A New Generalisation of Sam-Solai s Multivariate symmetric Arcsine Distribution of Kind-1*

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1 IOSR Journal o Mahemacs IOSRJM ISSN: Volume, Issue May-June 0, PP A New Generalsaon o Sam-Sola s Mulvarae symmerc Arcsne Dsrbuon o Knd-* Dr. G.S. Davd Sam Jayaumar. Dr.A.Solarau. 3 Mr.S.Dawood Al, 3 Asssan Proessor,Jamal Insue o Managemen,Truchraall Souh Inda, Inda. Assocae Proessor, De.o mahemacs, JamalMohamed,college,,Truchraall Souh Inda, Inda. Absrac: Ths aer roosed a new generalzaon o amly o Sarmanov ye Connuous mulvarae symmerc robably dsrbuons. More seccally he auhors vsualze a new generalzaon o Sam-Sola s Mulvarae symmerc arcsne dsrbuon o Knd- rom he unvarae case. Furher, we nd s Cumulaon, Margnal, Condonal dsrbuons, Generang uncons and also dscussed s secal case.the secal cases nclude he ransormaon o Sam-sola s Mulvarae symmerc arcsne dsrbuon o nd- no Mulvarae symmerc log arcsne dsrbuon o nd- and Mulvarae symmerc Inverse arcne dsrbuon o Knd-. I s ound ha he condonal varance o Sam-Sola s Mulvarae condonal symmerc arcsne dsrbuon s homoscedasc and he correlaon co-ecen among he random varables are smlar o Pearson s roduc momen correlaon co-ecen. Fnally area values are obaned or B-varae symmerc arcsne dsrbuon o nd- and Bvarae robably suraces and conours are vsualzed. Keywords: Sam-Sola s Mulvarae symmerc arcsne dsrbuon o nd-, homoscedasc, Mulvarae symmerc log arcsne dsrbuon o nd-, Mulvarae symmerc Inverse arcsne dsrbuon o Knd-. I. Inroducon The nroducon o rgonomerc dsrbuons n he sascal leraure s no unue; many auhors suded he dsrbuon beore hree decades.trgnomerc uncons are he age-old conces and he alcaon o hese n mahemacs and oher elds are dverse. Bu he enrance and usage o rgonomerc uncons n dsrbuons leraure s lle. Many auhors suded and roosed new orm o rgonomerc dsrbuons and le we see some o he oneers. Noron [975, 978] roosed he arcsne law and laer elaboraely suded he momen roeres o he dsrbuon. Smlarly Arnold e al [980] also nvesgaed some roeres o he arcsne dsrbuon and Ta acs[98] dscussed he arcsne law roosed by he aul L evy. On he oher hand, Kemerman e al [98] vsualzed he characerzaon o he arcsne dsrbuon as an neresng roery and Mchealwoodrooe [98] aled he arcsne law or he model selecon urose.moreover,koz a al [98],Cambans e al [983],Nadaraah e al[006] were roosed he crcular normal dsrbuons,αsymmerc dsrbuons,bea-trgnomerc dsrbuons whch ncludes he sn and arcsne dsrbuons are secal cases resecvely.fnally,as er he recen revew,marda [008,0]suded he Von-mses dsrbuon and mures o mulvarae sne dsrbuons wh reerence o he alcaons o bo-normacs. From he above dscussed revews, s clear; mos o he auhors roosed and suded he unvarae snearcsne rgonomerc dsrbuons, ece some auhors. Ths movaes he auhors o hs aer o roose a new orm o mulvarae generalzaon o arcsne dsrbuon.the srucure and he roeres o he mulvarae arcsne dsrbuon s dscussed n he ne and subseuen secons. II.Sam-Sola s Mulvarae symmerc arcsne Dsrbuon Denon.: Le X, X, X3, K X are he random varables ollowed Connuous unvarae symmerc arcsne dsrbuon wh mean 0 and varance / or all o. Then he densy o Mulvarae Sam-Sola s symmerc arcsne dsrbuon o Knd- s dened as,, 3 K, { r } Õ where ¹ - - Theorem.: -The cumulave dsrbuon uncon o he Sam-Sola s Mulvarae symmerc arcsne dsrbuon s dened by r 4 Page

2 A New Generalsaon O Sam-Sola s Mulvarae Symmerc Arcsne Dsrbuon O Knd-* 3 z z z3 z r u F Z, Z, Z, Z { u u } du 3 where ¹ - u Z - r z z z3 z z z z3 z r u u F Z, Z, Z, Z du u u du arcsn Z r - Z - Z / F Z, Z, Z3 K, Z Õ { } -- arcsn arcsn Z Z Theorem.3: The Probably densy uncon o Sam-Sola s Mulvarae symmerc arcsne dsrbuon o X on X, X3, K X s P r /, 3 K, r where ¹ - - r Proo: I s obaned rom,, 3 K, /, 3 K,, 3K, Theorem.4- Even order condonal momens o Sam-Sola s Mulvarae Condonal symmerc arcsne dsrbuon s G n n E /, 3 K, G n Proo: The odd momens o Mulvarae Condonal symmerc arcsne dsrbuon are zero and he even order momens are gven as n n 3K 3K E /,, /,, d - r n n - 3K E /,, d - r G n n E /, 3 K, G n From 6 and n,, 3, hen he hgher order condonal even momens o he dsrbuon are E /, 3 K, 4 3 E /, 3 K, E /, 3 K, Page

3 A New Generalsaon O Sam-Sola s Mulvarae Symmerc Arcsne Dsrbuon O Knd-* Theorem.5-I here are random varables, such ha random varables X, X, X3, K X condonally deends on he varables X, X, X 3, K X,hen he densy uncon o Sam-Sola s mulvarae condonal symmerc arcsne dsrbuon s { r } Õ -,, 3 K, /,, 3K, { r } å å where ¹ - - r Proo: Le he mulvarae condonal law or random varables X, X, X3, K X condonally deends on he varables X, X, X 3, K X s gven as,, 3, K,,, 3, K,, 3, K /,, 3, K,,, K,,, K /,,, K 3 3,, K, /,, K, 3 å å r Õ - K å å r Õ Õd { r } Õ 3 3 å å { r } where ¹ - - r III.Consans o Sam-Sola s Mulvarae symmerc arcsne Dsrbuon Theorem 3. The Margnal Co-varance and Poulaon Correlaon Co-ecen beween he random varables X and X s gven as r COV, Proo: Le he roduc momen o he Sam-Sola s mulvarae symmerc arcsne dsrbuon n erms o Covarance rom he orgn s gven as COV, K,, 3, K Õ d COV, K d r Õ - r COV, Remar 3.: The resul can be generalzed o he Co-varance beween he h and h random varable are gven as r COV, r COV, where ¹ - r Theorem 3.: The Momen generang uncon o Sam-Sola s Mulvarae symmerc arcsne dsrbuon s Page

4 A New Generalsaon O Sam-Sola s Mulvarae Symmerc Arcsne Dsrbuon O Knd-* r I I M,, I -- 8 K K,, 3, 3, Õ 0 I0 I0 where I 0 and I are he moded Bessel uncons. Proo: Le he momen generang uncon o he Mulvarae dsrbuon s gven as å M,, K e,,, K Õ d K K,, 3, 3, å M,, 3,,, 3, e r d K K K Õ Õ e e,, 3, 3, r K K K Õ K Õ M,, { d d From 9, observes e d I e d I From 9, 0, and by negraon r I I M,, I,, 3, 3, 0 K K Õ I0 I0 Theorem 3.3: The Cumulan o he Momen generang uncon o he Sam-Sola s Mulvarae symmerc arcsne dsrbuon s r I I C,, 3,,, 3, K log 0 log å I å å -- K I I 0 0 where I 0 and I are he moded Bessel uncons. Proo: I s ound rom C,, log M,,,, 3, K 3, K,, 3, K 3, K Theorem 3.4: The Characersc uncon o he Sam-Sola s Mulvarae symmerc arcsne dsrbuon s r J J,, 3,,, 3, 0 Õ J -- 3 K K J J 0 0 where J 0 and J are he Bessel uncons. Proo: Le he characersc uncon o a mulvarae dsrbuon s gven as å,, 3,,, 3,,, 3, K e K Õ d K K å,, 3,,, 3, e r d K K K Õ Õ e e,, 3, 3, d r d K K K Õ K Õ ,, { From 4, observes e d J e d J From 4, 5, 6 and by negraon r J J,,,, 3, 3, J0 K K Õ J0 J Page

5 A New Generalsaon O Sam-Sola s Mulvarae Symmerc Arcsne Dsrbuon O Knd-* IV.Some Secal Cases Resul 4.: From and r 0, hen here s no correlaon beween he random varables and he Sam Sola s mulvarae symmerc arcsne densy s reduced no roduc o he densy uncon o un-varae symmerc arcsne dsrbuons. Resul 4.: From and P, hen he densy o Sam-Sola s Mulvarae symmerc arcsne dsrbuon was reduced no, r where r Ths s called Sam-Sola s B-varae symmerc Arcsne dsrbuon o Knd- Resul 4.4: Below he dagram shows he B-varae robably surace o he Sam-Sola s B-varae sandard symmerc arcsne dsrbuon or varous values o oulaon correlaon coecen r -,-0.75,-0.5,-0.5, 0, 0.5, 0.5, 0.75 are gven. Resul 4.5: From and P, hen he Sam-Sola s B-varae Cumulave sandard symmerc arcsne dsrbuon s gven as arcsn Z arcsn Z F Z, Z r Z Z / Resul 4.6: Below he dagram shows he B-varae cumulave robably surace o he Sam-Sola s Bvarae cumulave symmerc arcsne dsrbuon or varous values o oulaon correlaon coecen r -,-0.75,-0.5,-0.5, 0, 0.5, 0.5, 0.75, are gven Page

6 A New Generalsaon O Sam-Sola s Mulvarae Symmerc Arcsne Dsrbuon O Knd-* Resul 4.7-Usng he soware Male verson 4, he able values rom -0.9 o 0.9 wh nerval value 0. or Sam-Sola s b-varae symmerc arcsne dsrbuon are obaned. Area under he Sam-Sola s B-varae symmerc arcsne suraces based on cumulave dsrbuon uncon or varous values o r s gven. Z Z r 47 Page

7 A New Generalsaon O Sam-Sola s Mulvarae Symmerc Arcsne Dsrbuon O Knd-* Resul 4.8: From and y e, hen he Sam-Sola s Mulvarae symmerc arcsne dsrbuon o Knd- ransormed no Sam-sola s Mulvarae symmerc log arcsne dsrbuon o Knd- and s densy s gven as y, y, y3 K, y r log y log y Õ y - log y where ¹ - e y e - Resul 4.9: From and y /, hen he Sam-Sola s Mulvarae symmerc arcsne dsrbuon o Knd- ransormed no Sam-sola s Mulvarae symmerc Inverse arcsne dsrbuon o Knd- and s densy s gven as r y, y, y3, y yy y y r where ¹ - - y V.Concluson The mulvarae generalzaon o Sam-Sola s mulvarae symmerc arcsne dsrbuon o nd- s havng some neresng eaures. A rs, he margnal un-varae dsrbuons o he Sam-Sola s Mulvarae symmerc arcsne dsrbuons are unvarae symmerc arcsne dsrbuons. Secondly, he Correlaon coecen o he roosed dsrbuon are smlar o Pearson s correlaon Co-ecen. The Condonal varance o Sam-Sola s Mulvarae Condonal symmerc arcsne dsrbuon s heeroscedasc n naure and hs eaure s a unue or he roosed dsrbuon. Thus he generalzaon o sarmonav amly o symmerc mulvarae dsrbuon oen he way or logcal eenson o he generalzaon o symmerc amly o all unvarae connuous robably dsrbuons. Fnally, he Muldmensonal Jacoban ransormaon o Mulvarae arcsne varables whch leads us o generalze he Sam-sola s Mulvarae symmerc log arcsne dsrbuon and Mulvarae symmerc Inverse arcsne dsrbuon. Reerences [] M. Abramowz and I. A. Segun Handboo o Mahemacal Funcons. Dover Publcaons, 964 [] I.S. Gradsheyn and I.M. Ryzh.Table o negrals, seres, and roducs. Academc Press, NewYor, 965 [3] R.M. Noron,On roeres o he arc sne law. Sanhya Ser. A, 37, 975, [4] R.M. Noron,Momen roeres and he arc sne law. Sanhya Ser. A, 40, 978,9-98 [5] B.C. Arnold and R.A. Groeneveld, Some roeres o he arcsne dsrbuon. J.Amer. Sas.Assoc., 75369, 980, [6] Ta acs, L, The arc sne law o Paul L evy. In Conrbuons o Probably. A Collecon o Paers Dedcaed o Eugene Lu acs J. Gan and V. K. Rohag, eds.,98, Academc Press, New Yor [7] J.H.B. Kemerman and M. Sbnsy, On he characerzaon o an neresng roery o he arcsne dsrbuon. Pacc J. Mah., 03, 98, [8] Mchael Woodrooe, On Model Selecon and he ARC Sne Laws, The Annals o Sascs. Volume 0, Number 4, 98, [9] Koz, S. and Johnson, N. L,Crcular normal dsrbuon. Encycloeda o Sascal Scences,, [0] S. Cambans, R. Keener, and G. Smons,On a α-symmerc mulvarae dsrbuons.j. Mulvarae Analyss, 3, 983, [] Johnson, N. L., Koz, S. and Balarshnan, N,Connuous Unvarae Dsrbuons., nd ed., Wley, New Yor, 994. [] Nadaraah, S. and Koz, S,Bea Trgonomerc Dsrbuon. Poruguese Economc Journal, Vol 5, No 3, 006, [3] K. V. Marda, G. Hughes, C. C. Taylor, and H. Sngh, A mulvarae von Mses dsrbuon wh alcaons o bonormacs.the Canadan Journal o Sascs,36, 008,99-09 [4] K. V. Marda, J. T. Ken, Z. Zhang, C. Taylor, and T. Hamelryc, Mures o concenraed mulvarae sne dsrbuons wh alcaons o bo-normacs. Submed, 0. [5] Solarau, Davd Sam and Kava Dev, A New Generalsaon o Sam-Sola s Mulvarae Lalace dsrbuon,global Journal o Mahemacal scences: Theory and Praccal,0 Vol.4, Number, 0, [6] Solarau, Davd Sam and Kava Dev, A New Generalsaon o Sam-Sola s Mulvarae--dsrbuon, Global Journal o Mahemacal scences: Theory and Praccal, Vol.4, Number, 0, [7] Solarau,Davd Sam and Kava Dev, A New Generalsaon o Sam-Sola s Mulvarae Cauchy dsrbuon o Tye-I, Global Journal o Theorecal and aled Mahemacal scences:vol., Number, 0, [8] Solarau, Davd Sam and KavaDev, A New Generalsaon o Sam-Sola s Mulvarae Cauchy dsrbuon o Tye-II, Global Journal o Theorecal and aled Mahemacal scences:vol., Number, 0, r 48 Page