In this paper, the bivariate Kummer-beta type IV distribution, which extends the Jones’
bivariate beta distribution, is discussed. The probability density functions of the product
and ratio of the components of this distribution are derived. Also, a shape analysis is done to
investigate the effect of the new parameter.http://www.satnt.ac.z

Bekker, Andriette, 1958-

Human, Schalk William

Jacobs, Rianne

Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie

UPSpace at the University of Pretoria

http://www.satnt.ac.za doi:10.4102/satnt.v30i1.103Refereraatopsomming’n Nuwe ontwikkeling tot die bivariate betaveldAuthors:Rianne Jacobs1Andriëtte Bekker1Schalk W.  Human1Affiliations:1Department of Statistics, University of Pretoria, South Africa  Correspondence to:Rianne JacobsEmail:rianne.jacobs@up.ac.za  Postal address:Private bag X20, Hatfield, Pretoria 0028, South AfricaHow to cite this abstract:Jacobs, R., Bekker, A. & Human, S.W., 2011, ‘’n Nuwe ontwikkeling tot die bivariate betaveld’, Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 30(1), Art. #103, 2 pages. doi: http://dx.doi.org/10.4102/satnt.v30i1.103Note:This abstract was initially presented as a paper at the annual Natural Sciences Student Symposium, presented under the protection of the Suid-Afrikaanse Akademie vir Wetenskap en Kuns. The symposium was held at the University of Pretoria on 05 November 2010. The following members formed part of the committee that was responsible for arranging the symposium:  Mr. R. Pretorius (Department of Geography, University of South-Africa), Dr E. Snyders (NECSA), Dr M. Landman (Department of Chemistry, University of Pretoria) and Dr W. Meyer (Department of Physics, University of Pretoria).A new development in the bivariate beta fieldIn this paper, the bivariate Kummer-beta type IV distribution, which extends the Jones’ bivariate beta distribution, is discussed.  The probability density functions of the product and ratio of the components of this distribution are derived.  Also, a shape analysis is done to investigate the effect of the new parameter.Page 1 of 2© 2011. The Authors.Licensee: AOSIS OpenJournals. This workis licensed under theCreative CommonsAttribution License.Kummer-tipe verdelings vorm ’n prominente deel van statistiese verdelingsteorie.  In die literatuur is daar verskeie Kummer-tipe verdelings voorgestel, soos die werk van Armero en Bayarri (1997), Ng en Kotz (1995), Gupta et al. (2001), Nagar en Gupta (2002) en Nagar en Cardeño (2001).  Aandag is gegee aan eenveranderlike, meerveranderlike en matriks-Kummerverdelings.In hierdie artikel ondersoek ons die bivariate Kummer-beta-tipe IV-verdeling wat ’n uitbreiding is van Jones se bivariate betaverdeling, waar die waarskynlikheidsdigtheidfunksie (wdf) van laasgenoemde gegee word deur:   [Verg. 1]Hierdie verdeling staan ook in die literatuur bekend as Jones se bivariate betaverdeling (Balakrishnan & Lai 2009:379) en is onafhanklik deur Jones (2001) en Olkin en Liu (2003) voorgestel.  Die wdf van die Kummer-beta-tipe IV-verdeling word gegee deur   [Verg. 2]met Hierdie verdeling is ontwikkel deur ’n Laplace transformasie van die bivariate beta-tipe IV. Die Laplace transformasie word gebruik om die normaliseringskonstante te bereken en bevat die Kummerfunksie, 1F1(.) (Gradshteyn 2007, Afdeling 9.2).  Die randdigtheidsfunksies van Vergelyking 2 word afgelei deur gebruik te maak van magreeksuitbreidings en die konfluente hipergeometriese reeks van twee veranderlikes, Φ1(.) (Gradshteyn 2007, Vgl 3.385).Die verdelings van die produk en kwosiënt van onafhanklike en afhanklike stogastiese veranderlikes het verskeie toepassings (sien bv. Nagar et al. (2009), Gupta en Nadarajah (2008), Joarder (2009), Pham-Gia en Turkkan (2002) en Pham-Gia (2000).  Eers ná 2002 geniet die produk en kwosiënt van Kummer-tipe verdelings aandag in die literatuur.  Die produk en kwosiënt van onafhanklike Kummer-beta-veranderlikes is deur Nagar en Zarrazola (2005) ondersoek en dié van onafhanklike Kummer-gamma-veranderlikes deur Morán-Vásquez en Nagar (2009).  Navorsing oor die produk en kwosiënt van Kummer-tipe verdelings is dus ’n huidige veld van navorsing. Ons betree ’n nuwe veld en kyk na die verdelings van die produk en kwosiënt van die afhanklike komponente van die bivariate Kummer-beta-tipe IV-verdeling.  Om hierdie digtheidsfunksies af te lei, word gebruik gemaak van die Mellin transformasie asook die inverse Mellin transformasie (Mathai 1993, Definisie 1.8).  Die uitdrukkings word verkry in terme van Meijer se G-funksie wat deesdae ’n bekende funksie in sagteware pakkette is.Deur verskillende positiewe sowel as negatiewe waardes vir die addisionele parameter ψ aan te neem, is dit uit die ondersoek duidelik dat ψ ’n noemenswaardige effek toon op die vorms van die nuwe verdelings asook op die korrelasie van die afhanklike komponente.f (x,y) =      1         xa–1 yb–1 (1 – x)b+c–1 (1 – y)a+c+1  ,   0 ≤ x, y ≤ 1                       B(a,b,c)                   (1 – xy)a+b+c                   f (x,y) = K   xa–1 yb–1 (1 – x)b+c–1 (1 – y)a+c+1 e –ψ(x+y)  ,   0 ≤ x, y ≤ 1                                                (1 – xy)a+b+c                   K–1 = ∑∞k=0     (a + b + c)k    1F1(a + k,a + b + c + k: – ψ) 1F1(a + k,a + b + c + k: – ψ)                            k!                               (B(a + c,b + k) B(b + c,a + k))–1    http://www.satnt.ac.za doi:10.4102/satnt.v30i1.103RefereraatopsommingTer opsomming is dit uit die studie duidelik dat die bivariate Kummer-beta-tipe IV-verdeling meer variasie in die vorm inbring teenoor dié bekende bivariate beta-tipe IV-verdeling.  Beta- en gammaverdelings word dikwels in ’n Bayes-opset gebruik, onder andere waar die betaverdeling as a priori-verdeling gebruik word en die gammaverdeling in toustaanmodelle ’n groot rol speel. Die moontlike toepassing van hierdie ‘buigbare’ bivariate Kummer-beta-tipe IV-verdeling in hierdie Bayes konteks moet ondersoek word.  Die produk en kwosiënt word dikwels in betroubaarheidstoepassings gebruik en ook hier moet die rol van die Kummer-beta-tipe IV ondersoek word.LiteratuurverwysingsArmero, C. & Bayarri, M.J., 1997, ‘A Bayesian analysis of queueing system with unlimited service’, Journal of Statistical Planning and Inference 58, 241–261.Balakrishnan, N. & Lai, C.D., 2009, Continuous Bivariate Distributions, 2nd edn, Springer, New York.Gradshteyn, I.S. & Ryzhik, I.M., 2007, Table of Integrals, Series, and Products, Academic Press, Amsterdam.Gupta, A.K., Cardeño, L. & Nagar, D.K., 2001, ‘Matrix-variate Kummer-Dirichlet distributions’, J. Appl. Math. 1(3), 117–139.Gupta, A.K. & Nadarajah, S., 2008, ‘Product moments of Downton’s bivariate exponential distribution’, Water Resource Management 22, 671–679.Joarder, A.H., 2009, ‘Moments of the product and ratio of two correlated chi-square variables’, Statistical Papers 50, 581–592.Jones, M.C., 2001, ‘Multivariate t and the beta distributions associated with the multivariate F distribution’, Metrika 54, 215–231.Libby,  D.L. & Novick, R.E., 1982, ‘Multivariate generalized beta distributions with applications to utility assessment’, Journal of Educational Statistics 7(4), 271–294.Mathai, A.M., 1993, A Handbook of Generalized Special Functions for Statistical and Physical Sciences, Clarendon Press, Oxford, UK.Morán-Vásquez, R.A. & Nagar, D.K., 2009, ‘Product and quotients of independent Kummer-gamma variables’, Far East Journal of Theoretical Statistics 27, 41–55Nagar, D.K. & Cardeño, L., 2001, ‘Matrix variate Kummer-Gamma distributions’, Random Oper. and Stoch. Equ. 9(3), 207–218.Nagar, D.K. & Gupta, A.K., 2002, ‘Matrix-variate Kummer-Beta distribution’, J. Austral. Math. Soc. 73, 11–25.Nagar, D.K., Orozco-Castañeda, J.M. & Gupta, A.K., 2009, ‘Product and quotient of correlated beta variables’, Applied Mathematics Letters, 22, 105–109.Nagar, D.K. & Zarrazola, E., 2005, ‘Distributions of the product and the quotient of independent Kummer-beta variables’, Sci. Math. Jpn., 61(1), 109–117.Ng, K.W. & Kotz, S., 1995, Kummer-gamma and Kummer-beta univariate and multivariate distributions, Research report 84, The University of Hong Kong.Olkin, I. & Liu, R., 2003, ‘A Bivariate beta distribution’, Statistics and Probability Letters 62, 407–412.Pham-Gia, T., 2000, ‘Distributions of the ratios of independent beta variables and applications’, Communications and Statistics-Theory and methods 29(12), 2693–2715.Pham-Gia, T. & Turkkan, N., 2002, ‘The product and quotient of general beta distributions’, Statistical Papers 43, 537–550.Page 2 of 2

A new development in the bivariate beta field

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A new development in the bivariate beta field

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