A NEW GENERALISATION OF SAM-SOLAI S MULTIVARIATE ADDITIVE GAMMA DISTRIBUTION*

A NEW GENERALISATION OF SAM-SOLAI S MULTIVARIATE ADDITIVE GAMMA DISTRIBUTION* Dr. G.S. Davd Sam Jayakumar, Assstant Profssor, Jamal Insttut of Managmnt, Jamal Mohamd Collg, Truchraall 620 020, South Inda, Inda. E-mal: samjaya77@gmal.com Dr. A. Solaraju, Assocat Profssor, Dt. of Mathmatcs, Jamal Mohamd Collg, Truchraall 620 020, South Inda, Inda. E-mal:Solarama@yahoo.co.n A.Sulthan, Rsarch scholar, Jamal Insttut of Managmnt, Truchraall 620 020 South Inda, Inda, E-mal: Sulthan90@gmal.com Abstract: Ths ar roosd a nw gnralzaton of boundd Contnuous multvarat symmtrc robablty dstrbutons. In ths ar, w vsualz a nw gnralzaton of Sam-Sola s Multvarat addtv Gamma dstrbuton from th un-varat two aramtrs Gamma dstrbuton. Furthr, w fnd ts Margnal, Multvarat Condtonal dstrbutons, Multvarat Gnratng functons, Multvarat survval, hazard functons and also dscussd t s scal cass. Th scal cass ncluds th transformaton of Sam-Sola s Multvarat addtv Gamma dstrbuton nto Multvarat Ch-squar dstrbuton, Multvarat Erlang dstrbuton, Two aramtr Multvarat Gamma dstrbuton, Multvarat Invrs Gamma dstrbuton, Multvarat log Gamma dstrbuton and Multvarat Nagakam-m dstrbuton. Morovr, t s found that th bvarat corrlaton btwn two Gamma random varabls urly dnds on th sha aramtr and w smulatd and stablshd slctd standard bvarat gamma corrlaton bounds from 900 dffrnt combnatons of valus for sha aramtr. Kywords: Sam-Sola s Multvarat Gamma dstrbuton, Transformaton, Multvarat Ch-squar dstrbuton, Multvarat Erlang dstrbuton, Two aramtr Multvarat Gamma dstrbuton, Multvarat Invrs Gamma dstrbuton, Multvarat log Gamma dstrbuton, Multvarat Nagakam-m dstrbuton, corrlaton bounds Introducton: Chryan (94) ntroducd a b-varat Gamma ty dstrbuton functon wth assumton of th Gamma random varabls ar corrlatd and smlarly Ramabhadran [95] roosd a multvarat Gamma ty dstrbuton n th xonntal famly of functons. Morovr Krshnamoorthy t al. (95) contnud th work of chryan, Ramabhadran and roosd a smlar ty of multvarat Gamma dstrbuton. On th othr hand, Sarmanov (968) roosd a gnralzd Gamma dstrbuton wth th assumton of symmtrcty among random varabls and Gavr (970) stablshd th mxtur of multvarat Gamma dstrbuton. Johnson t al (972, 2000) hghlghtd th Multvarat systm of Gamma dstrbuton and Dussauchoy t al (975) ntroducd a Multvarat Gamma dstrbuton whos margnal ar also followd a unvarat Gamma laws. Bckr t al(98) studd th xtnson of gamma dstrbuton for th bvarat cas and smlarly D Est(98) also dscrbd th Morgnstrn ty Gnralzaton of bvarat Gamma dstrbuton.kowalczyk t al(989) conductd a n-dth study about th rorts of Multvarat Gamma dstrbuton namly thr sha, stmaton of aramtrs and Matha(99,992) studd a dffrnt form of multvarat Gamma dstrbuton. Basd on th ast and rsnt ltraturs, th authors roosd a nw gnralzaton of boundd Contnuous multvarat symmtrc robablty dstrbutons wth scal rfrnc to th Gamma law and t s dscussd n th nxt scton. Thus th logcal gnralzaton of unvarat robablty dstrbuton for a Multvarat cas s an ntrstng task on th art of statstcans. Th gnralzaton of unvarat two aramtr Gamma dstrbuton to ts Multvarat cas basd on th addtv ty dstrbuton s dscussd. *Mathmatcs Subjct Classfcaton. Prmary 62H0; Scondary 62E5 Global Publshng Comany 39

G. S. DAVID SAM JAYAKUMAR, A. SOLAIRAJU, A. SULTHAN Scton : Sam-Sola s Multvarat Addtv Gamma dstrbuton Dfnton.: Lt X, X 2, X3, X b th random varabls followd Contnuous unvarat Gamma dstrbuton wth sha aramtr k and scal aramtr for all (= to ). Thn th dnsty functon of th Multvarat Sam Sola s addtv Gamma dstrbuton s dfnd as x f ( x, x, x, x ) {( ) ( )} k k x -- () whr 0, k 0. x Thorm.2: Th cumulatv dstrbuton functon of th Sam-Sola s Multvarat addtv Gamma dstrbuton s dfnd by x x2 x x 3 k u ( u ) 0 0 0 0 k F( x, x, x, x ) {( ) ( )} du du du du -- (2) Whr 0 u x, k 0 x k k u x u x k 0 F( x, x, x, x ) ( ){ ( du ) ( ))} (,, ) x x k F( x, x2, x3, x ) ( ){ ( ) ( )} x whr x k k u u ( x,, k ) du k 0 s th lowr ncomlt Gamma ntgral of th random varabl. Thorm.3: Th Probablty dnsty functon of Sam-Sola s Multvarat addtv Condtonal Gamma dstrbuton of X on X 2, X 3, X s P ( x ) {( ) ( )} k k k ( x) f ( x / x, x, x ) ( ) ( 2) k 2 x -- (3) whr 0 x, k 0 Proof: It s obtand from f ( x / x, x, x ) f ( x, x, x, x ) f ( x, x, x ) 2 3 Global Publshng Comany 40

A NEW GENERALIZATION Thorm.4: Man and Varanc of Sam - Sola s Multvarat addtv Condtonal Gamma dstrbuton ar k { ( k ) ( )} 2 k k ( x) E( x / x, x, x ) 2 ( x ) ( ) ( 2) k -- (4) V ( x / x, x, x ) E( x / x, x, x ) ( E( x / x, x, x )) 2 2 -- (5) whr k ( ) 2 { ( ) (2 x k k ) 2( )} 2 2 k E( x / x2, x3, x ) k ( x) ( ) ( 2)) k th Proof: Th n ordr momnt of th dstrbuton s 2 n n E( x / x, x, x ) x f ( x / x, x, x ) dx 0 k x {( ) ( )} n n k k ( ) 0 x 2 ( x ) E( x / x, x, x ) x dx ( ) ( 2) k ( nk ) ( x ) k { ( ){( n n ) ( )}} n k 2 k k ( x) E( x / x, x, x ) ( ) ( 2)) k 2 If n=, thn th Condtonal xctaton s k { ( k ) ( )} 2 k k ( x) E( x / x, x, x ) 2 ( x ) ( ) ( 2) k If n=2, thn th scond ordr momnt s k 2 { ( ) 2( k k ) 2( )} 2 2 k k ( x) E( x / x, x, x ) 2 ( x ) ( ) ( 2)) k Th condtonal varanc of th dstrbuton s obtand by Substtutng th frst and scond momnts n (5). Global Publshng Comany 4

G. S. DAVID SAM JAYAKUMAR, A. SOLAIRAJU, A. SULTHAN Thorm.5: If thr ar = (q + k) random varabls, such that q random varabls X, X, X, X X, X 2, X3, X q condtonally dnds on th k varabls,thn th dnsty functon of Sam-Sola s q q2 q3 qk multvarat addtv condtonal Gamma dstrbuton s ( x ) {( ) ( q k )} k q qk k q q q2 q3 qk k qk ( x ) f ( x, x, x, x / x, x, x, x ) ( ) ( k ) q k q x -- (6) whr 0, k 0 x Proof: Lt th multvarat condtonal law for q random varabls k varabls X, X, X, X q q2 q3 qk s gvn as X, X 2, X3, X q condtonally dndng on th f ( x, x, x, x / x, x, x, x ) q q q2 q3 qk f ( x, x, x, x, x, x, x, x ) q q q2 q3 qk f ( x, x, x, x ) q q2 q3 qk qk k x qk qk ( x ) {( ) ( q k )} k q q q2 q3 qk qk k x qk qk ( x ) q {( ) ( )} 0 0 0 k q k x q qk ( x ) {( ) ( q k )} k f ( x, x, x, x / x, x, x, x ) q q q2 q3 qk k qk ( x ) f ( x, x, x, x / x, x, x, x ) whr 0, k 0 x q k dx ( ) ( k ) q k Scton 2: Constants of Sam-Sola s multvarat addtv Gamma dstrbuton Thorm 2.: Th Margnal roduct momnts, Co-varanc and Poulaton Corrlaton Co-ffcnt btwn th Gamma random varabls X and X ar gvn as k E( x x ) k -- (7) ( k)( k2) COV ( x, x2) -- (8) ( k)( k2) ( x, x2) kk -- (9) whr ( x, x ) Global Publshng Comany 42

A NEW GENERALIZATION Proof: Assum that x and x 2 ar random varabls from Sam-Sola s multvarat addtv Gamma dstrbuton. Lt th roduct momnt of th dstrbuton s E( x x ) x x f ( x, x, x, x ) dx 0 0 0 Its Co-varanc s COV ( x, x ) E( x x ) E( x ) E( x ) -- (0) Thn k x ( x ) 0 0 0 k E( x x ) x x {( ) ( )} dx By valuaton, t follows that k E( x x ) k Th Margnal xctaton of Gamma varabls x and x 2 ar k / and k 2 / 2 rsctvly. Th Margnal Product momnt for E(x x 2 ) s obtand by substtutng th abov Margnal xctatons for x and x 2 n (0). Thus ( k)( k2) COV ( x, x2) -- () Corrlaton coffcnt of a dstrbuton s COV ( x, x2) ( x, x2) --(2a) It obsrvs that = k / and 2 = k 2 / 2 --(2b) From (), (2a) and (2b), t follows that ( k)( k2) ( x, x2) kk --(3) whr ( x, x ) Rmark 2.: Th Product momnts, Co-varanc and oulaton Corrlaton Co-ffcnt btwn th th and th j of Sam-Sola s multvarat addtv Gamma dstrbuton random varabl ar gvn as Global Publshng Comany 43

G. S. DAVID SAM JAYAKUMAR, A. SOLAIRAJU, A. SULTHAN k kj E( xx j) j ( k)( kj) COV ( x, x j) ( k)( kj) ( x, xj) kk j j --(4) --(5) --(6) whr j ; ( x, x ) Thorm 2.2: Th Momnt gnratng functon of Sam-Sola s Multvarat addtv Gamma dstrbuton s M t t t t k x, x2, x ( 3, x, 2, 3, ) ( ){( ( ) ) ( 2)} t t --(7) Proof: Lt th momnt gnratng functon of a Multvarat dstrbuton s gvn as tx M ( t, t, t t ) f ( x, x, x, x ) dx x, x2, x3, x, 0 0 0 tx k x ( ) x x, x2, x ( 3, x, 2, 3, ) {( ) ( )} 0 0 0 k M t t t t dx M t t t t k x, x2, x ( 3, x, 2, 3, ) ( ){( ( ) ) ( 2)} t t by ntgratng th abov quaton. Thorm 2.3: Th Cumulant of th Momnt gnratng functon of th Sam-Sola s Multvarat addtv Gamma dstrbuton s C t t t t k x, x2, x ( 3, x, 2, 3, ) log( ) log{( ( ) ) ( )} t t --(8) Proof: It s found from. C ( t, t, t t ) log( M ( t, t, t t )) x, x2, x3, x, x, x2, x3, x, Global Publshng Comany 44

A NEW GENERALIZATION Thorm 2.4: Th Charactrstc functon of th Sam-Sola s Multvarat addtv Gamma dstrbuton s j j k j ( t, t, t t ) ( ){( ( ) ) ( )} t t x, x2, x3, x, j j j j j j --(9) Proof: Lt th charactrstc functon of a multvarat dstrbuton s gvn as t x j j j x, x2, x ( 3, x t, t2, t3, t ) (, 2, 3, ) f x x x x dx j 0 0 0 j t x j j k j ( ) jxj j jx j j x, x2, x ( 3, x t, t2, t3, t ) {( ) ( )} j dx j 0 0 0 j j k j j j j k j ( t, t, t t ) ( ){( ( ) ) ( )} t t x, x2, x3, x, j j j j j j by ntgratng th abov quaton. Thorm 2.5: Th survval functon of th Sam-Sola s Multvarat addtv Gamma dstrbuton s (,, ) x x k S( x, x2, x3, x) ( )( ( ) ( )) x --(20) Proof: Lt th survval functon of a multvarat dstrbuton s gvn as S( x, x, x, x ) F( x, x, x, x ) x x2 x x 3 k ( ) u u 0 0 0 0 k S( x, x, x, x ) ( ( )) du du du du x k k u x u x k 0 S( x, x, x, x ) ( )( ( du ) ( )) (,, ) x x k S( x, x2, x3, x) ( )( ( ) ( )) x Whr varabl. x k k u u ( x,, k ) du k 0 s th lowr ncomlt Gamma ntgral of th random Global Publshng Comany 45

G. S. DAVID SAM JAYAKUMAR, A. SOLAIRAJU, A. SULTHAN Thorm 2.6: Th hazard functon of th Sam-Sola s Multvarat addtv Gamma dstrbuton s h( x, x, x, x ) k x x {( ) ( )} k (,, ) x x k x ( )( ( ) ( )) --(2) Proof: It s obtand from h( x, x, x, x ) f ( x, x, x, x ) S( x, x, x, x ) and S( x, x, x, x ) F( x, x, x, x ) Thorm 2.7: Th Cumulatv hazard functon of th Sam-Sola s Multvarat addtv Gamma dstrbuton s (,, ) x x k H( x, x2, x3, x) log( ( )( ( ) ( ))) x --(22) Proof: Lt th Cumulatv hazard functon of a multvarat dstrbuton s gvn 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 k H( x, x2, x3, x) log( ( )( ( ) ( ))) x Scton 3: Som Scal Cass Rsult 3.: Th un-varat margnal of th Sam-Sola s multvarat addtv Gamma dstrbuton s th un-varat two aramtr Gamma dstrbutons. Rsult 3.2: If P = n (), th Sam-Sola s multvarat addtv Gamma dnsty s rducd to dnsty of un-varat two aramtr Gamma dstrbuton. Rsult 3.3: If P = 2 n (), thn th dnsty of Sam-Sola s Multvarat Gamma dstrbuton was rducd nto x x f ( x, x ) ( ) k k k2 k2 2 k k2 ( x x2 ) --(23) Global Publshng Comany 46

A NEW GENERALIZATION whr 0, x x2, k k2,,, 0 Ths s calld th dnsty of Sam-Sola s B-varat addtv Gamma dstrbuton. Rsult 3.4: Th tabls, tabl 2 and B-varat robablty surfac for (23) show th slctd smulatd standard Bvarat corrlatons btwn two Gamma random varabls whch ar boundd btwn - and + calculatd from 900 dffrnt combnatons of sha aramtr ( k, k ). Tabl : Smulaton runs for slctd valus of sha aramtr wth oulaton corrlaton bounds Runs 8 86 802 873 28 435 268 497 238 442 K.3 3 3.9 2 0.5 2 0.5.3 0. 0.4 K 2 3 2.3.9 3 2 0.5 0.5 2 0..3 2.7 533 2.7 0.4 ( x, x ) -0.9898-0.7539-0.50 +0.5 +0.7488 +0.984 Tabl 2: Smulaton runs and combnaton of sha aramtrs of Bvarat Gamma dstrbuton whn ( x, x2) =0 Runs K K 2 Runs K K 2 Runs K K 2 8 0.3 309 0.7 562.2 3. 3 0.5 577 0.8 67.6 350.9 589.3 76 0.5 363 0. 60 3 05. 372.6 686. 07 0.4 375 0.9 687 0.7 0 2.2 386.8 733.5 2.3 46.4 764.5 30 0.8 425 0.2 784.4 42 2.7 426 2. 79 0. 54 0.3 429 2.9 800.5 62.8 433 0.4 807.2 88.2 460.8 83.7 92 2.8 484 835.3 226.9 500 3 844.7 270.7 52 2.5 856 0.2 273.9 52 0.9 884 2 278 2.6 53.6 888.4 280 2.4 537 0.6 895 0.6 290.3 547 - - - Global Publshng Comany 47

G. S. DAVID SAM JAYAKUMAR, A. SOLAIRAJU, A. SULTHAN Fgur 2 Fg. Fg.2 Fg.3 Fg. 4 Fg.5 Fg.6 Global Publshng Comany 48

A NEW GENERALIZATION Fg. 7 Fg.8 Fg.9 Fg.0 Fg. Fg.2 Global Publshng Comany 49

G. S. DAVID SAM JAYAKUMAR, A. SOLAIRAJU, A. SULTHAN Rsult 3.6: From () and f k =, thn th oulaton corrlaton btwn th th and th j random varabl s gvn as ( x, xj) =0 and th dnsty of Sam-Sola s Multvarat addtv Gamma dstrbuton s rducd as roduct of unvarat xonntal dstrbuton, and ts dnsty functon s f ( x, x2, x3, x) ( ) x x f ( x, x, x, x ) -- (24) whr 0 x 0 Rsult 3.7: From() and f =, th Sam-sola s Multvarat addtv Gamma dstrbuton s rducd nto Sam- sola s Multvarat on aramtr addtv Gamma dstrbuton wth aramtrs as k and ts dnsty functon s gvn k x f ( x, x2, x3, x) {( ) ( )} k x -- (25) k whr 0 k 0 k x Rsult 3.8: From () and f and, thn th Sam-sola s Multvarat addtv Gamma dstrbuton s rducd nto Sam-sola s Multvarat two aramtr addtv Gamma dstrbuton wth aramtrs, k and ts dnsty functon s gvn as f x x x x x k k (, 2, 3, ) {( ) ( )} k x -- (26) Rsult 3.9: From () and f n, k 2 2 whr 0, k 0 dstrbuton s modfd nto Sam-sola s Multvarat addtv ch-squar and ts dnsty functon s gvn as x, thn th Sam-sola s Multvarat two aramtr addtv Gamma 2 -dstrbuton wth n dgrs frdom Global Publshng Comany 50

A NEW GENERALIZATION n x 2 ( ) f ( x 2, x2, x3, x) ( ) {( ) ( )} 2 n 2 2 x --(27) whr 0 x n 0 k Rsult 3.0: From () and f, thn th Sam-sola s Multvarat two aramtr addtv Gamma dstrbuton s altrd nto Sam-sola s Multvarat addtv Erlang-k dstrbuton and ts dnsty functon s gvn as k ( kx) f ( x, x, x, x ) ( k ){( ) ( )} k kx -- (28) y x whr 0 x, k 0 Rsult 3.: From () and f, thn th Sam-sola s Multvarat two aramtr addtv Gamma dstrbuton s transformd nto Sam-sola s Multvarat addtv Invrs Gamma dstrbuton and ts dnsty functon s gvn as k ( ) y f ( y, y, y, y ) ( ){( ) ( )} k 2 y ( ) y -- (29) Rsult 3.2: From () and f y x whr 0, k 0 y, thn th Sam-sola s Multvarat two aramtr addtv Gamma dstrbuton s transformd nto Sam-sola s Multvarat addtv log-gamma dstrbuton and ts dnsty functon s gvn as k log y ( log y ) f ( y, y2, y3, y ) ( ){( ) ( )} y k -- (30) Rsult 3.3: From () and fk m, whr m and y, k 0 y x, thn th Sam-sola s Multvarat two aramtr addtv Gamma dstrbuton s transformd nto Sam-sola s Multvarat Nagakam-m dstrbuton and ts dnsty functon s gvn as Global Publshng Comany 5

G. S. DAVID SAM JAYAKUMAR, A. SOLAIRAJU, A. SULTHAN m m ( ) 2 m y m y m y m m f ( y, y, y, y ) 2 ( ( )) -- (3) whr 0 y, 0 m 0.5 Concluson: Th multvarat gnralzaton of two aramtr Gamma dstrbuton s an addtv form of Sam- Sola s gnralzaton havng som ntrstng faturs. At frst, th margnal un-varat dstrbutons of th Sam- Sola s Multvarat addtv Gamma dstrbuton ar un-varat and njoyd th symmtrc rorty. Scondly, th Poulaton Corrlaton co-ffcnt of th roosd dstrbuton s boundd btwn - and + for crtan valus of sha aramtr and th authors stablshd th smulatd standard bvarat corrlatons. Thrdly, th Condtonal varanc of Sam-Sola s Multvarat addtv condtonal Gamma dstrbuton s htroscdastc n natur and ths fatur s a unqu for th roosd dstrbuton. Fnally, th multvarat gnralzaton of two aramtr Gamma dstrbuton n an addtv form on th way for th sam addtv form of th gnralzaton of th xonntal famly of Sam-Sola s Multvarat Ch-squar dstrbuton, Multvarat Erlang dstrbuton, Multvarat Invrs Gamma dstrbuton, Multvarat log Gamma dstrbuton and Multvarat Nagakam-m dstrbuton. REFERENCES: []Chryan, K. C. A bvarat corrlatd gamma-ty dstrbuton functon, Journal of th Indan Mathmatcal Socty, 5, (94), 33-44. [2]Krshnamoorthy, A. S., and Parthasarathy, M. A multvarat gamma- ty dstrbuton, Annals of Mathmatcal Statstcs, 22, 549-557. (95), 3, 229 [3]Ramabhadran,V. R.A multvarat gamma-ty dstrbuton, Sankhya,, (95), 45-46 [4]Krshnaah, P. R., and Rao, M. M. Rmarks on a multvarat gamma dstrbuton, Amrcan Mathmatcal Monthly, 68, (96), 342-346. [5]Sarmanov, I. O.A gnralzd symmtrc gamma corrlaton, Doklady Akadm Nauk SSSR, 79, 279-285; Sovt Mathmatcs Doklady, 9, (968), 547-550. [6]Gavr, D. P. Multvarat gamma dstrbutons gnratd by mxtur, Sankhya, Srs A, 32, (970), 23-26. [7]Johnson, N. L., and Kotz, S. Contnuous Multvarat Dstrbutons, frst dton, Nw York: John Wly &: Sons, 972. [8]Dussauchoy, A., and Brland, R. A multvarat gamma dstrbuton whos margnal laws ar gamma, In Statstcal Dstrbutons n Scntfc Work, Vol. (G. P. Patl, S. Kotz, and J. K. Ord, ds.), (975),- 39-328 [9]Bckr, P. J., and Roux, J. J. J. A bvarat xtnson of th gamma dstrbuton, South Afrcan Statstcal Journal, 5, (98), -2. [0]D'Est, G. M. A Morgnstrn-ty bvarat gamma dstrbuton, Bomtrka, 68, (98). 339-340. []Kowalczyk, T., and Tyrcha, J.Multvarat gamma dstrbutons, rorts and sha stmaton, Statstcs, J. Math. Bol. 26, 465-475, 20, (989).465-474. [2]Matha, A. M., and Moschooulos, P. G. On a multvarat gamma, Journal of Multvarat Analyss, 39, (99).35-53. [3]Matha, A. M., and Moschooulos, P. G. A form of multvarat gamma dstrbuton, Annals of th Insttut of Statstcal Mathmatcs, 44, (992), 97-06. Global Publshng Comany 52