Slashed Exponentiated Rayleigh Distribution

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1 Revista Colombiana de Estadística July 15, Volume 38, Issue, pp. 453 to 466 DOI: Slashed Exponentiated Rayleigh Distribution Distribución Slash Rayleigh exponenciada Hugo S. Salinas 1,a, Yuri A. Iriarte,b, Heleno Bolfarine 3,c 1 Departamento de Matemática, Facultad de Ingeniería, Universidad de Atacama, Copiapó, Chile Instituto Tecnológico, Universidad de Atacama, Copiapó, Chile 3 Departamento de Estatística, IME, Universidad de Sao Paulo, Sao Paulo, Brasil Abstract In this paper we introduce a new distribution for modeling positive data with high kurtosis. This distribution can be seen as an extension of the exponentiated Rayleigh distribution. This extension builds on the quotient of two independent random variables, one exponentiated Rayleigh in the numerator and Beta(q, 1) in the denominator with q >. It is called the slashed exponentiated Rayleigh random variable. There is evidence that the distribution of this new variable can be more flexible in terms of modeling the kurtosis regarding the exponentiated Rayleigh distribution. The properties of this distribution are studied and the parameter estimates are calculated using the maximum likelihood method. An application with real data reveals good performance of this new distribution. Key words: Exponentiated Rayleigh Distribution, Kurtosis, Maximum Likelihood, Rayleigh Distribution, Slash Distribution. Resumen En este trabajo presentamos una nueva distribución para modelizar datos positivos con alta curtosis. Esta distribución puede ser vista como una extensión de la distribución Rayleigh exponenciada. Esta extensión se construye en base al cuociente de dos variables aleatorias independientes, una Raileigh exponenciada en el numerador y una Beta(q, 1) en el denominador con q >. La llamaremos variable aleatoria slash Rayleigh exponenciada. Hay evidencias que la distribución de esta nueva variable puede ser más flexible en términos de modelizar la curtosis respecto a la distribución Rayleigh exponenciada. Se estudian las propiedades de esta distribución y se calculan a Professor. hugo.salinas@uda.cl b Professor. yuri.iriarte@uda.cl c Professor. hbolfar@ime.usp.br 453

2 454 Hugo S. Salinas, Yuri A. Iriarte & Heleno Bolfarine las estimaciones de los parámetros utilizando el método de máxima verosimilitud. Una aplicación con datos reales revela el buen rendimiento de esta nueva distribución. Palabras clave: curtosis, distribución Rayleigh, distribución Rayleigh exponenciada, distribución Slash, máxima verosimilitud. 1. Introduction An up to date and detailed review of the exponentiated Weibull (EW) distribution is presented in Nadarajah, Cordeiro & Ortega (13) (see also, Cancho, Bolfarine & Achcar 1999). A random variable Z is said to follow the EW distribution if its cumulative distribution function and probability density function are given, respectively, by and F Z (z; α, β, θ) = (1 e (βz)θ ) α f Z (z; α, β, θ) = θαβ θ z θ 1 e (βz)θ (1 e (βz)θ ) α 1 for z >, α >, β > and θ >. The particular case θ = is the Burr type X distribution studied by various authors. In this case, one obtains by scale transformation the distribution of a random variable X that follows an exponentiated Rayleigh (ER) distribution if its density function is given by and F X (x; α, λ) = (1 e λx ) α f X (x; α, λ) = αλxe λx (1 e λx ) α 1 (1) respectively, for x >, λ > and α >. We write X ER(α, λ). In Gómez, Quintana & Torres (7), of slash elliptical distributions was introduced. This class of distributions can be seen as an extension of the class of elliptical distributions studied in Fang, Kotz & Ng (199). Genc (7) derived the univariate slash by a scale mixture of the exponential power distribution and investigated asymptotically the bias of the estimators. Wang & Genton (6) proposed the multivariate skew version of this distribution and examined its properties and inferences. A random variable Y follows a slash elliptical distribution with location parameter µ and scale parameter σ, denoted by Y SEl(t; µ, σ, g), if it can be represented as Y = σ X + µ, U 1/q where X El(, 1, g) and U U(, 1) are independent and q >. If Y SEl(, 1, q), then the density function of Y is given by f Y (y;, 1, q) = { q y y v q 1 q+1 g(v)dv, if y, q 1+q g(), if y =. () Revista Colombiana de Estadística 38 (15)

3 Slashed Exponentiated Rayleigh 455 In the canonic case, namely q = 1, () reduces to f Y (y;, 1, 1) = { G(y ) 1 y, if y, g(), if y =, where G(x) = x g(v) dv. The canonic case with g = φ where φ density function for the standard normal distribution, was studied by Rogers & Tukey (197) and by Mosteller & Tukey (1977). Arslan (8) discussed asymmetric versions of the slash elliptical family of distributions. Gómez, Olivares-Pacheco & Bolfarine (9) considered the slash elliptical family of distributions to extend the Birnbaum-Saunders family of distributions (Leiva, Soto, Cabrera & Cabrera 11). Olmos, Varela, Gómez & Bolfarine (1) made use of the slash elliptical family of distributions to extend the half-normal distribution. Olivares-Pacheco, Cornide-Reyes & Monasterio (1) used the slash elliptical family to extend the Weibull distribution. Iriarte, Gómez, Varela & Bolfarine (15) use the family of slash elliptical distributions to extend the Rayleigh distribution. The rest of this paper is organized as follows. In Section, we propose the new slash distribution and investigate its properties, including a stochastic representation. Section 3 discusses inference for model parameters, including moments and maximum likelihood estimation (MLE) for the parameters. Simulation studies are performed in Section 4 revealing good performance of the MLE. Section 5 gives a real illustrative application and reports the results indicating good performance in applied scenarios. Section 6 concludes our work.. Slashed Exponentiated Rayleigh Distribution.1. Stochastic Representation Definition 1. A random variable T has slashed exponentiated Rayleigh distribution if it can be represented as the ratio T = X W, (3) where X ER(α, λ) defined in (1) and W Beta(q, 1) are independent, α >, λ >, q >. We denote it as T SER(α, λ, q) Proposition 1. Let T SER(α, λ, q). Then, the density function of T is given by f T (t; α, λ, q) = αq λ q/ t (q+1) H(λt ; α, q) t, where α >, λ >, q > and H(x; α, q) is defined as H(x; α, q) = x u q/ e u ( 1 e u) α 1 du Revista Colombiana de Estadística 38 (15)

4 456 Hugo S. Salinas, Yuri A. Iriarte & Heleno Bolfarine Proof. Using the representation given in (3) and the Jacobian method, the density function associated with T is given by f T (t; α, λ, q) = qαλ 1 tw q+1 e λt w ( 1 e λt w ) α 1 dw and considering the change of variables u = λt w the result follows. Note 1. Particularly, if α = λ = q = 1, one obtains the exponentiated canonic slashed Rayleigh distribution, and is denoted as T SER(1, 1, 1). Then, the density function of the random variable T is given by π f T (t) = t G(t, 3/, 1), t. where G(x, α, β) = x β α Γ(α) uα 1 e βu du is the cumulative distribution function of the gamma distribution. Figure 1 shows some density functions of the slashed exponentiated Rayleigh distribution with various parameters. Density ER(3,1) SER(3,1,1) SER(3,1,) SER(3,1,3) SER(3,1,1) Density ER(1.5,.) SER(1.5,.,1) SER(1.5,.,) SER(1.5,.,3) SER(1.5,.,1) t Figure 1: Slashed exponentiated Rayleigh density for different values of its parameters t.. Properties In this subsection we deliver some basic properties of the slashed exponentiated Rayleigh distribution. Let T SER(α, λ, q), then 1. lim (α,q) (1, ) f T (t; α, 1/(σ ), q) = t t σ e σ ( ) (. lim f T (t; α, 1/(σ), q) = q(σ)q/ q + t α 1 t q+1 Γ G σ, q + ), 1 Revista Colombiana de Estadística 38 (15)

5 Slashed Exponentiated Rayleigh lim q f T (t; α, λ, q) = αλte λt ( 1 e λt) α 1 4. F T (k; α, λ, q) = P (T < k) = (1 e λk ) α k q f T (k; α, λ, q) where Γ(α) = u α 1 e u du is the gamma function. Note. Property 1 shows that as (α, q) (1, ) and λ = 1/(σ ), the slashed exponentiated Rayleigh converges to the ordinary Rayleigh distribution, i.e. T R(σ ) (see Johnson, Kotz & Balakrishnan 1994). Property shows that as α 1 and λ = 1/(σ) the slashed exponentiated Rayleigh distribution converges to the slashed Rayleigh distribution (Iriarte et al. 15). Property 3 shows that as q, the slashed exponentiated Rayleigh converges to the exponentiated Rayleigh distribution..3. Moments Proposition. Let T SER(α, λ, q), then the r th moments are given by µ r = E(T r ) = αq H(α, r), q > r, (4) λ r/ (q r) for r = 1,,... and H(α, r) := H( ; α, r) = u r/ e u (1 e u ) α 1 du. Proof. Using the stochastic representation given in (3), we have that ( ( µ r = E X ) r ) W = E(X r W r ) = E(X r )E(W r ) from where it follows that E (X r ) = u r/ e u (1 e u ) α 1 du are the λ r/ moments of the ER(α, λ) distribution and E(W r ) = q/(q r) with q > r. Corollary 1. Let T SER(α, λ, q), so that and E(T ) = V ar(t ) = αq λ α αh(α, 1) λ q (q 1), q > 1 ( ) H(α, ) q αqh (α, 1) (q 1), q > Corollary. Let T SER(α, λ, q), then the coefficients of asymmetry ( β 1 ) and kurtosis (β ) for q > 3 and q > 4 are, respectively, given by β1 = H(α,3) q 3 3αH(α,1)H(α,)q (q 1)(q ) + α H 3 (α,1)q (q 1) 3 ( ) 3/ αq H(α,) q αqh (α,1) (q 1) β = H(α,4) q 4 4αH(α,1)H(α,3)q (q 1)(q 3) + 6α H (α,1)h(α,)q (q 1) (q ) 3α3 H 4 (α,1)q 3 (q 1) ( ) 4 αq H(α,) q αqh (α,1) (q 1) Revista Colombiana de Estadística 38 (15)

6 458 Hugo S. Salinas, Yuri A. Iriarte & Heleno Bolfarine Note 3. Notice that as (α, q) (1, ), the asymmetry and kurtosis coefficients converge to (π 3) 4π (4 π) y 3 3π (4 π) respectively, which correspond to the corresponding coefficients for the Rayleigh distribution. As q, the asymmetry and kurtosis coefficients converge to γ 1 = H(α, 3) 3αH(α, 1)H(α, ) + α H 3 (α, 1) α (H(α, ) αh (α, 1)) 3/ and γ = H(α, 4) 4αH(α, 1)H(α, 3) + 6α H (α, 1)H(α, ) 3α 3 H 4 (α, 1) α(h(α, ) αh (α, 1)) the corresponding asymmetry and kurtosis coefficients for the exponentiated Rayleigh distribution, respectively. Figures and 3 depict graphs for the coefficients of asymmetry and kurtosis coefficients for variable SER for different values of the parameter q. Skewness ER q = 1 q = 6 q = 5 q = 4 Skewness ER q = 1 q = 6 q = 5 q = α Figure : Asymmetry coefficient SER density for different values of q. The solid line corresponds to the asymmetry coefficient of ER density. α Note 4. The function H : R + R + R can be approximated using power series about the point (α, r ) as follows: H(α, r) = n= 1 n! n 1+n =n n! n H(α, r ) (α α ) n1 (r r ) n, n 1! n! α n1 r n where n H(α,r ) α n 1 r n must be solved numerically. Revista Colombiana de Estadística 38 (15)

7 Slashed Exponentiated Rayleigh 459 Kurtosis ER q = 1 q = 6 q = 5 q = 4 Kurtosis ER q = 1 q = 6 q = 5 q = α Figure 3: Kurtosis coefficient SER density for different values of q. The solid line corresponds to the kurtosis coefficient of ER density. α 3. Inference In this section, we study moment and maximum likelihood estimators for parameters α, λ and q for the exponentiated slashed Rayleigh distribution Moments Estimators (ME) Inference The following proposition presents moment estimators for parameters α, λ and q. Proposition 3. Let T 1,..., T n a random sample from the distribution of the random variable T SER(α, λ, q). Then, the moment estimators ( α, λ, q) for (α, λ, q) with q > 3 can be calculated numerically from the following expressions ( ) α q λ = H ( α, 1) T ( q 1) (5) q = 3( d 1 d ) ± d 1 1 d1 d + 9 d ( d 1 d ) (6) where d 1 = α T H ( α,1), d T = α T 3 H 3 ( α,1) and T k is the k-th power sample mean, H( α,) T 3 H( α,3) k = 1,, 3. The terms d 1 and d depend only on the sample and the estimator α. Proof. Using (4) and replacing E(T ), E(T ) and E(T 3 ) with T, T and T 3, respectively, it follows that Revista Colombiana de Estadística 38 (15)

8 46 Hugo S. Salinas, Yuri A. Iriarte & Heleno Bolfarine T = T = T 3 = αh( α, 1) λ 1/ αh( α, ) λ αh( α, 3) λ 3/ q q 1, (7) q q, (8) q, q 3 q > 3, (9) From equation (7) is easy to obtain the expression for λ in (5). Then, by replacing λ in equations (8) and (9) we obtain ( q 1) q( q ) = d 1, (1) ( q 1) 3 q ( q 3) = d, (11) After an algebraic manipulation we get the equation ( d 1 d ) q 3( d 1 d ) q + d 1 = whose solutions are given in (6) such that d 1 1 d1 d + 9 d >. 3.. Maximum Likelihood (ML) Inference In this section, we consider the maximum likelihood estimation for parameters θ = (α, λ, q) of the SER model. Suppose t 1, t,..., t n is a random sample of size n from slashed exponentiated Rayleigh distribution. Then the log-likelihood function is given by log L(θ) = c(θ) (q + 1) n log(t i ) + n log(h(t i )), (1) where H(t i ) := H ( λt i ; α, q) and c(θ) = n log(α) + n log(q) nq log(λ). Maximum likelihood estimators are obtained by maximizing the likelihood function, which can be obtained by differentiating the log-likelihood function and solving the corresponding (score) equations. The likelihood equations are given by n n α + nq λ + n n q n n log(λ) log(t i ) + n H 1 (t i ) H(t i ) =, H (t i ) H(t i ) =, H 3 (t i ) H(t i ) =, Revista Colombiana de Estadística 38 (15)

9 Slashed Exponentiated Rayleigh 461 where H 1 (t i ) := λt α H(t i i) = u q/ e u (1 e u ) α 1 log(1 e u )du, H (t i ) := λ H(t i) = λ q/ t q+ H 3 (t i ) := q H(t i) = 1 λt i i e λt i (1 e λt i ) α 1, u q/ log(u) e u (1 e u ) α 1 du. Therefore, numerical algorithms are required for solving the score equations. One possibility is to employ the subroutine optim with the R Core Team (14). It is well known that as the sample size increases, the distribution of the MLE tends (under regularity conditions) to the normal distribution with mean (α, λ, q) and covariance matrix equal to the inverse of the Fisher (expected) information matrix. Due to the complexity of the likelihood function it is not possible to obtain its analytical expression. It is possible, however, to work with the observed information matrix, which is a consistent estimator for the expected information matrix. The observed information matrix follows from the Hessian matrix by replacing unknown parameters by their MLEs. Some algebraic manipulation yield the following Hessian matrix: I n(θ) = log L(θ) α log L(θ) λ α log L(θ) λ log L(θ) q α log L(θ) q λ log L(θ) q, such that log L(θ) α log L(θ) λ α log L(θ) q α log L(θ) α λ log L(θ) λ log L(θ) q λ log L(θ) α q = n α + n = = n n H 1 (t i ) H(t i ) n H 11 (t i ) n H(t i ) ( ) H1 (t i ), H(t i ) H 13 (t i ) H(t i ) n = log L(θ) λ α, = nq λ + n = n λ + n = log L(θ), q α H (t i ) H(t i ) n H 1 (t i )H (t i ) H, (t i ) H 1 (t i )H 3 (t i ) H, (t i ) H 3 (t i ) H(t i ) n ( ) H (t i ) H(t i ) H (t i )H 3 (t i ) H (t i ) Revista Colombiana de Estadística 38 (15)

10 46 Hugo S. Salinas, Yuri A. Iriarte & Heleno Bolfarine where log L(θ) λ q log L(θ) q = log L(θ), q λ = n n q + H 33 (t i ) H(t i ) n ( ) H3 (t i ), H(t i ) H 11 (t i ) := λt α H i 1(t i ) = u q/ e u (1 e u ) α 1 (log(1 e u )) du, H 1 (t i ) := e λt i (1 e λt i ) α 1 log(1 e u )du, λ H 1(t i ) = λ q/ t q+ i H 13 (t i ) := q H 1(t i ) = 1 H (t i ) := λ H (t i ) λt i u q/ log(u)e u (1 e u ) α 1 log(1 e u )du, = λq/ t q+ i e λt i (1 e λt i ) α 1 (q qe λt i λt i + αλt i e λt i ), λ(1 e λt i ) H 3 (t i ) := q H (t i ) = 1 λq/ t q+ i e λt i (1 e λt i ) α 1 (log(λ) + log(t i )), H 33 (t i ) := q H 3(t i ) = Simulation study λt i u q/ (log(u)) e u (1 e u ) α 1 du. In this section, we conduct a small scale simulation study illustrating the MLEs behavior for parameters α, λ and q in small and moderate sample sizes. One thousand random samples of sizes n =1, and 5 were generated from model SER(θ) for fixed parameters values. To generate T SER(θ) the following algorithm was used: 1. Generate U U(, 1). Compute X = log(1 U 1/α ) λ 3. Generate W Beta(q, 1) 4. Compute T = XW 1 MLEs can be obtained as described above using R Core Team (14). Empirical means and standard deviations are reported in Table 1 indicating good performances. Revista Colombiana de Estadística 38 (15)

11 Slashed Exponentiated Rayleigh 463 Table 1: Empirical means and SD s for the MLE s of α, λ and q. n = 1 α λ q α (SD) λ (SD) q (SD) (.314) 1.89 (.563) 1.41 (.155) (.6) 1.83 (.41) (.39) (.6) 1.54 (.361).4 (1.84) (1.178) 3.17 (1.154) (.49)..3 (.975) 3.17 (1.15).16 (.4).5.18 (.65) (.95).648 (.748) (1.173) (1.383).6 (.34) (3.874) 4.43 (1.).61 (.54) (3.93) 4.53 (1.14) (.94) n = α λ q α (SD) λ (SD) q (SD) (.174) 1.5 (.313) 1.4 (.11) (.15) 1.36 (.7) (.193). 1.9 (.14) 1.36 (.4).63 (.331) (.434) 3.11 (.686) (.161)..78 (.47) 3.85 (.663).51 (.88).5.7 (.361) 3.71 (.598).559 (.388) (1.844) 4.15 (.798).3 (.) (1.477) 4.94 (.699).551 (.31) (1.387) 4.45 (.638) 3.98 (.41) n = 5 α λ q α (SD) λ (SD) q (SD) (.115).968 (.197) 1.18 (.7) (.91) 1.8 (.156) (.118). 1.1 (.84) 1.1 (.147).7 (.188) (.48) 3.14 (.39) 1.58 (.1)..3 (.3) 3.4 (.37).15 (.15).5.3 (.14) 3.34 (.367).51 (.31) (1.98) (.793).31 (.1) (.8) 4.5 (.43).514 (.18) (.84) 4.38 (.438) 3.9 (.54) 5. Real Data Illustration We consider a data set to the life of fatigue fracture of Kevlar 49/epoxy which is subject to constant pressure at the 9% stress level until all failed, so we have complete data with the exact times of failure. For previous studies with this data set; see, Andrews & Herzberg (1985) and Barlow, Toland & Freeman (1984). Using results in Section 3.1, the following moment estimators were computed: α M =.57, λ M =.688 and q M = 6.587, which were used as starting values for MLEs. Table presents descriptive statistics where b 1 are b are the asymmetry and kurtosis coefficients, respectively. We note that the data set presents high positive asymmetry and also high kurtosis. Table 3 shows maximum likelihood estimators for the parameters of the exponentiated Rayleigh and slashed exponentiated Rayleigh distributions. The usual Akaike information criterion (AIC) introduced by Akaike (1974) and Bayesian information criterion (BIC) proposed Revista Colombiana de Estadística 38 (15)

12 464 Hugo S. Salinas, Yuri A. Iriarte & Heleno Bolfarine by Schwarz (1978) to measure the goodness of fit are also computed. It is known that AIC=k loglik and BIC=k log n loglik where k is the number of parameters in the model, n is the sample size and loglik is the maximized value of the likelihood function. For the ER model, AIC= () ( ) = and BIC= log(11) ( ) = Similarly, for the SER model, AIC= (3) ( 1.594) = and BIC= 3 log(11) ( 1.594) = In both cases, the SER model has the lowest values of AIC and BIC. Thus, the results show that the SER model fits better the data set. Figure 4 displays the fitted models using the MLEs. Table : Summary for stress-rupture data set. sample size mean variance b1 b Table 3: Maximum likelihood parameter estimates (with (SD)) of the ER and SER models for stress-rupture data set. Model α λ q loglik AIC BIC ER.31(.35).174(.33) SER.38(.47).686(.3).759(.88) Stress rupture Figure 4: Histogram and models fitted for stress-rupture data set: SER (solid line) and ER (dashed line). Hours 6. Concluding Remarks In this paper the slashed exponentiated Rayleigh distribution (SER) is studied. A random variable SER is the quotient between two independent random variables, Revista Colombiana de Estadística 38 (15)

13 Slashed Exponentiated Rayleigh 465 an exponentiated Rayleigh and the Beta(q, 1) random variable with q >. This proposal generalizes the exponentiated Rayleigh family and the Rayleigh family, among others. This generalization can be used for modeling positive data with high kurtosis. Moments and maximum likelihood estimation is discussed. A real data illustration revealed that the proposed model can be very useful in practical scenarios. [ Received: March 14 Accepted: January 15 ] References Akaike, H. A. (1974), A new look at statistical model identification, IEEE Transaction on Automatic Control 19(6), Andrews, D. F. & Herzberg, A. M. (1985), Data: a collection of problems from many fields for the student and research worker, Springer, New York. Arslan, O. (8), An alternative multivariate skew-slash distribution, Statistics & Probability Letters 78(16), Barlow, R. E., Toland, R. H. & Freeman, T. (1984), A Bayesian analysis of stressrupture life of Kevlar 49/epoxy spherical pressure vessels, in Proceedings of the Canadian conference in application statistics, Marcel Dekker, New York. Cancho, V. G., Bolfarine, H. & Achcar, J. A. (1999), A Bayesian analysis of the exponentiated-weibull distribution, Journal of Applied Statistical Science 8, 7 4. Fang, K. T., Kotz, S. & Ng, K. W. (199), Symmetric Multivariate and Related Distributions, Chapman and Hall, London-New York. Genc, A. (7), A generalization of the univariate slash by a scale-mixtured exponential power distribution, Communications in Statistics-Simulation and Computation 36, Gómez, H. W., Olivares-Pacheco, J. F. & Bolfarine, H. (9), An extension of the generalized Birnbaum-Saunders distribution, Statistics & Probability Letters 79(3), Gómez, H. W., Quintana, F. A. & Torres, F. J. (7), A new family of slash-distributions with elliptical contours, Statistics & Probability Letters 77(7), Iriarte, Y. A., Gómez, H. W., Varela, H. & Bolfarine, H. (15), Slashed Rayleigh distribution, Revista Colombiana de Estadística 38(1), Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994), Continuous Univariate Distributions, Wiley, New York. Revista Colombiana de Estadística 38 (15)

14 466 Hugo S. Salinas, Yuri A. Iriarte & Heleno Bolfarine Leiva, V., Soto, G., Cabrera, E. & Cabrera, G. (11), Nuevas cartas de control basadas en la distribución Birnbaum-Saunders y su implementación, Revista Colombiana de Estadística 34(1), Mosteller, F. & Tukey, J. W. (1977), Data Analysis and Regression: A Second Course in Statistics, Addison-Wesley Series in Behavioral Science: Quantitative Methods. Reading, Massachusetts. Nadarajah, S., Cordeiro, G. M. & Ortega, E. M. M. (13), The exponentiated Weibull distribution: A survey, Statistical Papers 54(3), Olivares-Pacheco, J. F., Cornide-Reyes, H. & Monasterio, M. (1), Una extensión de la distribución Weibull de dos parámetros, Revista Colombiana de Estadística 33(), Olmos, N. M., Varela, H., Gómez, H. W. & Bolfarine, H. (1), An extension of the half-normal distribution, Statistical Papers 53(4), R Core Team (14), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN * Rogers, W. H. & Tukey, J. W. (197), Understanding some long-tailed symmetrical distributions, Statistica Neerlandica 6, Schwarz, G. (1978), Estimating the dimension of a model, Annals of Statistics 6, Wang, J. & Genton, M. G. (6), The multivariate skew-slash distribution, Journal of Statistical Planning and Inference 136, 9. Revista Colombiana de Estadística 38 (15)