A Generalized Rayleigh Family of Distributions Based on the Modified Slash Model

symmetry S S Article A Generalized Rayleigh Family of Distributions Based on the Modified Slash Model Inmaculada Barranco-Chamorro 1,*,† , Yuri A. Iriarte 2,†, Yolanda M. Gómez 3,† , Juan M. Astorga 4,† and Héctor W. Gómez 2,*,†   Citation: Barranco-Chamorro, I.; Iriarte, Y.A.; Gómez, Y.M.; Astorga, J.M.; Gómez, H.W. A Generalized Rayleigh Family of Distributions Based on the Modified Slash Model. Symmetry 2021,13, 1226. https:// doi.org/10.3390/sym13071226 Academic Editor: Jinyu Li Received: 3 June 2021 Accepted: 4 July 2021 Published: 8 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2020 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/). 1Departamento de Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain 2Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile; [email protected] 3Departamento de Matemáticas, Facultad de Ingeniería, Universidad de Atacama, Copiapó 1530000, Chile; [email protected] 4Departamento de Tecnologías de la Energía, Facultad Tecnológica, Universidad de Atacama, Copiapó 1530000, Chile; [email protected] *Correspondence: [email protected] (I.B.-C.); hector[email protected] (H.W.G.) † These authors contributed equally to this work. Abstract: Specifying a proper statistical model to represent asymmetric lifetime data with high kurtosis is an open problem. In this paper, the three-parameter, modified, slashed, generalized Rayleigh family of distributions is proposed. Its structural properties are studied: stochastic representation, probability density function, hazard rate function, moments and estimation of parameters via maximum likelihood methods. As merits of our proposal, we highlight as particular cases a plethora of lifetime models, such as Rayleigh, Maxwell, half-normal and chi-square, among others, which are able to accommodate heavy tails. A simulation study and applications to real data sets are included to illustrate the use of our results. Keywords: generalized Rayleigh distribution; EM algorithm; kurtosis; maximum likelihood estimation; slashed generalized Rayleigh distribution 1. Introduction Vodua [ 1 ] proposed a two-parameter distribution for the analysis of positive data, the well-known generalized Rayleigh distribution. This model overcomes the problems found in practice, which are related to a Rayleigh distribution and pointed out by authors, such as Siddiqui [ 2 ] and Hirano et al. [ 3 ]. Recall that a random variable (rv) X follows a generalized Rayleigh distribution, denoted as X∼GR(θ , α) , if its probability density function (pdf) is given by the following: f(x;θ,α) = 2θα+1 Γ(α+1)x2α+1e−θx2,x>0, (1) where θ> 0 is a scale parameter, α>− 1 is a shape parameter and Γ(α) = R∞ 0uα−1e−udu is the gamma function. The cumulative distribution function (cdf) of the GR distribution can be expressed as follows: F(x;θ,α) = γ(α+1, θx2) Γ(α+1)=P(α+1, θx2),x>0, (2) Symmetry 2021,13, 1226. https://doi.org/10.3390/sym13071226 https://www.mdpi.com/journal/symmetry Symmetry 2021,13, 1226 2 of 18 where γ(α , x) = Rx 0uα−1e−udu is the incomplete gamma function and P(α+ 1, θx2) denotes the regularized gamma function [ 4 ]. The moments of X are easily derived by using the results in [5] (Section 3.478). In fact, the rth distributional moment is as follows: µr=E(Xr) = Γr 2+α+1 Γ(α+1)θr/2 ,r=1, 2, 3, ... (3) Following the approach proposed in [ 6 ], lifetime models to accommodate outliers were proposed in [ 7 ]. We focus on the approach proposed in [ 8 ], where an extension of the generalized Rayleigh distribution called the slashed generalized Rayleigh distribution was introduced. This model has a kurtosis coefficient, which exhibits a wider range of values than the kurtosis coefficient in the GR distribution; therefore, it is appropriate to fit the data sets with outliers. Recall that a rv T follows a slashed generalized Rayleigh distribution, denoted as T∼SGR(θ,α,q), if its stochastic representation is given by the following: T=X U1/q,q>0, (4) where X∼GR(θ , α) and U∼ U ( 0, 1 ) are independent. The associated pdf is given by the following: fT(t;θ,α,q) = qt−(q+1) Γ(α+1)θq/2 Γ2α+q+2 2Fθt2,2α+q+2 2, 1,t>0, (5) where θ> 0 is a scale parameter, α>− 1 is a shape parameter, q> 0 is a kurtosis parameter and F(x , α , β) = Rx 0 βα Γ(α)tα−1e−βtdt is the cdf of a gamma distribution with shape parameter α> 0 and rate parameter β> 0. If α= 0 and θ= ( 2 σ)−1 , the GR distribution reduces to the slashed-Rayleigh distribution. If α=− 1 / 2 and θ= ( 2 σ2)−1 , the GR distribution reduces to slashed-half normal distribution [9]. In this work, we propose a modified version of the SGR distribution, given in (4), which can be used as an alternative model to the GR and SGR distributions. We follow the idea proposed by [ 10 ], who introduced an extension of the normal distribution called modified slash (MS) distribution. A rv X follows a modified slash distribution, denoted as X∼MS(q), if it can be written as follows: X=Y W1/q,q>0, where Y∼ N ( 0, 1 ) and W∼Exp( 2 ) are independent. If q→∞ , then the MS distribution converges in law to a normal distribution. The paper is organized as follows. In Section 2the stochastic representation, the pdf and ordinary moments of the proposed model are given. Section 3is devoted to the family of distributions belonging to the generalized Rayleigh family of distributions based on the modified slash model. Among others, we can find here the modified slashed Rayleigh, half-normal, Maxwell and chi-square distributions. Recall that the unified treatment of a family of distributions is of interest in statistics; see, for instance, [ 11 – 13 ]. In Section 4 , the estimation of parameters is discussed by considering the moments and maximum likelihood approaches. Similar to [ 14 ], their asymptotic normality is established. In addition, we carry out a simulation study to illustrate the performance of the estimators. In Section 5, two applications to real data sets are presented, where we illustrate that the proposed distribution may provide a better fit than the generalized Rayleigh and slashed generalized Rayleigh distributions. Finally, the main conclusions of this paper are given. 2. The New Model In this section, some structural properties of the new model are given. These are the stochastic representation, pdf, hazard rate function and distributional moments. Symmetry 2021,13, 1226 3 of 18 2.1. Stochastic Representation Definition 1. A random variable T follows a modified slashed generalized Rayleigh distribution, denoted as T ∼MSGR(θ,α,q), if it can be expressed as follows: T=X −1 2log(Z)1/q,q>0, (6) where X ∼GR(θ,α)and Z ∼U(0, 1)are independent. Remark 1. 1. Let T∼MSGR(θ , α , q) . Then, T can be expressed as T=XY−1/q , where X∼GR(θ , α) and Y ∼Exp(2)are independent. 2. If q= 1then T follows a canonic modified slashed generalized Rayleigh (CMSGR) distribution, and we can write T=− 2 Xlog−1(Z) , where X∼GR(θ , α) and Z∼ U ( 0, 1 ) are independent. This is denoted as T ∼CMSGR(θ,α). The stochastic representation given in (6) can be used to generate values of T∼ MSGR(θ,α,q)by using the following algorithm: 1. Generate X∼GR(θ,α). 2. Generate Z∼U(0, 1). 3. Compute T=X−1 2log(Z)−1/q. 2.2. Probability Density Function To deal with the pdfs of generalized slash distributions, it is common to use hypergeometric functions as can be seen in [ 7 , 15 , 16 ]. In this subsection, we first introduce the special function, which is used in the expression of the pdf for the MSGR distribution, which is given in Proposition 1. Definition 2. For γ>0, a >0, r >0and s >0, let us consider the following: I(γ,a,r,s) = Z∞ 0xγ−1e−axr−sx dx (7) where I(·,·,·,·)is defined as in [17] (Equation 2.3.1.13). Proposition 1. Let T ∼MSGR(θ,α,q). Then, the pdf of T is given by the following: fT(t;θ,α,q) = 2qθ−q 2 Γ(α+1)tq+1Iα+q 2+1, 2 θq 2tq,q 2, 1,t>0, (8) where θ>0is a scale parameter, α>−1is a shape parameter and q >0is a kurtosis parameter. Proof. From (6), and by applying the change of variable method, the pdf of T is given by the following: fT(t;θ,α,q) = 4qθα+1 Γ(α+1)t2α+1Z∞ 0w2α+q+1e−θ(tw)2−2wqdw =2qθ−q 2 Γ(α+1)tq+1Z∞ 0uα+q 2e−u−2u θt2q 2 du . By considering I(·,·,·,·), the integral introduced in Definition 2, (8) is obtained. Symmetry 2021,13, 1226 4 of 18 Corollary 1. Let T ∼MSGR(θ,α,q). Then, the cdf of T is as follows: FT(t;θ,α,q) = 2q Γ(α+1)θq 2Zt 0u−(q+1)Iα+q 2+1, 2 θuq,q 2, 1du where Γ(·)is the gamma function and I(·,·,·,·)was defined in (7). As for the shape of the pdf in the MSGR model, we highlight that the pdf can be unimodal or monotonically decreasing. Plots of the pdf for certain values of the parameters are given in Figure 1. In Figure 1a,b, we can see that the parameter q is a kurtosis parameter since it mainly affects the right tail of the distribution. In Figure 1c, we can appreciate that the parameter θ is a scale-type parameter; the smaller the value of θ , the more spread out the distribution [ 18 ]. Finally, in Figure 1d, we can see that the parameter α is a shape parameter. 012345 0.0 0.2 0.4 0.6 0.8 (a): MSGR(θ=1, α=0.5, q) t Density function q ∞ 1 2 3 8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 (b): MSGR(θ=5, α=−0.5, q) t Density function q ∞ 1 2 3 8 012345 0.0 0.2 0.4 0.6 0.8 1.0 (c): MSGR(θ, α=1.5, q=3) t Density function θ 5 4 3 2 1 012345 0.0 0.2 0.4 0.6 0.8 1.0 (d): MSGR(θ, α=1.5, q=3) t Density function α −0.8 −0.5 0 1 2 Figure 1. Plots of the MSGR pdf for certain values of its parameters. 2.3. Reliability Analysis For non-negative rvs, it is of interest to know the reliability (or survival) function and the hazard rate function [ 19 , 20 ]. The reliability function, RT(t) = 1 −FT(t) , of a MSGR distribution is given by the following: RT(t;θ,α,q) = 1−2q Γ(α+1)θq 2Zt 0u−(q+1)I(u)du, (9) where I(u) = Iθ,α,q(u) = Iα+q 2+1, 2 θ q 2uq,q 2, 1is defined as in (7). The hazard rate function, defined by hT(t) = fT(t) RT(t), is given by the following: hT(t;θ,α,q) = t−(q+1)I(t) Γ(α+1)θ q 2 2q−Zt 0u−(q+1)I(u)du , where I(u)is as in (9) and I(t) = Iα+q 2+1, 2 θ q 2tq,q 2, 1, defined in (7). Figure 2displays some plots of the hazard rate function of the MSGR distribution for different values of its parameters. Symmetry 2021,13, 1226 5 of 18 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 MSGR(θ=1, α=−0.5, q) t Hazard rate q ∞ 1 5 10 15 0 1 2 3 4 0 1 2 3 4 5 6 MSGR(θ=1, α=2, q) t Hazard rate q ∞ 1 5 10 15 Figure 2. Plot of the hazard rate function for the MSGR distribution for different values of its parameters. Applications of interest in engineering of the Rayleigh models can be seen in [21–24] among others. In these applications, it can be of interest to apply our proposal. 2.4. Moments Next, we proceed to obtain the moments of the MGSR distribution. From these features, important characteristics of the model, such as the mean or expected value, variance, skewness and kurtosis coefficients are obtained. These results will also be the basis to apply the method of moments in Section 4. Proposition 2. Let T∼MSGR(θ , α , q) . Then, for r= 1, 2, ... and q>r the r th distributional moment is given by the following: µr=E(Tr) = 2r qΓq−r qΓr 2+α+1 Γ(α+1)θr 2 . (10) Proof. Using the stochastic representation given in Part 1 of Remark 1and the independence of Xand Y, it follows that µr=E(Tr)=E X Y1 q!r!=EXrY−r q=E(Xr)EY−r q, where EY−r q= 2 r qΓq−r q , q>r and E(Xr)=Γ(r 2+α+1) Γ(α+1)θr 2 are distributional moments of the Exp(2)and GR(θ,α)distributions, respectively. . Corollary 2. Let T∼MSGR(θ , α , q) . Then, the mean and variance are, respectively, the following: E(T) = 21/qΓq−1 qΓ(α+3/2) Γ(α+1)θ1/2 ,q>1, and Var(T) = 22/q Γ2(α+1)θΓ(α+1)Γ(α+2)Γq−2 q−Γ2(α+3/2)Γ2q−1 q,q>2. Symmetry 2021,13, 1226 6 of 18 Corollary 3. Let T∼MSGR(σ , q) . Then, the asymmetry ( β1 ) and kurtosis ( β2 ) coefficients are the following: β1=1 A3/2(α,q)Γ2(α+1)Γ(α+5/2)Γq−3 q−3Γ(α+3/2)Γ(α+2)Γ(α+1) ×Γq−1 qΓq−2 q+2Γ3(α+3/2)Γ3q−1 q,q>3, and β2=1 A2(α,q)Γ3(α+1)Γ(α+3)Γq−4 q−4Γ(α+3/2)Γ(α+5/2) ×Γ2(α+1)Γq−1 qΓq−3 q+6Γ2(α+3/2)Γ(α+1)Γ(α+2) ×Γ2q−1 qΓq−2 q−3Γ4(α+3/2)Γ4q−1 q,q>4, where A(α,q) = Γ(α+1)Γ(α+2)Γ(q−2 q)−Γ2(α+3 2)Γ2(q−1 q). Remark 2. Notice that β1 and β2 do not depend on θ because it is a scale parameter. Furthermore, as q →∞the β1and β2coefficients tend, respectively, to the following: β1GR =1 A3/2(α)hΓ2(α+1)Γ(α+5/2)−3Γ(α+3/2)Γ(α+2)Γ(α+1) + 2Γ3(α+3/2)i and β2GR =1 A2(α)hΓ3(α+1)Γ(α+3)−4Γ(α+3/2)Γ(α+5/2)Γ2(α+1) +6Γ2(α+3/2)Γ(α+1)Γ(α+2)−3Γ4(α+3/2), where A(α) = Γ(α+ 1 )Γ(α+ 2 )−Γ2(α+ 3 / 2 ) , which are the asymmetry and kurtosis coefficients of the GR distribution. Figure 3depicts plots for the asymmetry and kurtosis coefficients of the MSGR model. In the figure, we observe that the highest values of asymmetry and kurtosis are associated with small values of parameter q and large values of parameter α . In addition, when comparing the asymmetry and kurtosis coefficients of the GR and MSGR distributions, we observe that q has a great impact on the increase in kurtosis, which is why we refer to q as a kurtosis parameter. This point is clearly observed in Figure 4. Symmetry 2021,13, 1226 7 of 18 ALPHA 0 5 10 15 20 q 510 15 20 ASYMMETRY 0 5 10 15 20 ALPHA 0 5 10 15 20 q 5 10 15 20 25 KURTOSIS 0 50 100 Figure 3. Plots of the asymmetry and kurtosis coefficients for the MSGR model. 0 5 10 15 20 0 2 4 6 8 10 α Asymmetry MSGR q=6 q=5 q=4 q=3.5 GR 0 5 10 15 20 0 20 40 60 80 α Kurtosis MSGR q=7 q=6 q=5 q=4.5 GR Figure 4. Plots of the asymmetry and kurtosis coefficients for the GR and MSGR models. 3. The MSGR Family of Distributions In this section, we see that well-known models are associated with the MSGR family of distributions. Some of these results follow directly from (8) under certain reparameterizations, while other ones follow taking the limit when q→∞ . Similar to the techniques used in [25], the limit distribution of (1) when q→∞is obtained. Lemma 1. Let Tq∼MSGR(θ , α , q) . Then, Tq converges in distribution to X∼GR(θ , α) when q→∞. Proof. Let us consider the stochastic representation for Tqgiven in Remark 1: Tq=X Y1/q where X∼GR(θ,α)and Y∼Exp(2). First, we prove the convergence in probability of Y1/q to a constant as q→∞ . Since Y∼Exp(2)and q>0, we have the following: E[Y1/q] = Γ1+1 q 21/q, Var[Y1/q] = Γ1+2 q−Γ21+2 q 22/q. Symmetry 2021,13, 1226 8 of 18 By applying the Tchebychev inequality, we have the following: PhY1/q−E[Y1/q]>ei≤Var[Y1/q] e2∀e>0, (11) and since Var[Y1/q]→0 as q→∞then {Y1/q−E[Y1/q]}P −→ 0. Similarly, the sequence of real numbers behaves as follows: E[Y1/q] = Γ1+1 q 21/q−→ 1, q→∞. Therefore, Y1/q=Y1/q−E[Y1/q] + E[Y1/q]P −→ 1 as q→∞. By applying Slutsky’s theorem ([26] Corollary 2.3.2), we have the following: Tq=X Y1/q d −→ Xas q→∞ that is, Tqconverges in distribution to X∼GR(θ,α)distribution. Next, we see that well-known models are particular cases of the MSGR(θ , α , q) family of distributions. Specifically, for α= 0 and θ= ( 2 σ)−1 , the MSGR model is reduced to the modified slashed Rayleigh distribution [ 27 ]. If α=− 1 / 2 and θ= ( 2 σ2)−1 , then the MSGR reduces to the modified slashed half-normal distribution [ 28 ]. For α= 1 / 2 and θ=σ/ 2, we get the modified slashed Maxwell distribution. If α=σ/ 2 − 1 and θ= 1 / 2, we obtain the modified slashed chi-square distribution. Taking the limit as q→∞ in previous models, the Rayleigh, half-normal [ 29 ], Maxwell [ 30 ] and chi-square distribution are obtained [ 31 ]. Details about these related models are given in the next corollaries. The relationships among these models are summarized in Figure 5. Corollary 4 (Modified slashed Rayleigh model) . Let T∼MSGR( 1 /( 2 σ) , 0, q) . Then, T follows a modified slashed Rayleigh distribution, MSR(σ , q) , whose density function is given by the following: fT(t,σ,q) = 2q+2 2qσq 2 tq+1I q 2+1, 2q+2 2σq 2 tq,q 2, 1!,t>0, σ>0is a scale parameter, q >0is a kurtosis parameter [27]. As q→∞ then T converges in distribution to a Rayleigh distribution whose pdf is the following: fT(t) = t σe−t2 2σ,t>0. Corollary 5 (Modified slashed half-normal model) . Let T∼MSGR( 1 / 2 σ2 , 1 / 2, q) . Then, T follows a modified slashed half-normal distribution, MSHN(σ , q) , whose pdf is given by the following: fT(t;σ,q) = 2q+2 2qσq √πtq+1I q+1 2,2q+2 2σq tq,q 2, 1!,t>0, where σ>0is a scale parameter and q >0is a kurtosis parameter [28]. Symmetry 2021,13, 1226 9 of 18 As q→∞ , T converges in distribution to a half-normal distribution whose pdf is the following: fT(t) = 2 σφt σ,t>0, where φ(·)denotes the pdf of a N(0, 1)distribution. Corollary 6 (Modified slashed Maxwell model) . Let T∼MSGR(σ/ 2, 1 / 2, q) . Then, T follows a modified slashed Maxwell distribution, MSM(σ,q), whose pdf is given by the following: fT(t;σ,q) = 2 σq 24q √πtq+1I q+3 2,2q 2+1 σq 2tq,q 2, 1!,t>0, where σ>0is a scale parameter and q >0is a kurtosis parameter. As q→∞ , T converges in distribution to a Maxwell distribution whose pdf is the following: fT(t) = r2 πσ3/2t2e−σ 2t2,t>0. Corollary 7 (Modified slashed chi-square model) . Let T∼MSGR( 1 / 2, σ/ 2 − 1, q) . Then, T follows a modified slashed chi-square distribution, MSχ2 σ,q, whose pdf is given by the following: fT(t;σ,q) = 2q 2+1q Γσ 2tq+1I σ+q 2,2q 2+1 tq,q 2, 1!,t>0, where σ>0is a scale parameter and q >0is a kurtosis parameter. As q→∞ , T converges in distribution to a gamma distribution with shape parameter σ/ 2 and rate parameter 1/2 as follows: fT(t) = 1 2σ/2Γ(σ/2)tσ/2−1e−t 2,t>0. If σis a positive integer, then we have a chi-square distribution. MSGR(θ,α,q) θ=1/2;α=σ/2−1  q=1  α=0 α=1/2;θ=1/2σ2  α=1/2;θ=σ/2  q→∞  CMSCR(θ,α, 1) α=1 ** MSR(θ,q) q=1 uu q→∞;θ=(2λ2)−1  MSHN(σ,q) q→∞ 33 MSM(σ,q) q→∞  GR(θ,α) α=ν/2−1;θ=1/2  α=−1/2;θ=(2λ2)−1 }} α=1/2;θ=(2λ2)−1  α=1/2;θ=(2λ2)−1 __ MSχσ,q q→∞  CMSR(θ, 1) Rayleigh(λ)Maxwell(λ)Half-normal(λ)χ2 Figure 5. Relationships among distributions in the MSGR family. Symmetry 2021,13, 1226 16 of 18 Remission time (Months) Density 0 20 40 60 80 0.00 0.02 0.04 0.06 0.08 0.10 Figure 8. Fitted models by ML method for remission time data set: MSGR (solid line), SGR (dashed line), GR (dotted line). ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ●● ● 0 20 40 60 80 0 20 40 60 80 (a) Theorical Quantiles Sample Quantiles ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ●● ● ●●● ●● ● 0 20 40 60 80 0 20 40 60 80 (b) Theorical Quantiles Sample Quantiles ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ●● ● 0 20 40 60 80 0 20 40 60 80 (c) Theorical Quantiles Sample Quantiles Figure 9. QQ-plots: (a) MSGR model, (b) SGR model, (c) GR model. 6. Concluding Remarks In this article, a new generalization of the GR distribution, which is able to model positive data with high kurtosis, is introduced. The proposed distribution can be written as the ratio of two independent random variables. These are a GR distribution and a power of the exponential distribution. We derived some special cases that can be seen as twoparameter extensions of the Rayleigh, half normal, Maxwell and chi square distributions. The parameters were estimated by using moments and maximum likelihood methods. The method of moments estimates were used to start the maximum likelihood estimation through the Newton–Raphson procedure. The asymmetry and kurtosis coefficients were also obtained. The study of these coefficients illustrates the fact that the MSGR model can be used to describe positive right-skewed data with high kurtosis. Finally, the MSGR model was fitted to two real data sets. In both applications, the MSGR distribution presented a better fit than the GR and SGR models, which shows the potential applicability of our proposal. To conclude, we point out that these models can be used not only in reliability studies, but also in the fields of economics, for instance, to describe financial data, where models with heavy tails are very interesting. Author Contributions: Conceptualization, I.B.-C., Y.A.I. and Y.M.G.; Formal analysis, I.B.-C., Y.A.I., Y.M.G. and J.M.A.; Investigation, I.B.-C., Y.A.I., Y.M.G., J.M.A. and H.W.G.; Methodology, I.B.-C., Y.A.I., Y.M.G. and H.W.G.; Software, Y.A.I. and Y.M.G.; Supervision, I.B.-C., Y.M.G. and H.W.G.; Validation, I.B.C., Y.A.I. and 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. Symmetry 2021,13, 1226 17 of 18 Funding: The research of H.W. Gómez was supported by SEMILLERO UA-2021 project, Chile. 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