Miin-Jye Wen National Cheng Kung University, City Tainan, Taiwan, R.O.C. Key words: general moment, multivariate survival function, set partition

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1 A Multvarate Webull Dstrbuto Cheg K. Lee Charlotte, North Carola, USA M-Jye We Natoal Cheg Kug Uversty, Cty Taa, Tawa, R.O.C. Summary. A multvarate survval fucto of Webull Dstrbuto s developed by expadg the theorem by Lu ad Bhattacharyya (990). From the survval fucto, the probablty desty fucto, the cumulatve probablty fucto, the determat of the Jacoba Matrx, ad the geeral momet are derved. The proposed model s also appled to the tumor appearace data of female rats. Key words: geeral momet, multvarate survval fucto, set partto. Itroducto Lu ad Bhattacharyya (990) developed a ot survval fucto by lettg h ( ) h ( y ) be two arbtrary falure rate fuctos o [ 0, ), ad H ( x ) ad H ( ) correspodg cumulatve falure rate. Gve the stress Ss > 0, the ot survval fucto codtoed o s, as they defed, s { ( ) ( ) } F( x, y s) exp H x + H y s, x ad y ther where g measures the codtoal assocato of X ad Y. Further, based o the ot survval fucto, they proved a theorem that a bvarate survval fucto F( x, y s ) ca be derved wth the margals F x ad F y gve the assumpto that the Laplace trasform of the stress S exsts o [ 0, ) ad s strctly decreasg. From the theorem, they derved a bvarate Webull Dstrbuto x y F( x, y) exp +, λ λ where 0 <, 0 < λ, λ <, ad 0 <, <. Ths bvarate Webull Dstrbuto s exactly the same as developed by Hougaard (986). By the same steps, the theorem ca be expaded to more tha two radom varables, ad, therefore, a multvarate survval fucto of Webull dstrbuto s costructed as

2 x x x Sx (, x,..., x ) exp , () λ λ λ where measures the assocato amog the varables, 0 <, 0 < λ, λ,, λ <, ad 0 <,,, <. Ths model ca also be derved by usg a copula costructo whch the geerator s ( log (). ) (Frees ad Valdez, 998). Equato s smlar to the geue multvarate Webull dstrbuto developed by Crowder (989) who studed aother verso exteded from the geue multvarate Webull dstrbuto. I ths paper, we mathematcally tesvely studed the proposed multvarate Webull model of Equato by gvg the probablty desty fucto secto, the Jacoba matrx secto 3, the geeral momet secto 4 ad a applcato secto 5.. Probablty Desty Fucto of The Multvarate Webull Dstrbuto The multvarate probablty desty fucto f ( x, x,..., x ) of a multvarate dstrbuto fucto ca be obtaed by dfferetatg the multvarate survval fucto wth respect to each varable. L (997) has show that Sx (, x,..., x ) f ( x, x,..., x ) ( ). x x... x Usg L s dervato ad oe of the specal cases of the multvarate Faa d Bruo formula by Costate ad Savts (996), the probablty desty fucto s x x x f( x, x,..., x ) exp λ λ λ x x x λ λ λ λ λ λ P ( ) x x x ( ) Ps (, ) λ λ λ where s the umber of summads of the th partto of such that + + +, > 0, ; s equal... P s the total, () to ( ) ( ) +, the fallg factoral of (Kuth, 99); ( ) umber of parttos of ; P (, ) s s the total umber of set parttos of the set S {,,} correspodg to the th partto of. The specfc way of parttog

3 ad S s gve by McCullagh ad Wls (988). I ther paper, parttos of are creasg umber of summads ad orderg all the summads verse lexcographc order whe a partto has the same umber of summads, ad S {,, }s parttoed by lstg the blocs from the largest to the smallest ad by breag the tes of equal szed blocs by orderg them lexcographcally ad the umber of blocs a set partto s equal to the umber of summads of the correspodg partto of. For example, the total umber of blocs of the partto of S correspodg to the th partto s ad the umbers of elemets each bloc are equal to,,,. 3. The Jacoba Matrx Smlar to the dervato of the Bvarate Webull Dstrbuto by Lu ad Bhattacharyya (990), let( y, y,..., y ) z z z z + z z z + z z z + z z,,...,,( z + z z ) where z x λ, z x λ,, z x λ. The, x y y λ, x y y λ,, x y y λ, ( ) λ x y y y y. Note that z, z,..., z > 0, ad y y... y z+ z z z+ z z z > 0. z+ z z The Jacoba matrx s (3) J x x x y y y x x x y y y x x x y y y.

4 Let C(,) be the th row ad th colum the Jacoba matrx, the λ y C(,) C(,) C(,) λ y y y λ,,,,-,,,,,-, ( y y y ) y ( y y y ) y λ C(,), C(,)0 whe,,,-,,,-.,,,,-, The determat of the Jacoba matrx ca be obtaed usg Gaussa elmato to costruct a upper tragle matrx. The, the determat s equal to the product of the dagoal elemets. C(, ) C(, ) J C(, ) C(, ) C(, ) ( ) λλ λy y y y y y y After the dervato of the Jacoba, the PDF terms of y, y,, y, g( y, y,, y ) f y y, y y,, y, ( ) y λ λ y y y λ y λ J P( ) y Ps, exp( y) Γ y y... y y y... y f( y ), (5) ( ) ( ) ( ) ( ) ( ) where y, y,, y has a Drchlet dstrbuto wth the probablty desty equal to ( ) y y... y ( y y... y ) Γ, ad f ( y ) ( ) P( ) Γ. (4) y ( ) Ps(, ) exp( y) (6) has a mxture dstrbuto of the expoetal dstrbuto ad Gamma dstrbuto. Equato (6) ca be rewrtte as

5 P( ) exp( ) y y ( ) Γ( ) Ps(, ). Γ( ) Whe t s tegrated over the rage of y, t becomes P( ) ( ) ( ) ( ) Ps(, ) ( ) Γ. Γ That s the weghts of y are summed to. The probablty desty fucto of y s the mxed Gamma dstrbuto by Dowto (969). Followg hs dervato, the cumulatve desty fucto of y s f ( y ) Γ( ) P( ) ( ) y ( ) ( ) Ps ( ) y ( ) (). (7) Γ, exp( ) Γ Γ 4. The Geeral Momet The geeral momet of x, x,, x s E x x x E y y ( ) λ y y λ y y λ y y y y λ λ λ λ ( ) Ey y y y y y E y + + The, Ey y y ( ) y y y Γ( )... ( ) y y y y y y dy dy Γ( ) Γ + Γ + Γ + r r r Γ whch s the Drchlet tegral (Rao, 954.) For E y + +, let c E y c, the

6 ( ) P( ) c y y ( ) Ps(, ) exp( y) dy Γ 0 ( ) P( ) c+ ( ) Ps (, ) y exp( y) dy 0 Γ P( ) ( ) Ps(, ) ( c ) ( ) Γ + Γ P( ) ( ) Ps (, ) ( c) c ( ) Γ Γ (8) where c s the rsg factoral defed as c( c+ ) ( c+ ) by Kuth (99). Cosderg Ps (, ) the above equato, t s the total umber of set parttos correspodg to the th partto of such that + + +,,,, > 0. It has bee show by McCullaph ad Wls (988) that! Ps (, )!!! m! m! m! where m, m,, md are the umber of each dstct summad. The, the product Ps (, )!!!! m! m! m! d!!!! m! m! m!!!!!!! d ( ) ( ) ( )! m! m! md! d!!,! m! m! md!! where m! m! md! s the umber of permutatos of,,, of every possble order. Whe sum Ps (, ) over the same value of, say,, the, Ps (, )

7 !!!!!! m m md!!!!!! m m md whch equals C(,, ), the C-umbers defed by Charalambdes (977). Note that the summato s over all the permutatos of wth. Usg the equalty( ) c ( c) c E y ( ) (Goldma, Joch, Reer ad Whte, 976), P( ) ( ) Ps (, ) Γ( c) c Γ P ( ) ( c) Ps (, ) ( c) ( ) Γ Γ ( c) Ps (, ) ( c) ( ) Γ Γ ( c) C(,, )( c) Γ Γ ( ) ( ) Γ( ) Γ( ) Γ ( c)( c) ( c)( ) ( c) (usg equato.3 by Charalambdes, 977) Γ (usg the formula by Goldma et al. 976) ( ) ( c )( c ) ( c ) ( c ) + + Γ + Γ +. Therefore, the geeral momet of x, x,, x s E x x x + + λ λ λ ( ) Ey y y y y y Ey λ λ λ Γ( ) Γ + Γ + Γ + r r r Γ

8 ( ) Γ ( ) Γ λ λ λ Γ + Γ + Γ + Γ r r r Γ (9) From the geeral momet, the expectato ad the varace of ay radom varable, ad the covarace ad the correlato coeffcet of ay umber of radom varables ca be derved. 5. Applcato We aalyze the data publshed by Matel, Bohdar ad Cmear (997). The data (Table ) cotas 50 ltters of female rats wth oe drug-treated ad two cotrol rats each letter. The same data was also aalyses by Hougaard (986) usg a bvarate Webull dstrbuto. We assume the tme to the appearace of tumor of the treatmet group, the cotrol group ad the cotrol group are Webull dstrbuted. Table dsplays the couts of combatos of cesorg status of ltters. The results of parameter estmates ad stadard errors based o the secod dervatves valued at the maxmzed loglelhood fucto are Table 3. The estmates for the 3 shape parameters are sgfcatly greater tha wth sgfcat level equal to 0.05 dcatg that the 3 group have a mootocally creasg hazard fucto. The estmate of the assocato parameter s ot sgfcatly dfferet from dcatg that the tme to the tumor occurrece amog the 3 groups are ot assocated. Wthout cosderg the stadard errors, by equato (9), the correlato coeffcet of the treatmet group ad cotrol group s The correlato coeffcet of the treatmet group ad the cotrol group s The correlato coeffcet of the two cotrol groups s Table. Tme to tumor appearace wees of treatmet group (T) ad cotrol groups (C, C). Ltter T C C Ltter T C C

9 deote rght cesored tmes Table. Number of ltters of varous cesorg status combatos Cesorg Status (T, C, C) Number of Ltters (0, 0, 0) (0, 0, ) 3 (0,, 0) 6 (, 0, 0) (0,, ) (, 0, ) (,, 0) (,, ) 3 Tumor occurrece s deoted as 0 ad cesored s deoted as Table 3. Maxmum lelhood estmates ad stadard errors. Parameter Estmate Stadard Error Scale (T) Shape (T) Scale (C) Shape (C) Scale (C) Shape (C)

10 Refereces Charalambdes, CH. A. (977) A New Kd of Numbers Appearg The -Fold Covoluto of Trucated Bomal ad Negatve Bomal Dstrbutos. SIAM Joural o Appled Mathematcs 33, Costate, G. M. & Savts, T. H. (996). A Multvarate Faa D Bruo Formula wth Applcatos. Amerca Mathematcal Socety 358, Crowder, M. (989). A Multvarate Dstrbuto wth Webull Coecto. Joural of the Royal Statstcal Socety, Seres B 5, No., Dowto, F. (969). A Itegral Equato Approach to Equpmet Falure. Joural of the Royal Statstcal Socety, Seres B 3, No., Frees, E. W. & Valdez, E. A. (998). Uderstadg Relatoshps Usg Copulas. North Amerca Actuaral Joural, -5. Goldma, J. R., Joch, J. T., Reer, D. L. & Whte, D. E. (976). Roo Theory. II, Boards of Bomal Type. SIAM Joural o Appled Mathematcs 3, Hougaard, P. (986). A Class of Multvarate Falure Tme Dstrbutos. Bometra 73, Lu, J-C. & Bhattacharyya, G. (990) Some New Costructos of Bvarate Webull Models. Aals of The Isttute of Statstcal Mathematcs 4, McCullagh, P. M. & Wls, A. R. (988). Complemetary Set Parttos. Proceedg of the Royal Socety of Lodo 45, Matel, N., Bohdar, N. R. & Cmera, J. L. (977). Matel-Haeszel aalyses of lttermatched tme-to-respose data, wth modfcatos for recovery of terltter formato. Cacer Research 37, Kuth, D. E. (99). Two Notes o Notato. The Amerca Mathematcal Mothly 99, L, C. L. (997). A Model for Iformatve Cesorg, Ph. D. Dssertato, The Uversty of Alabama at Brmgham. Rao, S. K. L. (954). O The Evaluato of Drchlet s Itegral. The Amerca Mathematcal Mothly 66, Rayes, W. S. & Srvasa, C. (994). Depedece Propertes of Geeralzed Louvlle Dstrbutos o the Smplex. The Joural of the Amerca Statstcal Assocato Vol. 89, No. 48,