.W. Gómez was supported by Semillero UA-2022 project, Chile. Data Availability Statement: Dataset studied in Section 6.1 can be found at http://lib.stat.cmu. edu/datasets/wind.data, (last accessed on 12 July 2022). The dataset in Section 6.2 has been taken from [27]. Acknowledgments: The authors want to express their gratitude to referees, whose comments helped to improve significantly this paper. Conflicts of Interest: The authors declare no conflict of interest. Appendix A In this Appendix details are given about the models under consideration in Section 3. These models are: • Skew-Normal, X∼SN(λ), introduced in [1], whose pdf is f(x;λ) = 2φ(x)Φ(λx),x∈R,λ∈R. (A1) The cdf is F(x;λ) = Φ(x)−2O(x,λ),x∈R,λ∈R, (A2) where O(·,λ)denotes the Owen function [17]. • Generalized Gamma, GG(a,d,p), introduced in [16], whose pdf is f(x;a,d,p) = 1 Γd pp adxd−1e−(x/a)p,x>0, a,d,p>0 . (A3) The cdf is F(x;a,d,p) = Γzd p Γd p(A4) where z= (x/a)pand Γzd p=Zz 0v(d/p)−1e−vdv, see [29]. Mathematics 2022,10, 3740 19 of 19 References 1. Azzalini, A. 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