A new mixed beta distribution and structural properties with applications

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1 Sogklaakar J. Sc. Tchol. 37 (, 97-08, Ja. - F Orgal Artcl A w xd ta dstruto ad structural proprts wth applcatos Tpagor Isuk, Wa Bodhsuwa 2, ad Urawa Jaroratku * Dpartt of Appld Statstcs, Faculty of Appld Scc, Kg Mogkut s Uvrsty of Tchology North Bagkok, Bag Su, Bagkok, 0800 Thalad. 2 Dpartt of Statstcs, Faculty of Scc, Kastsart Uvrsty, Chatuchak, Bagkok, 0900 Thalad Rcvd: 27 Ju 204; Accptd: 2 Dcr 204 Astract I ths papr, w troduc a w sx-paratr dstruto, aly Bta Expotatd Wull Posso (BEWP whch s otad y copoudg tw th xpotatd Wull Posso ad ta dstrutos. W propos ts asc structural proprts such as dsty fucto ad ots for ths w dstruto. W r-xprss th BEWP dsty fucto as a EWP lar coato, ad us ths to ota ts ots. I addto, t also cotas svral su-odls that ar wll kow. Morovr, w apply th axu lklhood thod to stat paratrs, ad applcatos to ral data sts show th suprorty of ths w dstruto y coparg th ftss wth ts su-odls. Kywords: ta xpotatd wull posso, ta-g dstruto. Itroducto For or tha a dcad, Wull dstruto has appld xtsvly ay aras ad or partcularly usd th aalyss of lft data for rlalty grg or ology (R, Howvr, th Wull dstruto has a wakss for odlg phoo wth o-ooto falur rat. Thrfor Mudholkar ad Srvastava (993 proposd th xpotatd Wull (EW dstruto that s a xtso of th Wull faly, otad y addg a scod shap paratr. Th t s flxl to odl survval data whr th falur rat ca crasg, dcrasg, athtu shap, or uodal (Mudholkar t al., 995. Lt W a rado varal of th EW dstruto. Th th cuulatv dstruto fucto (cdf ad proalty dsty fucto (pdf of W ar gv y w F w, w 0, * Corrspodg author. Eal addrss: ur@kut.ac.th whr a,,q >0 ad w w f w w, w 0, rspctvly. For survval aalyss, thr ar portat aalytcal fuctos such as survval ad hazard rat fuctos gv y ad S w w w w w hw w. rspctvly. Rctly, ay rsarchrs hav attptd to odfy EW dstruto wth dffrt tchqus y usg EW as th asl dstruto to dvlop or flxlty. Pho t al. (202 proposd gaa xpotatd Wull, Sgla t al. (202 studd ta gralzd Wull (BGW, Cordro t al. (203 troducd th ta xpotatd Wull (BEW ad xpotatd Wull Posso (EWP

2 98 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, 205 was proposd y Mahoud ad Spahdar (203. I ths papr, w propos a w flxl sx-paratr dstruto calld Bta Expotatd Wull Posso (BEWP dstruto. Th purpos of ths study s to crat a w dstruto y xg EWP dstruto ad th ta dstruto. So proprts of ths w dstruto wll vstgatd. Th BEWP dstruto s dvlopd y usg th Bta-G dstruto class that was troducd y Eug t al. (2002 who also proposd th ta oral (BN dstruto. Th, usg th Bta-G dstruto class was appld to crat a w dstruto xtsvly. For xapl, Nadaraah ad Gupta (2004 proposd th ta Frcht (BF dstruto ad Nadaraah ad Kotz (2004 studd th ta Gul (BGu dstruto. Nadaraah ad Kotz (2006 troducd th ta xpotal (BE dstruto, L t al. (2007 proposd th Bta Wull (BW, Mahoud (20 proposd th Bta Gralzd Parto (BGP, BGW or BEW ad Prcot t al. (203 studd th Bta Wull Posso (BWP. Th EWP dstruto fts th skwd data (Mahoud ad Spahdar, 203 ad t s usful for solvg copltary rsks prol (Basu ad Kl, 982 th prsc of latt rsks, th ss that thr s o forato aout whch factor s rsposl for th copot falur ad oly th axu lft valu aog all rsks s osrvd. Mxg th EWP dstruto wth th ta dstruto causs th two addtoal shap paratrs whch srv to cotrol skwss ad tal wghts of EWP dstruto. As a rsult, BEWP dstruto s th gralzd dstruto that has a wd varty trs of shap of th dstruto, so t s a flxl altratv for applcatos grg ad ology. I grg applcatos, th BEWP dstruto ca ployd rlalty aalyss, such as product rlalty ad syst rlalty. Prcot t al. (203 appld th BWP dstruto to th atac data o actv rpar ts for aror coucato. I addto, for ology or dcal scc, w ay apply to survval aalyss.g. Mudholkar t al. (996 appld th gralzd Wull dstruto fttg th ral survval t data of th patts who wr gv radato thrapy ad chothrapy fro had ad ck cacr clcal tral. Dasgupta t al. (200 studd th charactrstcs of coroary artry calcu whch s a arkr of coroary artry dsas. Ths appars to a Wull dstruto ad a Wull rgrsso odl was proposd to xa factors flucg th dsas. Ortga t al. (203 dvlopd th ta Wull dstruto to th log-ta Wull dstruto ad studd th log-ta Wull rgrsso odl wth applcato to prdct rcurrc of prostat cacr. Th rst of ths papr s orgazd th followg squc. Scto 2 dscusss aout th xpotatd Wull Posso (EWP dstruto that s usd as th asl to dvlop th BEWP dstruto. Th proalty dsty fucto (pdf ad cuulatv dsty fucto (cdf ar troducd Scto 3. Scto 4 gvs a suary of su-odls of BEWP th for of tal ad chart whr svral su-odls ar wll kow. Scto 5 dscusss th ot gratg fucto (gf ad th ot. I Scto 6 w apply th axu lklhood thod to stat paratrs, ad Scto 7 copars th su-odls of th BEWP dstruto y th applcatos to ral data sts. So cocludg rarks ar gv Scto Th Expotatd Wull Posso dstruto Lt W, W2, W3,..., W z dpdt ad dtcally dstrutd rado varals fro xpotatd Wull dstruto wth pdf w w f w;,, w, w 0, ad Z, whch s dpdt fro W s, a rado varal fro zro trucatd Posso dstruto wth proalty ass fucto (pf z p z; z, 0, z,2,3,... whr s th gaa fucto. Prcot t al. (203 dscrd th odl for X W, W2,..., Wz ad X = ax W, W2,..., W z that ca usd sral ad paralll syst wth dtcal copots, whch appar ay dustral applcatos ad ologcal orgass. For ths odl, w df. W assu th falur occurs aftr all Z factors hav actvatd. Th w ota ( z g x z;,, z w u u u, x 0, x whr u ad th argal pdf of X s, 0, u g x x u u x ad th cdf of X s u G x. (2 Th w tak ths pdf ad cdf of EWP to th asl for cratg th w Bta-G dstruto th xt scto. W apply th trprtato of th EWP fro Adads ad Loukas (998, that th falur (of a dvc, for xapl occurs du to th prsc of a ukow ur, Z, of tal dfcts of th sa kd (a ur of scoductors fro a dfctv lot, for stac. Th W s rprst thr lfts ad ach dfct ca dtctd oly aftr causg falur, whch cas t s rpard prfctly. 3. Th Bta Expotatd Wull Posso dstruto Dfto : Lt F(x th cdf of a rado varal X. Accordg to Eug t al. (2002, th cdf for a gralzd class of dstrutos ca dfd as th logt of ta rado varal y (

3 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, Gx a, 0, (3 0 F x w w dw x whr a,>0. G(x dots th proalty dstruto of th gralzd class of dstruto. Th pdf of X s gv y (for or dtals s Mood t al. (974 whr a f x g x G x G x, x 0, (4 g x dg x. dx Thor : Lt X a rado varal of th BEWP dstruto wth paratrs,,,, a ad. Th pdf of X s dfd y f x whr u u a u x u u Ba, (5 u x Proof: Sply y usg Dfto, w ota th pdf of X y susttutg g(x ad G(x fro Eqs.( ad (2 to Eq.(4. Th Eq.(5 s th pdf of BEWP dstruto as th followg proprty. a u u u x u u f xdx dx 0 0 Ba, u lt v, t ca rwrtt as 0 Ba, a v v f xdx dv Not that, w ca df th xpaso of th pdf as th lar coato of EWP dsty fucto as whr,, f x s g( x;,,, (6 s 0 o s (+, (, ( - + ad s Lt a o-tgr ral ur ad w. Gx a F x w w dw a x 0,,, 0 By usg th spcal cas of oal thor ( w, w 0 r. hc Gx a F x w ( w dw Ba, 0 0 ( G x a 0 0 a a c ( a, G x, (7 whr c ( a, (. If s a tgr, th dx a stops at -. W ota f(x for tgr a Cosdr th cas whr th powr of G(x s a o-tgr G x whr d ( ( ( G x 0 (, th w susttut!! G x for G x a Eq. (7, w ota ad F x 0 r ( a, G x f x g(x ( +r ( a, G x 0 whr r ( a, c ( a, d (a+ or whr s 0 f x s ( +g(xg x 0, r ad w ca xprss pdf of BEWP ds- truto trs of a lar coato of EWP dstruto as u u x u u f x s ( o s ( + ( ( - + u ( - + x u u, s ( + (, u, x u u, 0 o ( - +

4 00 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, o s g( x;,,,,, whr ad s, s ( +, (, ( - + To show th varous of shaps of th dstruto, so spcfd paratrs of th BEWP dstruto ad thr dsty fuctos ar provdd Fgur : (a Fx paratrs = 4, = 0.5, =.5, = 0.5 ad vary paratrs a ad, ad ( Fx paratrs a = 2, = 2 ad vary paratrs,, ad. Thus, th BEWP dstruto ca sutal for fttg to varous shaps of data, for xapl whr th paratrs a = 2, = 2, = 4, = 0.5, =.5, = 0.5. Ths dstruto s sutal for fttg skwd data ad t s sutal for fttg uodal data wh th paratrs a = 2, = 2, = 0.5, = 2, = 0., = 5. Accordg to Fgur, BEWP dstruto ca a faly of dstrutos cotag 32 su-odls whch wll dscussd Scto 4. Thor 2: Lt X a rado varal of a BEWP dstruto wth paratrs,,,,a ad. Th cdf of X s gv y or whr u ( /( a (8 0 F x w w dw u F x I ( a, (9 u x ( /( Proof: Sply y usg Dfto aga, w ca df th cdf of X y rplacg G(x fro Eq.(2 Eq.(3, hc th cdf of BEWP dstruto s as otad Eq.(8. Not that, w ca df th xpaso of th cdf as th lar coato of EWP dsty fucto gv y F(x s G( x;,,, (0 0 0 y tgratg f(x Eq.(6 x,, F(x s g( x;,,,,, s x, Su-odls s,,, u,, u x u u,,, s G( x;,,, Ths w dstruto cossts of a total of 32 sudstruto odls as show Fgur 2 ad Tal. I Tal, th su-odls assocatd wth Posso dstruto, ar assgd whch asd o th paralll copots syst coply wth BEWP dstruto that s udr th sa assupto. For, w also rfr to th Rfrcs colu ad ark wth th astrsk syol (* Tal. 5. Mot Gratg Fucto ad Mot Thor 3: Lt X a rado varal of a BEWP dstruto wth paratrs,,,, a ad. Th ot gratg fucto (gf of X ca gv y dx Fgur. Dsty fucto of th BEWP dstruto.

5 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, Fgur 2. Th su-odl chart of BEWP dstruto Tal. Th su-odl tal of BEWP dstruto. 5 paratrs paratrs Dstruto F(x Rfrcs a. Bta Expotatd a 2 F( x I 2 a, ( x Raylgh Posso (BERP 2. Bta Expotatd a Expotal Posso (BEEP F ( x I a, ( x 3. Bta Wull Posso a F( x I a, Prcot t al. (203* ( x (BWP 4. Bta Expotatd a 0 F( x I a, Wull (BEW 4 paratrs 5. Bta Raylgh Posso a 2 (BRP Sgla t al. (202, x ( Cordro t al. (203 F( x I a, 2 x ( 6. Bta Expotal Posso a F( x I x a, (BEP 7. Gralzd Wull a Posso (GWP F( x ( ( x a x ( 8. Expotatd Wull F( x Mahoud ad Posso (EWP Spahdar (203

6 02 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, 205 Tal. Cotud paratrs Dstruto F(x Rfrcs a 9. Gralzd Expotatd a 0 Wull (GEW F( x x a 0. Bta Expotatd a 2 0 F( x I ( a, ( x 2 Raylgh (BER Cordro t al. (203a. Bta Expotatd a 0 F( x I a, x Barrto-Souza t al. Expotal (BEE ( Bta Wull (BW a 0 F ( x I a, L t al. (2007 x 3 paratrs 2 a x ( 3. Gralzd Raylgh a 2 F( x Posso (GRP x a ( 4. Gralzd Expotal a F( x Barrto-Souza, ad Posso (GEP Crar-Nto (2009 ( 5. Expotatd 2 F( x Mahoud ad Spahdar Raylgh Posso (ERP ( Expotatd Expotal Posso (EEP 7. Wull (WP 8. Expotatd Wull 0 (EW 9. Gralzd Wull(GW a Gralzd Expotatd a 2 0 Raylgh(GER 2. Gralzd Expotatd a 0 Expotal(GEE F( x F( x x 2 ( x ( x x Prcot t al. (203*, Rstæ ad Nadaraah (204*, Mahoud ad Spahdar (203 Hat t al. (20*, Lu ad Sh (202*, Mahoud ad Spahdar (203 F( x ( Mudolkar ad Srvastava (993 cocd wth GW a x F( x Mudolkar ad Srvastava (993 cocd wth EW x 2 a F( x Cordro t al. (203a F( x a x 22. Bta Raylgh (BR a 2 0 F( x I 2 a, x 23. Bta Expotal(BE a 0 F ( x I, x a 2 paratrs 24. Raylgh Posso (RP Expotal Posso (EP 26. Wull (W 0 F ( x F( x x ( 2 x ( Nadaraah ad Kotz (2006 Mahoud ad Spahdar (203 Kus (2007*, Cacho t al. (20 x Mudolkar ad Srvastava (993 F( x (

7 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, Tal. Cotud paratrs Dstruto F(x Rfrcs a 27. Expotatd Raylgh 2 0 (ER 28. Gralzd Raylgh (GR a Expotatd Expotal 0 (EE 30. Gralzd Expotal a 0 (GE 3. Raylgh(R Expotal(E 0 x 2 F( x Kudu ad Raqa (2005 cocd wth GR 2 a Kudu ad Raqa (2005 cocd wth ER F( x x x Gupta ad Kudu (999 cocd wth GE a Gupta ad Kudu (999 cocd wth EE 2 Johso t al. (994 Johso t al. (994 F( x ( F( x x F( x x F( x x X M t t ( 0 whr,, l,, ad = 0,,2,, l, 0 0 Proof: To fd th gf of BEWP dstruto, w apply th dfto of gf to th lar coato of EWP dsty fucto as tx, ;,,,, M t s g x dx X BEWP s M, X EWP, t;,,,, whr Mahoud ad Spahdar (203 drvd that,, l M X t l EWP, W ca rduc to t. 0 0 l0!! l M X t,,,,, BEWP l t, l,, 0 0 whr s,, l,, l,, l.,!! l Ad w ca rduc aga to M X t t BEWP 0 whr,, l,, ad = 0,,2,, l, 0 0 Thor 4: Lt X a rado varal of a BEWP dstruto wth paratrs,,,,a ad. Th ot of X ca wrtt as E X s!,, l l,,, l 0 0 l (2 Proof: To fd th ot of BEWP dstruto, w apply th dfto of ot aga to th lar coato of EWP dsty fucto as, ;,,,, E X s x g x dx BEWP s E X ;,,,,, EWP, whr Mahoud ad Spahdar (203 drvd that EEWP X l!, l., 0 0 l So w ca ota th ot of BEWP dstruto as E X, s, l l.,,, l 0 0! l Th w ca fd varac, skwss ad kurtoss of rado varal X y usg th wll-kow rlatoshp of ach ot. 6. Paratr Estato I ths scto, w suppos that th sapl sz was T draw fro BEWP dstruto, ad lt π,,,, a, th paratr vctor. Th th log-lklhood fucto of

8 04 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, 205 BEWP s gv y,,,,, log log log log log, l a B a whr u u x u u a log log u log a log x. So th lts of scor vctor l l l l l l U,,,,, a whr l log u u T a u u log u u u log u u x u logx log x x l log log log u x u log x log u x a x u log x log u u x x u log x log u l u u u x x x u a x u u x x u x u u u l a a u u u u u u l u a a log log a l u a log log Γ ( x whr x s th dgaa fucto. Th axu Γ( x T lklhood stator π ˆ ˆ, ˆ, ˆ, ˆ, aˆ, ˆ s th soluto to th aov scor quatos that ar calculatd y usg Nwto-Raphso thod R packag (R Cor Ta, Applcatos I ths scto, to rval th suprorty of BEWP dstruto, w ft a BEWP odl to two ral data sts fro th applcato of EWP (Mahoud ad Spahdar, 203. Th frst applcato, w study th skwd data rprstg strgths of.5 c glass frs, asurd at th Natoal Physcal Laoratory, Eglad, whch ar gv Tal 2. Ufortuatly, th uts of asurt ar ot gv th papr. For th scod applcato, w xa th data showg th strss-ruptur lf of Kvlar 49/poxy strads (ut: hours whch wr suctd to costat sustad prssur at th 90 strss lvl utl all had fald as dsplayd Tal 3. W ft th BEWP dstruto to aov two data sts ad copar th ftss wth ts su-odls that ar BWP, BEW, BEE, EWP, EW, EE cludg Wull dstruto y cosdrg th p-valu of Kologorov-Srov (K-S statstcs. Th axu lklhood stats of th paratrs, th K-S statstcs ad th corrspodg p-valu for th fttd odls ar show for data sts I ad II Tals 3 ad 4, rspctvly. Graphcal approach s th aothr way to xprss ths data sts ft wth ths dstruto. W also prst th coparso of th prcal cdf wtth ach statd cdf Fgur 3. It shows th fttg of data to proposd odls. Th proalty plots of th BEWP dstruto corrspodg to data sts I ad II Fgur 4 dcat that (a ost data l aroud th straght l spcally th ddl 50% of th data, ad ( th frst 75% of th data l o th straght l ad th last 25% of th data l aov, whch suggsts a slght rght-skwss. It ss rasoal to ttatvly coclud that oth data sts ar BEWP dstruto. 8. Cocluso For ths papr, a w sx-paratr dstruto, aly BEWP s studd. It s otad y copoudg ta ad xpotatd Wull Posso dstrutos. W troduc ts asc athatcal proprts such as dsty fucto. W show that th pdf of BEWP dstruto ca xprssd th lar coato for of EWP dstruto cludg ts ots. Morovr, t also cotas th ay su-odls that ar wll kow. Fally, w hav appld th Tal 2. Strgths of.5 c glass frs

9 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, Tal 3. Strss-ruptur lf of Kvlar 49/poxy strads (ut: hours Tal 4. MLE ad K-S statstcs wth corrspodg p-valus for th strgths of.5 c glass frs. Fttg Dstruto Paratrs a K-S p-valu BEWP BWP BEW BEE EWP EW EE Wull axu lklhood thod to stat paratrs ad ft th BEWP dstruto to two ral data sts. W copard th rsults wth ts su-odls such as BWP, BEW, BEE, EWP, EW, EE ad Wull dstruto. Th rsults showd that BEWP dstruto provds a ttr ft tha xstg xturs of th EW or Wull dstruto ad so wllkow su-odls. Ackowldgts Th authors would lk to thak th rfrs ad th dtor for valual cots ad suggstos to prov ths work. Rfrcs Adads, K. ad Loukas, S A lft dstruto wth dcrasg falur rat. Statstcs ad Proalty Lttrs. 39, Barrto-Souza, W. ad Crar-Nto, F A gralzato of th xpotal-posso dstruto. Statstcs ad Proalty Lttrs. 79, Barrto-Souza, W., Satos, A.H.S. ad Cordro, G.M Th ta gralzd xpotal dstruto. Joural of Statstcal Coputato ad Sulato. 80, Basu, A.P. ad Kl, J.P So rct rsults coptg rsks thory. Survval Aalyss. 2, Cacho, V.G., Louzada-Nto, F. ad Barrga, G.D.C. 20. Th Posso-xpotal lft dstruto. Coputatoal Statstcs ad Data Aalyss. 55, Cordro, G.M., Crsto, C.T., Hashoto, E.M., ad Ortga, E.M.M. 203 (a. Th ta gralzd Raylgh dstruto wth applcatos to lft data. Statstcal paprs. 54, Cordro, G.M., Gos, A.E., da-slva, C.Q. ad Ortga, E.M.M. 203 (. Th ta xpotatd Wull dstruto. Joural of Statstcal Coputato ad Sulato. 83, Dasgupta, N., X, P., Chy, M.O., Brolg, L. ad Jr., C.H.M Th Spoka hart study: wull rgrsso ad coroary artry dsas. Coucatos Statstcs Sulato ad Coputato. 29,

10 06 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, 205 Fgur 3. Coparso tw pral cdf ad statd cdf of data sts I ad II.

11 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, Fgur 3. Coparso tw pral cdf ad statd cdf of data sts I ad II. (Cotud Fgur 4. Th proalty plot of th BEWP dstruto of data sts I ad II.

12 08 T. Isuk t al. / Sogklaakar J. Sc. Tchol. 37 (, 97-08, 205 Tal 5 MLE ad K-S statstcs wth corrspodg p-valus for th strss-ruptur lf of Kvlar 49/poxy strads (ut: hours Fttg Dstruto Paratrs a K-S p-valu BEWP BWP BEW BEE EWP EW EE Wull Eug, N., L, C. ad Faoy, F Bta-Noral dstruto ad ts applcatos. Coucatos Statstcs - Thory ad Mthods. 3, Gupta, R.D. ad Kudu, D Gralzd xpotal dstrutos. Australa ad Nw Zalad Joural of Statstcs. 4, Hat, F., Khorra, E. ad Rzakhah, S. 20. A w thrparatr agg dstruto. Joural of Statstcal Plag ad Ifrc. 4, Johso, N.L., Kotz, S. ad Balakrsha, N Cotuous Uvarat dstrutos. Wly ad Sos, Nw York, U.S.A.,Vol., pp. 456,494. Kudu, D. ad Raqa, M.Z Gralzd Raylgh dstruto : dffrt thods of statos. Coputatoal Statstcs ad Data Aalyss. 49, Kus, C A w lft dstruto. Coputatoal Statstcs ad Data Aalyss. 5, L, C., Faoy, F. ad Oluolad, O Bta-Wull dstruto: So proprts ad applcatos to csord data. Joural of Modr Appld Statstcal Mthods. 6, Lu, W. ad Sh, D A w copoudg lf dstruto: th Wull-Posso dstruto. Joural of Appld Statstcs. 39, Mahoud, E. 20. Th ta gralzd Parto dstruto wth applcato to lft data. Mathatcs ad Coputrs Sulato. 8, Mahoud, E. ad Spahdar, A Expotatd Wull- Posso dstruto: Modl, proprts ad applcatos. Mathatcs ad Coputrs Sulato. 92, Mood, A.M., Grayll, F.A. ad Bos, D.C.974. Itroducto to th thory of statstcs. McGraw-Hll, Nw York, U.S.A., pp.532. Mudolkar, G.S. ad Srvastava, D.K Expotatd Wull faly for aalyzg athtu falur-rat data. IEEE Trasactos o Rlalty. 42, Mudholkar, G.S., Srvastava, D.K. ad Frr, M Th Expotatd Wull Faly: A raalyss of th us-otor-falur data. Tchotrcs. 37, pp Mudholkar, G.S., Srvastava, D.K. ad Kolla, G.D A gralzato of th Wull dstruto wth applcato to th aalyss of survval data. Joural of th Arca Statstcal Assocato. 9, Nadaraah, S., Gupta, A.K Th ta Frécht dstruto. Th Far East Joural of Thortcal Statstcs. 4, Nadaraah, S., ad Kotz, S Th ta Gul dstruto. Mathatcal Prols Egrg. 0, Nadaraah, S., ad Kotz, S Th ta xpotal dstruto. Rlalty Egrg ad Syst Safty. 9, Ortga, E.M.M., Cordro, G.M., ad Katta M.W Th log-ta Wull rgrsso odl wth applcato to prdct rcurrc of prostat cacr. Statstcal Paprs. 54, Prcot, A., Blas, B. ad Cordro, G.M Th ta Wull Posso dstruto. Chla Joural of Statstcs. 4, Pho, L.G.B., Cordro, G.M., ad Nor, J.S Th Gaa-Expotatd Wull dstruto. Joural of Statstcal Thory ad Applcatos. (4, R Cor Ta, 202. R: A laguag ad vrot for statstcal coputg. R Foudato for Statstcal Coputg, Va, Austra. R, H Th Wull Dstruto: A Hadook, Chapa ad Hall/CRC, U.S.A., pp Rstæ, M.M. ad Nadaraah, S A w lft dstruto. Joural of Statstcal Coputato ad Sulato. 84, Sgla, N., Ja, K. ad Shara, S.K Th Bta Gralzd Wull dstruto: Proprts ad applcatos. Rlalty Egrg ad Syst Safty. 02, 5-5.