Generalized modified slash Birnbaum–Saunders distribution

symmetry S S Article Generalized Modified Slash Birnbaum–Saunders Distribution Jimmy Reyes 1, Inmaculada Barranco-Chamorro 2, Diego I. Gallardo 3and Héctor W. Gómez 1,* 1Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile; jimmy[email protected] 2Departamento de Estadística e Investigación Operativa, Universidad de Sevilla, 41000 Sevilla, Spain; [email protected] 3Departamento de Matemática, Facultad de Ingeniería, Universidad de Atacama, Copiapó 1530000, Chile; [email protected] *Correspondence: hector[email protected]; Tel.: +56-552-637-278 Received: 15 October 2018; Accepted: 3 December 2018; Published: 6 December 2018 Abstract: In this paper, a generalization of the modified slash Birnbaum–Saunders (BS) distribution is introduced. The model is defined by using the stochastic representation of the BS distribution, where the standard normal distribution is replaced by a symmetric distribution proposed by Reyes et al. It is proved that this new distribution is able to model more kurtosis than other extensions of BS previously proposed in the literature. Closed expressions are given for the pdf (probability density functio), along with their moments, skewness and kurtosis coefficients. Inference carried out is based on modified moments method and maximum likelihood (ML). To obtain ML estimates, two approaches are considered: Newton–Raphson and EM-algorithm. Applications reveal that it has potential for doing well in real problems. Keywords: Birnbaum–Saunders distribution; generalized modified slash distribution; kurtosis; maximum likelihood; EM-algorithm 1. Introduction The BS distribution was introduced by Birnbaum and Saunders [ 1 , 2 ]. The aim of this distribution is to model the fatigue in lifetime of certain materials. Nowadays, its use has spread to other contexts such as economic and environmental data. In these new applications, it is quite common to find real datasets in which a BS model with heavier tails would be suitable. Slash models are a good option to deal with this kind of situations, in which heavy tails are a serious problem for the data analyst. This is the main reason slash distributions have received a great deal of attention during the last decades. In this context, we face the problem of improving BS distribution by introducing a generalization able to model more kurtosis than other slash extensions previously proposed in the literature. In these extensions, the emphasis is on kurtosis because, as Moors [ 3 ] pointed out, the presence of heavy tails produces high kurtosis. Next, we briefly describe the BS-model and the most relevant slash precedents of our proposal. 1.1. Birnbaum–Saunders Distribution If a random variable (rv) T> 0 follows a BS distribution with shape parameter α> 0 and scale parameter β>0, T∼BS(α,β), then Tcan be expressed as T=β α 2Z+rα 2Z2+1!2 (1) Symmetry 2018,10, 724; doi:10.3390/sym10120724 www.mdpi.com/journal/symmetry Symmetry 2018,10, 724 2 of 18 where Z∼N( 0, 1 ) . From Equation (1), T is a monotone transformation of Z , and its cumulative distribution function (cdf) FTis FT(t) = Φ(w(t)) (2) with Φ(·)the cdf of a N(0, 1)distribution and w(t) = wα,β(t) = 1 α st β−rβ t!=2 αsinh ln st β!,t>0. (3) The probability density function (pdf) of Tis fT(t;α,β) = t−3 2(t+β) 2αβ1 2 φ(w(t)) (4) where φ(·)is the pdf of a N(0, 1)distribution (Johnson et al. [4]). As properties (see, for instance, Leiva [ 5 ]), we highlight that the BS distribution is continuous, unimodal and positively skewed (asymmetry to right). β is the median of the distribution. α is a shape parameter that modifies the skewness and kurtosis of the distribution. As α tends to zero, the BS distribution tends to be symmetrical around β and its variability decreases. On the other hand, as α increases, the BS distribution exhibits heavier tails. 1.2. Slash Methodology To use the BS distribution for modeling data with outliers, Gómez et al. [ 6 ] and Reyes et al. [ 7 ] proposed extensions of BS model based on the slash (S) and modified slash (MS) distribution. In this way, they got extensions of BS distribution with a high kurtosis coefficient. The canonic slash distribution was introduced by Rogers and Tukey [ 8 ]. This model is defined as the ratio of a N( 0, 1 ) and an independent uniform U( 0, 1 ) distribution. It is proposed as a model for bell-shaped data with heavier tails than the corresponding normal distribution. Their theoretical properties can be seen, for instance, in Rogers and Tukey [ 8 ] or Johnson et al. [ 4 ]. The slash model, denoted as S, in which a kurtosis parameter qis introduced, is defined as S=Z U1 q (5) with Z∼N( 0, 1 ) independent of U∼U( 0, 1 ) and q> 0. Based on the representation given in Equation (5), Reyes et al. [ 7 ] proposed the modified slash (MS) distribution in which the variable at the denominator of Equation (5) is replaced by an exponential distribution of parameter 2, that is, U∼Exp( 2 ) . The MS model exhibits greater kurtosis than the S model. A new extension, called generalized modified slash (GMS) distribution, was introduced recently by Reyes et al. [ 9 ]. These authors proposed a new slash model where the denominator in Equation (5) is a Gamma distribution of parameters (2 q , q ) with q> 0. The GMS model generalizes the MS model. As main features of the GMS model, we highlight that is a bell-shaped distribution, symmetrical with respect to zero, and exhibits a greater level of kurtosis than its predecessors, thus it can be of interest to study the distribution of the BS extension obtained when Z∼N( 0, 1 ) in Equation (1) is replaced by a GMS distribution with kurtosis parameter q> 0. This proposal is a generalization of the papers by Gómez et al. [ 6 ] and Reyes et al. [ 7 ] where slash versions of the BS distribution were considered based on the slash and modified slash distribution, called slash Birnbaum–Saunders (SBS) and modified slash Birnbaum–Saunders (MSBS), respectively. This paper is outlined as follows. In Section 2, the stochastic representation of the generalized modified slash Birnbaum–Saunders (GMSBS) distribution is introduced, and its probability density function, properties, moments, skewness and kurtosis coefficients are obtained. Section 3is devoted Symmetry 2018,10, 724 3 of 18 to estimation methods: modified moment and maximum likelihood estimation (an iterative method and the EM-algorithm are proposed). Section 4assesses the performance of the MLE using the EM algorithm via a simulation study. Two practical applications are given in Section 5. 2. GMSBS Distributions In this section, the stochastic representation of a GMSBS distribution is introduced. A closed expression for its pdf is obtained and its properties are studied in depth. Motivated by Equation (1), the stochastic expression proposed for a GMSBS distribution is T=β α 2X+rα 2X2+1!2 ,α>0, β>0, (6) where X follows a Generalized Modified Slash distribution, X∼GMS( 0, 1, q) , q> 0. It is then said that T follows a GMSBS distribution with parameters α , β , and q , T∼GMSBS(α , β , q) . Similar to the BS distribution, α is a shape parameter and β is a scale parameter. It is shown below that the new parameter q allows us to control the kurtosis and skewness of this new model and to obtain distributions with greater level of kurtosis than other slash Birnbaum–Saunders models. This fact allows us to model real datasets in which a BS-model can be appropriate but we have heavy tails, especially on the right. 2.1. Probability Density Function Since T , introduced in Equation (6), is given as a function X with X∼GMS( 0, 1, q) , to obtain the distribution of T, we need the pdf of X, which is given in next lemma. Lemma 1. Let X∼GMS( 0, 1, q) be defined as X=Z/V with Z∼N( 0, 1 ) independent of V∼Ga( 2 q , q) , q>0. Then, the pdf of X is fX(x;q) = (2q)q Γ(q)Z∞ 0vqe−2qvφ(xv)dv,x∈R(7) where φ() denotes the pdf of a N(0, 1)distribution. Proof. It can be seen in Reyes et al. [9]. Lemma 2. The following closed expression for Equation (7) can be given fX(x;q) =        q √8π,if x =0 q2q/2 √2π 1 |x|q+2U1+q 2,3 2,2 x2,if x 6=0 (8) where U(·) denotes the confluent hypergeometric function of the second kind (Abramowitz and Stegun [ 10 ], p. 505). Proof. It can be seen in Reyes et al. [9]. Proposition 1. Let T ∼GMSBS(α,β,q). Then, the pdf of T is fT(t;α,β,q) = 2q−1qqt−3/2(t+β) αβ1/2Γ(q)Z∞ 0vqe−2qvφ(w(t)v)dv, (9) with w(t) = 1 αqt β−qβ tand t >0. Symmetry 2018,10, 724 4 of 18 Proof. From Equation (6) FT(t) = FX(w(t)) (10) where FT(·)and FX(·)denote the cdfs of Tand X, respectively, and w(t)is given in Equation (3). From Equation (10), the following relationship for the pdf’s of Tand Xfollows fT(t) = fX(w(t))w0(t), (11) where fX(·)denotes the pdf of a X∼GMS(0, 1, q). Finally, from Equation (11), Lemma 1and Equation (3), the expression proposed in Equation (9) is obtained. Corollary 1. From Equation (11) and (8), we have the following closed expression for fT() fT(t;α,β,q) =        1 αβ√8π,if t =β 2q/2qq+2 √8π (β+t) |t−β|q+2αq+1β(q+1)/2t(q−1)/2 U1+q 2,3 2,2q2α2βt (t−β)2,if t 6=β (12) with t >0,α>0,β>0and q >0. The next corollary relates the new model, proposed in Equation (6), to other slash models previously introduced in the literature. Corollary 2. For q= 1, the pdf given in Equation (12) reduces to the pdf of a modified slash Birnbaum–Saunders distributions, MSBS(α,β, 1)proposed in Reyes el al. [11]. Proof. This corollary follows from the fact that, for q= 1, a Ga( 2,1 ) distribution reduces to an exponential, Exp(2), and the stochastic representation proposed in Equation (6). Figure 1illustrates the effect of the parameter q on the tails of our proposal. Plots given in this figure compare the pdfs of several GMSBS models for different values of q . Specifically, the pdfs of a GMSBS( 0.3, 2, q) distribution for q= 8, 3, 1 are given. Note that a greater level of kurtosis is observed for small values of q. These appreciations are formalized in Section 2.3 where moments are obtained. 2.2. Properties In this subsection, some properties of GMSBS distributions are deduced. Proposition 2. Let T ∼GMSBS(α,β,q), with α>0,β>0, q >0. Then, 1. Let tpbe the pth quantile of T, 0<p<1. tp=β α 2xp+rα 2xp2+1!2 (13) where xpdenotes the pth quantile of X ∼GMS(0, 1, q). In particular, the median of T is β, t0.5 =β. 2. ∀b>0, bT ∼GMSBS(α,bβ,q). 3. T−1∼GMSBS(α,β−1,q). Proof. (1). Equation (13) follows from the fact that Equation (6) is a one-to-one transformation from R to R+. Symmetry 2018,10, 724 5 of 18 On the other hand, t0.5 =β since X∼GMS( 0, 1, q) is a symmetric distribution around zero, and therefore x0.5 =0. (2) and (3) are immediate from Proposition 1by properly using the change-of-variable technique. 0 1 2 3 4 5 6 0.05 0.10 0.15 0.20 0.25 0.30 0.35 t density function GMSBS(α=0.3, β=2, q=5) GMSBS(α=0.3, β=2, q=3) GMSBS(α=0.3, β=2, q=2) Figure 1. GMSMS(α=0.3, β=2, q)pdfs for different values of q. Next, we show that if q→∞ then GMSBS(α , β , q) model approaches to a Birnbaum–Saunders distribution. The subscript qis included in the notation to highlight this fact. Proposition 3. Let Tq∼GMSBS(α,β,q). Then, Tqconverges in law to T ∼BS(2α,β)as q →∞, that is TqL −→ T,q→∞,where T ∼BS(2α,β). (14) Proof. See Appendix A, Proof of Proposition 3. Proposition 3means that, for large q , GMSBS(α , β , q) model can be approached by a Birnbaum– Saunders distribution. 2.3. Moments Formulae for the moments of order r,r∈Z+, in a GMSBS distribution are given next. Proposition 4. Let T ∼GMSBS(α,β,q). For r ∈Z+, E[Tr]exists if and only if q >2r and E[Tr] = βrr ∑ y=02r 2yy ∑ s=0y sα 22(r+s−y)2(r+s−y)q2(r+s−y)[2(r+s−y)]!Γ(q−2(r+s−y)) (r+s−y)!Γ(q). (15) Proof. See Appendix A, Proof of Proposition 4. Remark 1. From Equation (15), note that E[Tr] is a polynomial in β of degree r , in α of degree 2 r (only even powers are obtained), and coefficients that involve rational functions of q (with numerator and denominator of the same degree). Next, some non-central moments for the GMSBS distribution are given. These expressions involve the Pochhammer symbol or rising factorial, defined for a>0 and k∈Z+as (a)k=a(a+1)(a+2). . . (a+k−1) = Γ(a+k) Γ(a). (16) Symmetry 2018,10, 724 6 of 18 Corollary 3. Let T ∼GMSBS(α,β,q)and µr=E[Tr]. Then µ1=βh2α2q2 (q−2)2+1i,q>2(mean or expected value of T), µ2=β2h24α4q4 (q−4)4+8α2q2 (q−2)2+1i,q>4, µ3=β3h480α6q6 (q−6)6+144α4q4 (q−4)4+18α2q2 (q−2)2+1i,q>6, µ4=β4h13440α8q8 (q−8)8+3840α6q6 (q−6)6+480α4q4 (q−4)4+32α2q2 (q−2)2+1i,q>8. Proof. The proposed results follow from Proposition 4and Equation (16). Aditional details can be seen in Appendix A, Proof of Corollary 3. From Corollary 3, it follows that the variance of Tis Var(T) = 4β2hα2c2(q) + α4c4(q)i,q>4 (17) where c2(q) = q2 (q−2)2 c4(q) = 6q4 (q−4)4−q4 {(q−2)2}2. The skewness coefficient, pβ1 , and the kurtosis coefficient, β2 , can be computed by using the previous expressions and the relationships pβ1=µ3−3µ1µ2+2µ3 1 (µ2−µ2 1)3/2 . β2=µ4−4µ1µ3+6µ2 1µ2−3µ4 1 (µ2−µ2 1)2. Next, the behavior of pβ1and β2as functions of the kurtosis parameter qis studied. Although the convergence in law, in general, does not imply the convergence in moments, in this case, we have such convergence as q→∞ . The notation β1(q) and β2(q) is used. The next corollary states explicit results for pβ1(q) and β2(q) , if q→∞ , along with others that help us to understand the behavior of these features. The explicit expressions of pβ1(q) and β2(q) , given in Appendix A, Equation (A12), are used. Corollary 4. (1) Limit behavior of skewness coefficient lim q→6qβ1(q) = ∞ lim q→∞qβ1(q) = 48α+352α4 {4(1+5α2)}3/2 that is, if q→∞ then the skewness coefficient of a GMSBS(α , β , q) tends to the skewness coefficient of a BS(2α,β)distribution. (2) Limit behavior of kurtosis coefficient Symmetry 2018,10, 724 7 of 18 lim q→8β2(q) = ∞ lim q→∞β2(q) = 3+24α2+ (372α2+41) (20α2+4)2. that is, if q→∞ then the kurtosis coefficient of a GMSBS(α , β , q) tends to the kurtosis coefficient of a BS(2α,β)distribution. Proof. The proposed results follow from expressions for β1(q) and β2(q) given in Appendix A, Equation (A12), and the moments µr, given in Equation (15). Remark 2. Interpretation of parameters in a GMSBS(α,β,q)model. (i) In the GMSBS model, as in the Birnbaum–Saunders distribution, β> 0is a scale parameter, which is also the median of the distribution (see Equation (6) and Proposition 2). (ii) It can be seen in Leiva [ 5 ] that in the Birnbaum–Saunders distribution α> 0is a shape parameter that modifies the skewness and kurtosis of the distribution. As α tends to zero, the BS distribution tends to be more symmetrical around its median β and its variability decreases. The expressions of the skewness coefficient pβ1 , given in Equations (A12) and (17), suggest that αhas a similar interpretation in the GMSBS model. (iii) As for the parameter q> 0, it is proven through this paper that controls the kurtosis and skewness coefficient in the GMSBS model, in such a way that allows us to obtain models with greater level of kurtosis than other slash BS distributions, previously introduced in the literature. As graphical aid, to show the way in which α and q determine the asymmetry and kurtosis of a GMSBS(α , β , q) model, see plots in Figure 2. Without loss of generality, the scale parameter is taken equal to one, β= 1. They illustrate the way in which the asymmetry and kurtosis coefficients depend on both parameters. Plots in Figure 2suggest that, on the one hand, for increasing values of α , the asymmetry and kurtosis increase. On the other hand, if α is fixed, asymmetry and kurtosis coefficients are decreasing functions of q. 6 8 10 12 14 16 18 20 0 10 20 30 40 50 q β1 α = 0.25 α = 1 α = 5 8 10 12 14 16 18 20 0 200 400 600 800 1000 q β2 α = 0.25 α = 1 α = 5 Figure 2. Skewness and kurtosis coefficients for GMSBS(α , β= 1, q) model as function of q taking α=0.25, 1 and 5. These considerations motivate that GMSBS distribution can be used for modeling more kurtosis than other slash Birnbaum–Saunders distributions previously introduced in the literature such as SBS and MSBS densities. Figure 3displays the GMSBS pdf plot along with MSBS and SBS densities. Note that the right tail of the GMSBS distribution is heavier than the tails of the other ones. Symmetry 2018,10, 724 8 of 18 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 t density function GMSBS(α=0.3, β=0.8, q=3) MSBS(α=0.3, β=0.8, q=3) SBS(α=0.3, β=0.8, q=3) Figure 3. Comparison of right tails of densities for GMSBS, MSBS and SBS models for the same value for parameters α,βand q. 3. Estimation Let T1 , . . . , Tn be a simple random sample (srs) from T∼GMSBS(α , β , q) , n> 3. In this section, we face the problem of estimating (α , β , q) . Next, we propose a couple of techniques to tackle this problem. 3.1. Modified Moment Estimation Following Ng et al. [ 12 ], a modified method moment based on Property (3) given in Proposition 2 is next introduced. Thus, we propose to equal E[T],E[T2], and E[1/T]to their corresponding sample moments, that is E[T] = β1+2α2q2 (q−2)2=¯ T(18) E[T2] = β21+8α2q2 (q−2)2 +24α2q4 (q−4)4=m2(19) E1 T=1 β1+2α2q2 (q−2)2=R(20) where ¯ T=∑n i=1Ti n,m2=∑n i=1T2 i n, and R=∑n i=11 Ti n. Note that R=H−1with H=n ∑n i=11 Ti the sample harmonic mean. The solutions of previous equations for q> 4 and α> 0 are called the modified moment (MM) estimators, denoted as bαMM,b βMM, and b qMM. 3.2. Maximum Likelihood Estimation Given a srs T1 , T2 , ..., Tn from a GMSBS(α , β , q) distribution and t1 , t2 , ..., tn their observations, by applying Equation (9), the log-likelihood function is l(α,β,q) = n ∑ i=1 log fT(ti;α,β,q) =n(q−1)log2+nq log q−3 2 n ∑ i=1 log ti+ n ∑ i=1 log(ti+β)−nlog Γ(q)−nlog α−n 2log β + n ∑ i=1 log G(wi)with G(wi) = R∞ 0vqe−2qvφ(wiv)dv and wi=1 αqti β−qβ ti. Symmetry 2018,10, 724 9 of 18 To maximize l(α , β , q) in (α , β , q) , consider the first derivatives of l(α , β , q) with respect to α , β and q , denoted as ˙ lα , ˙ lβ and ˙ lq , respectively. From ˙ lα= 0, ˙ lβ= 0 and ˙ lq= 0, we obtain the likelihood equations, whose expressions are given in Appendix B, and can be solved by using iterative Newton–Raphson methods. Let us denote by d(ti) = R∞ 0vq+2e−2qvφ(wiv)dv G(wi) . Then, the following iterative process can be proposed for k≥0 bα(k+1)=1 n(n ∑ i=1 ti b β(k)+b β(k) ti−2!d(k)(ti))1/2 (21) b β(k+1)=         1 2(bα(k))2 n ∑ i=1 tid(k)(ti) n 2b β(k)− n ∑ i=1 1 ti+b β(k)+1 2(bα(k))2 n ∑ i=1 1 ti d(k)(ti)          1/2 (22) b q(k+1)=exp (ψ(q(k))−1−ln 2 −1 n n ∑ i=1 n ∑ i=1 G(k) 3(wi) G(k)(wi))(23) which needs starting values bα(0) , b β(0) and b q(0) to start the recursion. As initial values, the modified moment estimators, previously proposed, can be considered. Remark 3. (1) In Equations (21)–(23), d(k)(·) , G(k)(·) and G(k) 3(·) denote these expressions evaluated at bα(k) , b β(k)and b q(k). The expression of G3(·)can be seen in Appendix B. (2) It can be seen in Leiva [ 5 ] p. 41 that in the Birnbaum–Saunders model, BS(α , β) , the iterative equations for the MLEs of bαand b βare bα(k+1)=1 n(n ∑ i=1 ti b β(k)+b β(k) ti−2!)1/2 (24) b β(k+1)=         1 2(bα(k))2 n ∑ i=1 ti n 2b β(k)− n ∑ i=1 1 ti+b β(k)+1 2(bα(k))2 n ∑ i=1 1 ti          1/2 . (25) The effect of introducing the generalized modified slash variable on the BS(α , β) model can be appreciated by comparing Equations (21) and (22) to Equations (24) and (25). 3.3. ML Estimation Using EM-Algorithm Taking advantage of the stochastic representation of the GMSBS model, we can develop a more attractive iterative method to find the MLEs based on the EM algorithm (Dempster et al. [ 13 ]). This is a well-known tool when unobserved (missing) data or latent variables are present while modeling. This algorithm enables the computationally efficient determination of the ML estimates when iterative procedures are required. Looking at the stochastic representation of a generalized modified slash distribution given in Equation (6), we note that the scale factor V depends on the parameter q , thus we consider a re-parameterization to get the EM-algorithm in the GMSBS model. Then, the resulting stochastic representation for Tcan be expressed as T=βα 2X+rα 2X2+12, (26) Symmetry 2018,10, 724 16 of 18 where X∼GMS( 0, 1, q) is such that E[X2j] = 2jq2j(2j)!Γ(q−2j) Γ(q) for q> 2 j , as can be seen in Reyes et al. [ 9 ]. Taking j=r+s−y, the result proposed in Equation (15) is obtained. Proof of Corollary 3.From Proposition 4, it is straightforward that µ1=βh2α2q2Γ(q−2) Γ(q)+1i,q>2 µ2=β2h24α4q4Γ(q−4) Γ(q)+8α2q2Γ(q−2) Γ(q)+1i,q>4 µ3=β3h480α6q6Γ(q−6) Γ(q)+144α4q4Γ(q−4) Γ(q)+18α2q2Γ(q−2) Γ(q)+1i,q>6 µ4=β4h13440α8q8Γ(q−8) Γ(q)+3840α6q6Γ(q−6) Γ(q)+480α4q4Γ(q−4) Γ(q)+32α2+q2Γ(q−2) Γ(q)+1i,q>8, and, thus, the proposed results follow by using the notation introduced in Equation (16). Corollary A1 (Central moments).Let T ∼GMSBS(α,β,q). Then, (1) The variance of T, Var(T), was given in Equation (17). (2) The central moment of order 3 is E[(T−µ1)3] = β3hα4d4(q) + α6d6(q)i,q>6, (A4) where d4(q) = 72 q4 (q−4)4−24 q4 {(q−2)2}2(A5) d6(q) = 480 q6 (q−6)6−144 q6 (q−4)4(q−2)2 +16 q6 {(q−2)2}3. (A6) (3) The central moment of order 4 is E[(T−µ1)4] = β8hα4f4(q) + α6f6(q) + α8f8(q)i,q>8, (A7) where f4(q) = 48 q4 (q−4)4 (A8) f6(q) = 1920 q6 (q−6)6−576 q6 (q−4)4(q−2)2 +96 q6 {(q−2)2}3(A9) f8(q) = 13440 q8 (q−8)8−3840 q8 (q−6)6(q−2)2 +576 q8 (q−4)4{(q−2)2}2(A10) −48 q8 {(q−2)2}4. (A11) Proof. They are obtained by considering the results given in Corollary 3and the following relationships Var(T) = µ2−µ2 1 E[(T−µ1)3] = µ3−3µ1µ2+2µ3 1 E[(T−µ1)4] = µ4−4µ1µ3+6µ2 1µ2−3µ4 1 Proposition A1 (Skewness and kurtosis coefficient) . For T∼GMSBS(α , β , q) distribution the skewness, pβ1, and kurtosis, β2, coefficients can be calculated as Symmetry 2018,10, 724 17 of 18 pβ1=µ3−3µ1µ2+2µ3 1 (µ2−µ2 1)3/2 β2=µ4−4µ1µ3+6µ2 1µ2−3µ4 1 (µ2−µ2 1)2 From previous expressions, we have that pβ1=α 8 d4(q) + α2d6(q) {c2(q) + α2c4(q)}3/2 (A12) β2=1 16 f4(q) + α2f6(q) + α4f8(q) {c2(q) + α2c4(q)}2(A13) Appendix B. Likelihood Equations The likelihood equations are n ∑ i=1 G1(wi) G(wi)=n α(A14) n ∑ i=1 G2(wi) G(wi)=n 2β− n ∑ i=1 1 ti+β(A15) n ∑ i=1 G3(wi) G(wi)=−n(1+log 2 +log q) + nψ(q)(A16) where G1(wi) = ∂G(wi) ∂α =1 α3ti β+β ti−2Z∞ 0vq+2e−2qvφ(wiv)dv G2(wi) = ∂G(wi) ∂β =1 2α2β2 1 ti (t2 i−β2)Z∞ 0vq+2e−2qvφ(wiv)dv G3(wi) = ∂G(wi) ∂q=Z∞ 0(ln v−2v)vqe−2qvφ(wiv)dv and ψ(q) = Γ0(q)/Γ(q)is the digamma function. References 1. Birnbaum, Z.W.; Saunders, S.C. A New Family of Life Distributions. J. Appl. Probab. 1969 ,6, 319–327. [CrossRef] 2. Birnbaum, Z.W.; Saunders, S.C. 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