Research Journal of Pure Algebra -2(12), 2012, Page: Available online through ISSN

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1 Research Journal of Pure Algebra (, 0, Page: Avalable onlne through ISSN A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION* Dr. G. S. Davd Sam Jayakumar* Assstant Professor, Jamal Insttute of Management, Jamal Mohamed College, Truchraall 60 00, South Inda, Inda Dr. A. Solaraju Assocate Professor, Det. of Mathematcs, Jamal Mohamed College, Truchraall 60 00, South Inda, Inda A. Sulthan Research scholar, Jamal Insttute of Management, Truchraall South Inda, Inda (Receved on: 0; Revsed & Acceted on: 03 ABSTRACT Ths aer roosed a new generalzaton of bounded Contnuous multvarate symmetrc robablty dstrbutons. More secfcally the authors vsualzes a new generalzaton of SamSola s Multvarate addtve Exonental dstrbuton from the unvarate exonental dstrbuton.further,we fnd ts Margnal, Multvarate Condtonal dstrbutons, Multvarate Generatng functons, Multvarate survval, hazard functons and also dscussed t s secal cases. The secal cases ncludes the transformaton of SamSola s Multvarate addtve exonental dstrbuton nto Multvarate Inverse exonental dstrbuton, Multvarate Webull dstrbuton, Multvarate Power law dstrbuton, Multvarate chsquare dstrbuton wth two d.f, Multvarate Raylegh dstrbuton, Multvarate Pareto dstrbuton, Multvarate logstc dstrbuton, Multvarate Generalzed extreme value dstrbuton and Multvarate Benktander webull dstrbuton. Moreover, the bvarate correlaton between any two exonental random varables found to be 0.5 and t s ndeendent from the Covarance. Smlarly, we smulated and establshed a symmetrc matrx of Covarances based on dfferent combnatons of values for arameters. Keywords: SamSola s Multvarate Exonental dstrbuton, Multvarate Inverse exonental dstrbuton, Multvarate Webull dstrbuton, Multvarate Power law dstrbuton, Multvarate chsquare dstrbuton, Multvarate Raylegh dstrbuton, Multvarate Pareto dstrbuton, Multvarate logstc dstrbuton, Multvarate Generalzed extreme value dstrbuton and Multvarate Benktander Webull dstrbuton. *Mathematcs Subject Classfcaton: Prmary 6H0; Secondary 6E5. INTRODUCTION: The exonental dstrbuton was extensvely studed n the ast and t s used n the feld of Relablty theory, queung theory and also aled to test the lfe length of equment, machneres etc. In recent tmes, statstcans showed the nterest to generalze the unvarate dstrbutons to ts multvarate case and exonental dstrbuton s not an exceton to ts multvarate generalzaton. Some authors gave more concentraton to the bvarate generalzaton of exonental dstrbuton wth dfferent assumtons resectve to ther felds. Gumbel (960, Freud (96 Marshal and Olkn (967, Downton(970, Block et al(975,975b,977, Frdayandatl (977, Raftery(984, Cowan(987, Sarkar (987 Arnold and Strauss(988, Hayakawa(994 and Kotz et al(999 were extensvely studed the Bvarate generalzaton of exonental dstrbuton and gve dfferent forms and shaes of Bvarate exonental dstrbuton. Smlarly these authors also attemted to gve the multvarate generalzaton of the exonental dstrbuton. Krshnamoorthy and Parathsarathy(95 studed the multvarate exonental dstrbuton as a secal case of a multvarate Gamma tye dstrbuton and Gumbel (96 roosed a new form of multvarate exonental dstrbuton. Lkwse Marshalland Olkn (967a, 995, Essary(974, Block (975a, Raftery(994, Lndleyand s I n g urwalla (986, Ghurye(987,O cnnede. et.al (989,sngurwalla and youngren (993 ntroduced new form of multvarate exonental dstrbuton and some authors roosed alternate form of multvarate exonental dstrbuton and aled t to the secfc feld of alcaton resectvely. Ths aer deals wth the new and alternate generalzaton of multvarate exonental dstrbuton and the authors dscussed ts roertes n the next secton. *Corresondng author: Dr. G. S. Davd Sam Jayakumar*, Assstant Professor, Jamal Insttute of Management, Jamal Mohamed College,Truchraall 60 00, South Inda, Inda Research Journal of Pure Algebra (, Dec

2 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. SECTION : SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION Defnton.: Let X, X, X3, X be the random varables followed Contnuous unvarate exonental dstrbuton wth arameter for all (= to, then the densty functon of the Multvarate SamSola s addtve exonental dstrbuton s defned as where 0 0 x x x x f( x, x, x, x {( e ( e ( }( e ( Theorem.: The cumulatve dstrbuton functon of the Sam s Multvarate addtve exonental dstrbuton s defned by x x x x 3 u u u F( x, x, x, x {( e ( e ( }( e du du du ( du where 0 u x x x ( ( x e e x x ( e ( e F( x, x, x, x ( e { ( } Theorem.3: The Probablty densty functon of SamSola s Multvarate addtve Condtonal exonental dstrbuton of X on X X X s where 0 x,, P 3 f( x / x, x, x 0 x x x e {( e ( e ( } x x {( e ( e ( 3} (3 Proof: It s obtaned from f( x / x, x, x f( x, x, x, x f( x, x, x 3 Theorem.4: Mean and Varance of Sam Sola s Multvarate addtve Condtonal exonental dstrbuton are x x {( ( ( } e e Ex ( / x, x, x x x {( ( ( 3} e e (4 V( x / x, x, x E( x / x, x, x ( E( x / x, x, x (5 where x x 3 {( ( ( } e e 4 x x Ex ( / x, x, x {( e ( e ( 3} th Proof: The n order moment of the dstrbuton s n n 0 E( x / x, x, x x f( x / x, x, x dx 0, RJPA. All Rghts Reserved 37

3 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. x x x e {( e ( e ( } n n 0 x x {( e ( e ( 3} E( x / x, x, x x dx ( nk ( x k { ( {( n n ( }} n k k k ( x Ex ( / x, x, x If n=, then the Condtonal exectaton s If n=, then the second order moment s ( ( k x x {( ( ( } e e Ex ( / x, x, x x x {( ( ( 3} e e x x 3 {( ( ( } e e 4 x x Ex ( / x, x, x {( e ( e ( 3} The condtonal varance of the dstrbuton s obtaned by Substtutng the frst and second moments n (5. Theorem.5: If there are = (q + k random varables, such that q random varables X, X, X3, X condtonally q deends on the k varables Xq, Xq, Xq3, X,then the densty functon of SamSola s multvarate addtve qk condtonal Exonental dstrbuton s q qk (6 q x qk x x qk ( e {( e ( e ( qk} f( x, x, x3, xq / xq, xq, xq3, xqk qk qk x x k q {( e ( e ( k} where 0 0 x q Proof: Let the multvarate condtonal law for q random varables varables X, X, X, X q q q3 qk s gven as f( x, x, x, x / x, x, x, x q q q q3 qk X, X, X3, Xq condtonally deendng on the k f( x, x, x3, xq, xq, xq, xq3, xqk f( x, x, x, x q q q3 qk f( x, x, x, x / x, x, x, x q q q q3 qk qk qk x x qk qk x qk ( e {( e ( e ( qk} qk qk x x qk qk q x qk ( {( ( ( } e e e q k dx f( x, x, x, x / x, x, x, x q q q q3 qk where 0 0 x q qk q x qk x x qk ( e {( e ( e ( qk} qk qk x x k q {( e ( e ( k} q 0, RJPA. All Rghts Reserved 37

4 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. SECTION : CONSTANTS OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION Theorem.: The Margnal roduct moments, Covarance and Poulaton Correlaton Coeffcent between the Exonental random varables X and X are gven as 3 E( xx 4 (7 COV ( x, x (8 4 ( x, x (9 4 Proof: Assume that X and X are random varables from SamSola s multvarate addtve exonental dstrbuton. Let the roduct moment of the dstrbuton s Its Covarance s Then E( xx xx f( x, x, x, x dx COVx (, x Exx ( Ex ( Ex ( (0 x x x E( xx xx{( e ( e ( }( e dx By evaluaton, t follows that 3 E( xx 4 The Margnal exectaton of exonental varables E( xx moment for Thus X and X are /λ and /λ resectvely. The Margnal Product s obtaned by substtutng the above Margnal exectatons for COV ( x, x 4 X and X n (0. ( Correlaton coeffcent of a dstrbuton s It observed that σ = /λ and σ = /λ COV ( x, x ( x, x (a (b From (, (a and (b, t follows that ( x, x 4 (3 th Remark.: The Product moments, Covarance and oulaton Correlaton Coeffcent between the and th j of SamSola s multvarate addtve exonental dstrbuton random varable are gven as (4 3 E( xx j 4 j (5 COV ( x, x j 4 j (6 ( x, xj 4 0, RJPA. All Rghts Reserved 373

5 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. Theorem.: The Moment generatng functon of SamSola s Multvarate addtve exonental dstrbuton s ( t t Mx, x, x3, x( t, t, t 3, t ( { ( ( } t ( t t (7 Proof: Let the moment generatng functon of a Multvarate dstrbuton s gven as tx M x, x, x ( 3, x t, t, t3, t e f ( x, x, x3, x dx tx x x x x, x, x ( 3, x,, 3, {( ( ( }( M t t t t e e e e dx ( t t Mx, x, x3, x( t, t, t3, t ( { ( ( } t ( t t by ntegratng the above equaton. Theorem.3: The Cumulant of the Moment generatng functon of the SamSola s Multvarate addtve exonental dstrbuton s ( t t Cx, x, x3, x( t, t, t3, t log( log{ ( ( } t ( t t (8 Proof: It s found from. C ( t, t, t t log( M ( t, t, t t x, x, x3, x, x, x, x3, x, Theorem.4: The Characterstc functon of the SamSola s Multvarate addtve exonental dstrbuton s j ( j t j j t j ( t, t, t, t ( { ( ( } t ( t t x, x, x3, x (9 j j j j j j j j j Proof: Let the characterstc functon of a multvarate dstrbuton s gven as t x j j j,, ( t 3,, t, t3, t e f ( x, x, x3, x dx x x x x j j t x j j jx j jxj j jx j j x, x, x ( 3, x t, t, t3, t {( ( ( }( e e e je dx j j j j j ( j t j j t j ( t, t, t, t ( { ( ( } t ( t t x, x, x3, x j j j j j j j j j by ntegratng the above equaton. Theorem.5: The survval functon of the SamSola s Multvarate addtve exonental dstrbuton s x x ( ( x e e Sx (, x, x 3, x ( e { ( } x x ( e ( e (0 Proof: Let the survval functon of a multvarate dstrbuton s gven as S( x, x, x, x = F( x, x, x, x 0, RJPA. All Rghts Reserved 374

6 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. x x x x 3 λu λ u λu = = λ = = = S( x, x, x, x {( e ( e ( }( e du du du du x x ( ( x e e x x ( e ( e Sx (, x, x, x ( e { ( } Theorem.6: The hazard functon of the SamSola s Multvarate addtve exonental dstrbuton s x x x {( e ( e ( }( e x x ( ( x e e e x x ( e ( e hx (, x, x, x ( ( { ( } Proof: It s obtaned from hx (, x, x, x = f( x, x, x, x Sx (, x, x, x and S( x, x, x, x = F( x, x, x, x Theorem.7: The Cumulatve hazard functon of the SamSola s Multvarate addtve exonental dstrbuton s e e H( x, x, x, x log( ( e { ( } x x ( ( x x x ( e ( e ( Proof: Let the Cumulatve hazard functon of a multvarate dstrbuton s gven as H( x, x, x, x = log( F( x, x, x, x H( x, x, x, x = log( S( x, x, x, x x x ( ( x e e x x ( e ( e H( x, x, x, x log( ( e { ( } SECTION 3: SOME SPECIAL CASES Result 3.: The unvarate margnal of the SamSola s multvarate addtve exonental dstrbuton s the unvarate two arameter exonental dstrbutons. Result 3.: From ( and If P=, the SamSola s multvarate addtve exonental densty s reduced to densty of unvarate exonental dstrbuton. Result 3.3: From ( If P=, then the densty of SamSola s Multvarate Exonental dstrbuton was reduced nto f( x, x (e e 4 e e (3 x x ( x x ( x x where 0 x, x,, 0 Ths s called the densty of SamSola s Bvarate addtve Exonental dstrbuton. Result 3.4: Table and Bvarate robablty surface for (3 show the selected smulated standard Bvarate Covarances between any two exonental random varables calculated for dfferent combnatons of arameters values(,. j 0, RJPA. All Rghts Reserved 375

7 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. Table: Standard Covarances between two exonental random varables for dfferent values of arameters λ and λ j λ / λ j Where j 0, RJPA. All Rghts Reserved 376

8 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. Fg Fg Fg 3 Fg 4 Fg 5 Fg6 Fg 7 Fg 8 0, RJPA. All Rghts Reserved 377

9 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. Fg 9 Fg 0 Fg Fg Fg 3 Fg 4 Fg 5 0, RJPA. All Rghts Reserved 378

10 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. Result 3.5 From ( and If y / x, then the Samsola s Multvarate addtve exonental dstrbuton transformed nto Samsola s Multvarate addtve Inverse exonental dstrbuton of Knd wth arameter and ts densty functon s gven as ( ( y y y y (4 f( y, y, y, y {( e ( e ( }( e where 0, 0 y k Result 3.6From ( and If =and y x, the Samsola s Multvarate addtve exonental dstrbuton transformed nto Samsola s Multvarate addtve Webull dstrbuton wth Parameters (, k and ts densty functon s gven as y k ( ( y k y k ( k k f( y, y, y, y {( e ( e ( }( ( k y e where 0 y (5, k 0 x Result 3.7From ( and If y e / k, then the Samsola s Multvarate addtve exonental dstrbuton reduced nto Samsola s Multvarate Power law dstrbuton wth arameters, k and ts densty functon s gven as f( y, y, y3, y {( ( ky ( ( ky ( }( k y (6 where y, k 0 0 k Result 3.8 From ( and If /,then the Samsola s Multvarate addtve Exonental dstrbuton modfed nto Samsola s Multvarate addtve chsquare dstrbuton wth degrees of freedom and ts densty functon s gven as x ( x x ( f( x, x, x3, x {( e ( e ( }( e (7 where 0 x Result 3.9 From ( and If / and y x, then the Samsola s Multvarate addtve exonental dstrbuton transformed nto Samsola s Multvarate addtve Raylegh dstrbuton wth arameter σ and ts densty functon s gven as y y ( y ( y f( y, y, y3, y {( e ( e ( }( e (8 where 0 y 0 x Result 4.0 From (If y ke, then the Samsola s Multvarate addtve Exonental dstrbuton transformed nto Samsola s Multvarate addtve Pareto dstrbuton wth arameters (, k and ts densty functon s gven as k k k f( y, y, y3, y ( (9 y y y where k y, k 0 0, RJPA. All Rghts Reserved 379

11 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. Result 4.From(If x e and y log x e,then the Samsola s Multvarate addtve Exonental dstrbuton transformed nto Samsola s Multvarate logstc dstrbuton of Knd wth arameters and and ts densty functon s gven as y y y e e e ( y/ ( y/ ( y/ e e ( e y,, 0 f( y, y, y, y {( ( ( ( }( (30 where Result 4. From ( If and y log x, then the Samsola s Multvarate addtve exonental dstrbuton transformed nto Samsola s Multvarate Generalzed extreme value dstrbuton of Knd wth arameters and and ts densty functon s gven as ( y/ ( y / {( / e y } ( y / e e e f( y, y, y3, y {( e ( e ( }(, 0 where y (3, Result 4.3From ( If y x,then the Samsola s Multvarate addtve exonental dstrbuton transformed nto Samsola s Multvarate Benktander webull dstrbuton of Knd wth arameter and ts densty functon s gven as ( y ( y ( y f( y, y, y3, y {( e ( e ( }( e (3 where, 0 y CONCLUSION The multvarate generalzaton of exonental dstrbuton n an addtve form of SamSola s generalzaton havng some nterestng features. At frst, the margnal unvarate dstrbutons of the SamSola s Multvarate addtve Exonental dstrbuton are unvarate and enjoyed the symmetrc roerty. Secondly, the Poulaton Correlaton coeffcent of any two exonental random varables s found to be 0.5 and correlaton coeffcent s ndeendent from ts Covarance. Moreover,the authors smulated and roosed standard Covarance matrx for dfferent values of arameters. Fnally, the multvarate generalzaton of exonental dstrbuton n an addtve form oen the way for the same addtve form of the generalzaton of Multvarate Inverse exonental dstrbuton, Multvarate Webull dstrbuton, Multvarate Power law dstrbuton, Multvarate chsquare dstrbuton wth two d.f, Multvarate Raylegh dstrbuton, Multvarate Pareto dstrbuton, Multvarate logstc dstrbuton, Multvarate Generalzed extreme value dstrbuton and Multvarate Benktander webull dstrbuton. REFERENCES [] Krshnamoorthy, A. S., and Parthasarathy, M. (95. A multvarate gamma tye dstrbuton, Annals of Mathematcal Statstcs,, ; Correcton, 3, 9. [] Gumbel, E. J. (960. Bvarate exonental dstrbutons, Journal of the Amercan Statstcal Assocaton, 55, [3] Freund, J. (96. A bvarate extenson of the exonental dstrbuton, Journal of the Amercan Statstcal Assocaton, 56, [4] Gumbel, E. J. (96. Multvarate exonental dstrbutons, Bulletn of the Internatonal Statstcal Insttute, 39, [5] Marshall, A. W., and Olkn, I. (967a. A multvarate exonental dstrbuton, Journal of the Amercan Statstcal Assocaton, 6, , RJPA. All Rghts Reserved 380

12 Dr. G. S. Davd Sam Jayakumar* et al./ A NEW GENERALISATION OF SAMSOLAI S MULTIVARIATE ADDITIVE EXPONENTIAL DISTRIBUTION*/RJPA (, Dec.0. [6] Marshall, A. W., and Olkn, I. (967b. A generalzed bvarate exonental dstrbuton, Journal of Aled Probablty, 4, 930. [7] Downton, F. (970. Bvarate exonental dstrbutons n relablty theory, Journal of the Royal Statstcal Socety, Seres B, 3, [8] Block, H. W., and Basu, A. P. (974. A contnuous bvarate exonental extenson, Journal of the Amercan Statstcal Assocaton, 69, [9] Esary, J. D., and Marshall, A. W. (974. Multvarate dstrbutons wth exonental mnmums, Annals of Statstcs,, [0] Block, H. W., Paulson, A. S., and Kohberger, R. C. (975. A class of bvarate dstrbutons, rernt. [] Block, H. W. (975a. Contnuous multvarate exonental extensons, n Relablty and Fault Tree Analyss (R. E. Barlow, J. B. Fussell, and N. D. Sngurwalla, eds., Phladeha, Pennsylvana: Socety for Industral and Aled Mathematcs. [] Block, H. W. (975b. Infnte dvsblty of a bvarate exonental extenson and mxtures of bvarate exonental dstrbutons, Research Reort 75, Deartment of Mathematcs, Unversty of Pttsburgh, Pttsburgh, PA. [3] Block, H. W. (977. A characterzaton of a bvarate exonental dstrbuton, Annals of Statstcs, 5, [4] Frday, D. S., and Patl, G. P. (977. A bvarate exonental model wth alcatons to relablty and comuter generaton of random varables, n Theory and Alcatons of Relablty, Vol. (C P. Tsokos and I. N. Shm, eds., , New York: Academc Press. [5] Raftery, A. E. (984. A contnuous multvarate exonental dstrbuton, Communcatons n Statstcs Theory and Methods, 3, [6] Raftery, A. E. (985. Some roertes of a new contnuous bvarate exonental dstrbuton, Statstcs and Decsons, Sulement, [7] Lndley, D. V., and Sngurwalla, N. D. (986. Multvarate dstrbutons for the lfe lengths of comonents of a system sharng a common envronment, Journal of Aled Probablty, 3, [8] Cowan, R. (987. A bvarate exonental dstrbuton arsng n random geometry, Annals of the Insttute of Statstcal Mathematcs, 39, 03 [9] Ghurye, S. G. (987. Some multvarate lfetme dstrbutons, Advances n Aled Probablty, 9, [0] Sarkar, S. K. (987. A contnuous bvarate exonental dstrbuton, Journal of the Amercan Statstcal Assocaton, 8, [] Arnold, B. C, and Strauss, D. (988. Bvarate dstrbutons wth exonental condtonals, Journal of the Amercan Statstcal Assocaton, 83, [] O'Cnnede, C. A., and Raftery, A. E. (989. A contnuous multvarate exonental dstrbuton that s multvarate hase tye, Statstcs & Probablty Letters, 7, [3] Sngurwalla, N. D., and Youngren, M. A. (993. Multvarate dstrbutons nduced by dynamc envronments, Scandnavan Journal of Statstcs, 0, 56. [4] Hayakawa, Y. (994. The constructon of new bvarate exonental dstrbutons from a Bayesan ersectve, Journal of the Amercan Statstcal Assocaton, 89, [5] Marshall, A. W, and Olkn, I. (995. Multvarate exonental and geometrc dstrbutons wth lmted memory, Journal of Multvarate Analyss, 53, 05. [6] Kotz, S., and Sngurwalla, N. D. (999. On a bvarate dstrbuton wth exonental margnals, Scandnavan Journal of Statstcs, 6, Source of suort: Nl, Conflct of nterest: None Declared 0, RJPA. All Rghts Reserved 38