EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 5, 2017, 1124-1134 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On Trigonometric Moments of the Stereographic Semicircular Gamma Distribution Phani Yedlapalli1,∗, A.J.V.Radhika2, S.V.S.Girija3, A.V.Dattatreya Rao4 1 Department of Basic Science, Shri Vishnu Engineering College for Women, Bhimavaram, A.P-534202, India 2 Department of Mathematics, University College of Engineering, Acharya Nagarjuna University, Guntur, A.P, India 3 Department of Mathematics, Hindu College, Guntur, A.P, India 4 Department of Statistics, Acharya Nagarjuna University, Guntur,A.P ,India. Abstract. Phani (2013) constructed a good number of circular and semicircular models induced by inverse stereographic projection. Minh and Farnum (2003) and Toshihiro Abe et al (2010) proposed a new method to derive circular distributions from the existing linear models. In this paper, a new semicircular model, which is coined as Stereographic Semicircular Gamma distribution is derived by inducing modified inverse stereographic projection on Gamma distribution. This distribution generalizes Stereographic Semicircular Exponential model (Phani et al (2013)) and the density and distribution functions of proposed model admit closed form. Explicit expressions for trigonometric moments are derived by applying Meijer’s G- function and the new semicircular model is extended to construct Stereographic l- axial Gamma distribution. 2010 Mathematics Subject Classifications: 60E05, 62H11 Key Words and Phrases: Circular Model, Directional Data, Meijer’s G-function, Stereographic Projection, Trigonometric Moments. 1. Introduction Directions in two dimensions can be represented as points on the circumference of a unit circle and models for representing such data are called circular distributions. Quite a lot of work was done on circular models defined on the unit circle (Fisher, 1993; Jammalamadaka and Sen Gupta (2001); Mardia and Jupp, (2000)) and recent publications (Dattatreya Rao et al (2007), Girija (2010), Phani et al (2012)). Most of these models are applicable for dealing circular data. To fit/model certain practical data sets, it is not required to go for full circular models but semicircular/arc models are adequate. Some recent ∗Corresponding author. Email addresses: phaniyedlapalli23@gmail.com (P. Yedlapalli) ajv.radhika09@gmail.com (A.J.V.Radhika), svs.girija@gmail.com (S.V.S.Girija) avdrao@gmail.com (A.V.Dattatreya Rao) http://www.ejpam.com 1124 c© 2017 EJPAM All rights reserved. P. Yedlapalli, A.J.V.Radhika, S.V.S.Girija, A.V.D. Rao / Eur. J. Pure Appl. Math, 10 (5) (2017), 1124-1134 1125 papers (Guardiola (2004), Byoung et al (2008) and Phani et al (2013)) address this issue and provided some methodology for constructing distributions suitable for modeling these types of data. For example, when sea turtles emerge from the ocean in search of a nesting site on dry land, a random variable having values on a semicircle is very much sufficient for modeling such data. Given the angles of initial heading and departure, to trace the debris of aircraft lost problem, semicircular models need to be used. A few more examples of semicircular data is available in Ugai et al (1977). A little attention is paid on the study of semicircular distributions. The aim of the present article is to contribute towards filling this gap. In this paper, the modified inverse stereographic projection is used to define a new semicircular model, coined as the Stereographic Semicircular Gamma distribution which generalizes the Stereographic Semicircular Exponential Model (Phani et al (2013)). Explicit expressions for trigonometric moments are derived in terms of Meijer’s G- function and it is extended to the Stereographic l - axial Gamma distribution for modeling axial data. In section 2, methodology of modified inverse stereographic projection is presented. Section 3 is devoted to introduce the proposed distribution and to present graphs of probability density function for various values of parameters. In section 4, the first four trigonometric moments in terms of Meijer’s G- function are derived and the proposed model is extended to l- axial distributions for modeling axial data in section 5. 2. Methodology of Modified Inverse Stereographic Projection (Phani et al (2012)) Modified inverse stereographic projection is defined by a one to one mapping given by T (θ) = x = vtan ( θ 2 ) , where x ∈ (−∞,∞), θ ∈ [−π, π), v > 0. Suppose x is randomly chosen on the interval (−∞,∞). Let F (x) and f (x) denote the cumulative distribution and probability density functions of the random variable X respectively. Then T−1 (x) = θ = 2tan−1 ( x v ) by Toshihiro Abe et al (2010) is a random point on the unit circle. Let G (θ) and g (θ) denote the cumulative distribution and probability density functions of this random point θ respectively. Then G (θ) and g (θ) can be written in terms of F (x) and f (x)using the following theorem. Theorem 2.1. For v > 0, i) G (θ) = F ( vtan ( θ 2 )) ii) g (θ) = v ( sec2( θ2) 2 ) f ( vtan ( θ 2 )) By applying this modified inverse stereographic projection on linear models with sup- port on R+ (R), new distributions mapped onto [0, π) ([−π, π)) are derived to study semi- circular (circular) data. P. Yedlapalli, A.J.V.Radhika, S.V.S.Girija, A.V.D. Rao / Eur. J. Pure Appl. Math, 10 (5) (2017), 1124-1134 1126 3. Stereographic Semicircular Gamma Model Gamma distribution plays a prominent role in actuarial science. Here an attempt is made to construct stereographic version of semicircular Gamma distribution by inducing inverse stereographic projection. A random variable X on the real line is said to have gamma distribution with index parameterc > 0, scale parameter λ > 0 and location parameter α if the probability density, cumulative distribution and characteristic functions of X are respectively given by (1) f(x) = (x−α)c−1 λcΓ(c) exp ( −(x−α) λ ) for λ, c > 0, x > 0 and α > 0 (2)F (x) = Γ( xλ)(c) Γ(c) forλ, c > 0, x > 0 (3) φX(t) = 1 (1−iλt)c where t ∈ R By applying inverse stereographic projection defined by a one to one mapping x = v tan ( θ 2 ) , v > 0, 0 ≤ θ < π a Stereographic Semicircular Gamma Distribution is obtained A semicircular random variable θ is said to follow Stereographic Semicircular Gamma Distribution with index parameter c > 0, location parameter µ and scale parameter σ > 0 denoted by SSCG (µ, σ, c) if the probability density and cumulative distribution functions are respectively given by g(θ) = 1 2σcΓ(c) sec2 ( θ 2 )( tan ( θ 2 ) − µ )c−1 exp ( − 1 σ ( tan ( θ 2 ) − µ )) where 0 ≤ θ < π, c > 0, σ = λ v > 0 and µ = α v (3.1) G(θ) = Γt(c) Γ(c) where t = 1 σ tan ( θ 2 ) , θ ∈ [0, π) (3.2) Special case: If c = 1 in (3.1) reduces to the density function of Stereographic Semi- circular Exponential distribution (Phani et al (2013)). 4. Trigonometric moments of Stereographic Semicircular Gamma distribution The characteristic function of the stereographic semicircular gamma model is φθ(p) = π∫ 0 eipθg(θ)dθ = π∫ 0 eipθ 1 2σcΓ(c) sec2 ( θ 2 ) ( tan ( θ 2 ) − µ )c−1 exp ( − 1 σ ( tan ( θ 2 ) − µ )) dθ The integration is not tractable. But trigonometric moments can be derived by ap- plying the Meijer G function [Gradshteyn and Ryzhik (2007)]. The trigonometric mo- ments of the distribution are given by {ϕp : ±1,±2,±3, . . . } where ϕp = αp + i βp with P. Yedlapalli, A.J.V.Radhika, S.V.S.Girija, A.V.D. Rao / Eur. J. Pure Appl. Math, 10 (5) (2017), 1124-1134 1127 Figure 1: Graphs of probability density function of Stereographic Semicircular gamma distribution for various values of σ and c = 1.5 αp = E(cos pθ) and βp = E(sin pθ) being the pth order cosine and sine moments of the random angle θ, respectively and are required to study the population characteristics. Theorem 4.1: The trigonometric moments αp = E(cos pθ) and βp = E(sin pθ), for p = 1, 2, 3, 4 of the stereographic semicircular gamma distribution with µ = 0, are given as follows α1 = 1− 1 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − c 2 , 0, 1 2 ) β1 = 1 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣ 1−c 2 1−c 2 , 0, 1 2 ) α2 = 1 + 4 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 1 − c 2 , 0, 1 2 ) − 4 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − c 2 , 0, 1 2 ) P. Yedlapalli, A.J.V.Radhika, S.V.S.Girija, A.V.D. Rao / Eur. J. Pure Appl. Math, 10 (5) (2017), 1124-1134 1128 β2 = 2 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − c 2 , 0, 1 2 ) − 4 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣∣ − (c+1) 2 − (1−c) 2 , 0, 1 2 ) α3 = 1− 16 3σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 2 − c 2 , 0, 1 2 ) − 24 σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 1 − c 2 , 0, 1 2 ) − 9 σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − c 2 , 0, 1 2 ) β3 = 3 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣ 1−c 2 1−c 2 , 0, 1 2 ) − 8 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣∣ − (c+1) 2 − (1−c) 2 , 0, 1 2 ) α4 = 1 + 32 3σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 3 − c 2 , 0, 1 2 ) + 80 σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 1 − c 2 , 0, 1 2 ) − 64 σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 2 − c 2 , 0, 1 2 ) − 16 σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − c 2 , 0, 1 2 ) β4 = 4 σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ 1−c 2 1−c 2 , 0, 1 2 ) − 8 σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c+1 2 1−c 2 , 0, 1 2 ) − 16 σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c+1 2 3−c 2 , 0, 1 2 ) − 32 3σc √ π Γ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c+3 2 3−c 2 , 0, 1 2 ) where ∞∫ 0 x2v−1 ( u2 + x2 )Q−1 e−µxdx = u2v+2Q−2 2 √ πΓ (1−Q) G31 13 ( µ2u2 4 ∣∣∣∣ 1− v 1−Q− v, 0, 1 2 ) (4.1) for | arg uπ| < π 2 , Reµ > 0 and Rev > 0 and G31 13 ( µ2u2 4 ∣∣∣∣ 1− v 1−Q− v, 0, 1 2 ) is called the Meijer’s G-function (Gradshteyn and Ryzhik, 2007, formula no. 3.389.2). Proof. ϕp = π∫ 0 cos(pθ)dθ + i ∞∫ 0 sin(pθ)g(θ)dθ = αp + iβp where αp = 1 2σcΓ(c) π∫ 0 cos(pθ) sec2 ( θ 2 )( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ P. Yedlapalli, A.J.V.Radhika, S.V.S.Girija, A.V.D. Rao / Eur. J. Pure Appl. Math, 10 (5) (2017), 1124-1134 1129 βp = 1 2σcΓ(c) π∫ 0 sin(pθ) sec2 ( θ 2 )( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ To derive the first order cosine and sine moments α1 = 1 2σcΓ(c) π∫ 0 cos(θ) sec2 ( θ 2 )( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ Consider the transformation x = tan ( θ 2 ) , cos θ = 1− 2x2 1+x2 and formula (4.1) α1 = 1 σcΓ(c) π∫ 0 [ 1− 2x2 1+x2 ] xc−1e 1 σ xdx = 1− 2 σcΓ(c) π∫ 0 xc+1 ( 1 + x2 )−1 e 1 σ xdx α1 = 1− 1 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − c 2 , 0, 1 2 ) β1 = 1 2σcΓ(c) π∫ 0 sin(θ) sec2 ( θ 2 )( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ Consider the transformation x = tan ( θ 2 ) , sin θ = 2x 1+x2 and formula (4.1) β1 = 1 σcΓ(c) π∫ 0 [ 2x 1+x2 ] xc−1e 1 σ xdx = 2 σcΓ(c) π∫ 0 xc ( 1 + x2 )−1 e 1 σ xdx β1 = 1 σc √ π Γ (c) G31 13 ( 1 4σ2 ∣∣∣∣ 1−c 2 1−c 2 , 0, 1 2 ) To derive the second order cosine and sine moments α2 = 1 2σcΓ(c) π∫ 0 cos(2θ) sec2 ( θ 2 )( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ P. Yedlapalli, A.J.V.Radhika, S.V.S.Girija, A.V.D. Rao / Eur. J. Pure Appl. Math, 10 (5) (2017), 1124-1134 1130 Consider the transformation x = tan ( θ 2 ) , cos 2θ = 1 + 8x4 (1+x2)2 − 8x2 (1+x2) and formula (4.1) α2 = 1 σcΓ(c) ∞∫ 0 [ 1 + 8x4 (1+x2)2 − 8x2 (1+x2) ] xc−1e− 1 σ xdx = 1 σcΓ(c) ∞∫ 0 xc−1e− 1 σ xdx+ 8 σcΓ(c) ∞∫ 0 xc+3 (1+x2)2 e− 1 σ xdx− 8 σcΓ(c) ∞∫ 0 xc+1 (1+x2) e− 1 σ xdx α2 = 1 + 4 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 1 − c 2 , 0, 1 2 ) − 4 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − c 2 , 0, 1 2 ) β2 = 1 2σcΓ(c) π∫ 0 sin 2θ sec2 ( θ 2 ) ( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ Consider the transformation x = tan ( θ 2 ) , sin 2θ = 4x (1+x2) − 8x3 (1+x2)2 and by formula (4.1) β2 = 1 σcΓ(c) ∞∫ 0 [ 4x (1+x2) − 8x3 (1+x2)2 ] xc−1e− 1 σ xdx = 4 σcΓ(c) ∞∫ 0 xc (1+x2) e− 1 σ xdx− 8 σcΓ(c) ∞∫ 0 xc+2 (1+x2)2 e− 1 σ xdx β2 = 2 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ 1−c 2 1−c 2 , 0, 1 2 ) − 4 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣∣ − (c+1) 2 (1−c) 2 , 0, 1 2 ) To derive the third cosine and sine moments α3 = 1 σc2Γ(c) π∫ 0 cos 3θ sec2 ( θ 2 )( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ Consider the transformation x = tan ( θ 2 ) , cos 3θ = 1− 32x6 (1+x2)3 + 48x4 (1+x2)2 − 18x2 (1+x2) α3 = 1 σcΓ(c) ∞∫ 0 [ 1− 32x6 (1+x2)3 + 48x4 (1+x2)2 − 18x2 (1+x2) ] xc−1e− 1 σ xdx = 1− 32 σcΓ(c) ∞∫ 0 xc+5(1 + x2)−3e− 1 σ xdx+ 48 σcΓ(c) ∞∫ 0 xc+3(1 + x2)−2e− 1 σ xdx − 18 σcΓ(c) ∞∫ 0 xc+1(1 + x2)−1e− 1 σ xdx P. Yedlapalli, A.J.V.Radhika, S.V.S.Girija, A.V.D. Rao / Eur. J. Pure Appl. Math, 10 (5) (2017), 1124-1134 1131 α3 = 1− 16 3σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 2 − c 2 , 0, 1 2 ) + 24 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 1 − c 2 , 0, 1 2 ) − 9√ πσcΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − c 2 , 0, 1 2 ) β3 = 1 2σcΓ(c) π∫ 0 sin 3θ sec2 ( θ 2 )( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ Consider the transformation x = tan ( θ 2 ) , sin 3θ = 6x (1+x2) − 32x3 (1+x2)3 β3 = 1 σcΓ(c) ∞∫ 0 [ 6x (1+x2) − 32x3 (1+x2)3 ] xc−1e− 1 σ xdx = 6 σcΓ(c) ∞∫ 0 xc(1 + x2)−1e−σxdx− 32 σcΓ(c) ∞∫ 0 xc+2(1 + x2)−3e− 1 σ xdx β3 = 3 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ 1−c 2 1−c 2 , 0, 1 2 ) − 8 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − (c+1) 2 1−c 2 , 0, 1 2 ) To derive the fourth cosine and sine moments α4 = 1 2σcΓ(c) π∫ 0 cos 4θ sec2 ( θ 2 )( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ Consider the transformation x = tan ( θ 2 ) , cos 4θ = 1 + 128x8 (1+x2)4 + 160x4 (1+x2)2 − 256x6 (1+x2)3 − 32x2 (1+x2) and formula (4.1) α4 = 1 + 128 σcΓ(c) ∞∫ 0 xc+7(1 + x2)−4e− 1 σ xdx+ 160 σcΓ(c) ∞∫ 0 xc+3(1 + x2)−2e− 1 σ xdx − 256 σcΓ(c) ∞∫ 0 xc+5(1 + x2)−3e− 1 σ xdx− 32 σcΓ(c) ∞∫ 0 xc+1(1 + x2)−1e− 1 σ xdx = 1 + 128 σcΓ(c) ∞∫ 0 x2( c2+4)−1(1 + x2)−3−1e− 1 σ xdx+ 160 σcΓ(c) ∞∫ 0 x2( c2+2)−1(1 + x2)−1−1e− 1 σ xdx − 256 σcΓ(c) ∞∫ 0 x2( c2+3)−1(1 + x2)−2−1e− 1 σ xdx− 32 σcΓ(c) ∞∫ 0 x2( c2+1)−1(1 + x2)0−1e− 1 σ xdx P. Yedlapalli, A.J.V.Radhika, S.V.S.Girija, A.V.D. Rao / Eur. J. Pure Appl. Math, 10 (5) (2017), 1124-1134 1132 α4 = 1 + 32 3σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 3 − c 2 , 0, 1 2 ) + 80 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 1 −c, 0, 1 2 ) − 64 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − 2 − c 2 , 0, 1 2 ) − 16 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − c 2 − c 2 , 0, 1 2 ) β4 = 1 2σcΓ(c) π∫ 0 sin 4θ sec2 ( θ 2 ) ( tan ( θ 2 ))c−1 e− 1 σ tan( θ2)dθ sin 4θ = 8x (1+x2) − 16x3 (1+x2)2 − 64x3 (1+x2)3 + 128x5 (1+x2)4 and formula (4.1) β4 = 8 σcΓ(c) ∞∫ 0 xc(1 + x2)−1e− 1 σ xdx− 16 σλΓ(c) ∞∫ 0 xc+2(1 + x2)−2e− 1 σ xdx − 64 σcΓ(c) ∞∫ 0 xc+2(1 + x2)−3e− 1 σ xdx+ 128 σcΓ(c) ∞∫ 0 xc+4(1 + x2)−4e− 1 σ xdx = 8 σcΓ(c) ∞∫ 0 x2( c2+ 1 2)−1(1 + x2)0−1e− 1 σ xdx− 16 σcΓ(c) ∞∫ 0 x2( c2+ 3 2)−1(1 + x2)−1−1e− 1 σ xdx − 64 σcΓ(c) ∞∫ 0 x2( c2+ 3 2)−1(1 + x2)−2−1e− 1 σ xdx+ 128 σcΓ(c) ∞∫ 0 x2( c2+ 5 2)−1(1 + x2)−3−1e− 1 σ xdx β4 = 4 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ 1−c 2 1−c 2 , 0, 1 2 ) − 8 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − (c+1) 2 1−c 2 , 0, 1 2 ) − 16 σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − (c+1) 2 3−c 2 , 0, 1 2 ) − 32 3σc √ πΓ(c) G31 13 ( 1 4σ2 ∣∣∣∣ − (c+3) 2 3−c 2 , 0, 1 2 ) On the similar lines the higher-order moments can be obtained . 5. Stereographic -l-axial Semicircular Gamma Distribution The above proposed model is extended to the l-axial distribution, which is applicable to any arc of arbitrary length say 2π l for l ∈ N, . It is possible and useful to extend the Stereographic Semicircular Gamma distribution to construct the Stereographic-l-axial Gamma distribution. The density function of stereographic semicircular gamma distribution by using the transformation φ = 2θ l , l ∈ N. on the probability density function of φ is given by g(φ) = 1 2σcΓ(c) sec2 ( lφ 4 )( tan ( lφ 4 ))c−1 exp ( − 1 σ ( tan ( lφ 4 ))) (5.1) REFERENCES 1133 0 < φ < 2π l , σ > 0, c > 0 and l = 1, 2, . . . It is coined as Stereographic -l-axial Gamma Distribution Case (1) When l = 1 , in the probability density function (5.1), we get the density function g(φ) = 1 2σcΓ(c) sec2 ( φ− µ 4 )( tan ( φ− µ 4 ))c−1 exp ( − 1 σ ( tan ( φ− µ 4 ))) (5.2) 0 < φ < 2π, σ > 0 and c > 0 It is named by us as Stereographic Circular Gamma Distribution. Case (2) When l = 2, the probability density function (5.1) is the same as that of Stereographic Semicircular Gamma Distribution. Case (3) When l = 2 and c = 1, in the probability density function (5.1) we get the den- sity function of Stereographic Semicircular Exponential Distribution [Phani et al (2013)]. 6. Conclusion In this paper, we derived the semicircular distribution induced by modified inverse stereographic projection on Gamma distribution is discussed and named it by us as Stere- ographic Semicircular Gamma distribution. The density and distribution function of Stere- ographic semicircular gamma distribution admit explicit forms, as do trigonometric mo- ments. As this distribution is asymmetric, it is suitable for modeling skewed directional data. References [1] Byoung, J.A and Hyoung M.K.,A New Family of Semicircular Models: The Semi- circular Laplace Distributions, Communications of the Korean Statistical Society Vol.15(2008), 775-781. 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