1. Introduction. Joseph Nderitu Gitahi *, John Kung u, Leo Odongo

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1 Arica Joural of Tortical ad Applid Statistics 07; 6(3: 4-49 ttp:// doi: 0.648/.atas SSN: (Prit; SSN: (Oli Estiatio of t Paratrs of Poisso-Epotial Distributio Basd o Progrssivly Typ Csorig Usig t Epctatio Maiizatio (E Algorit Josp Ndritu Gitai *, Jo Kug u, Lo Odogo Dpartt of Statistics ad Actuarial Scic, Kyatta Uivrsity (KU, Nairobi, Kya Eail addrss: drituosp3@gail.co (J. N. Gitai, okugu08@yaoo.co (J. Kug u, lo_odogo@yaoo.co.uk (L. Odogo * Corrspodig autor To cit tis articl: Josp Ndritu Gitai, Jo Kug u, Lo Odogo. Estiatio of t Paratrs of Poisso-Epotial Distributio Basd o Progrssivly Typ Csorig Usig t Epctatio Maiizatio (E Algorit. Arica Joural of Tortical ad Applid Statistics. Vol. 6, No. 3, 07, pp doi: 0.648/.atas Rcivd: Marc 6, 07; Accptd: April 6, 07; Publisd: April 7, 07 Abstract: Tis papr cosidrs t paratr stiatio probl of tst uits fro Poisso-Epotial distributio basd o progrssivly typ rigt csorig sc. T aiu likliood stiators (MLEs for Poisso-Epotial paratrs ar drivd usig Epctatio Maiizatio (EM algorit. EM-algorit is also usd to obtai t stiats as wll as t asyptotic variac-covariac atri. By usig t obtaid variac-covariac atri of t MLEs, t asyptotic 95% cofidc itrval for t paratrs ar costructd. Troug siulatio, t bavior of ts stiats ar studid ad copard udr diffrt csorig scs ad paratr valus. t is cocludd tat for a icrasig sapl siz; t stiatd valu of t paratrs covrgs to t tru valu, t variacs dcras ad t widt of t cofidc itrval bco arrowr. Kywords: Poisso-Epotial Distributio, Progrssiv Typ Csorig, Maiu Likliood Estiatio, EM Algorit. troductio t statistical litratur o ca fid urous distributios for odllig lif ti data. lif ti study, potial distributio is o of t ost discussd distributios du to its siplicity ad asy atatical aipulatios. Howvr, its us is iappropriat i tos situatios wr associatd azard rat is ot costat. A ubr of lif ti distributios avig o-costat azard rat ar availabl i t litratur.g., Gaa, Wibull, Epotiatd Epotial tc. Ts distributios ar gralizatio of Epotial distributio ad possss icrasig, dcrasig or costat azard rat dpdig o t valu of t sap paratrs ad rduc to potial distributio for tir spcific coics of t sap paratr. A odificatio i potial distributio was proposd by Kus [] to gt a dcrasig failur rat distributio by fidig t distributio of t iiu of idpdtly, idtically ad potially distributd rado variabls wr is rado followig zro trucatd poisso distributios. Sic t distributio is obtaid troug t copoudig of poisso ad potial. Furtr Barrto ad Cribari [] gralizd t distributio proposd by Kus by icludig a powr paratr. Caco t al. [3] proposd a w faily of distributio, calld Poisso-Epotial (PE distributio avig icrasig failur rat. T distributio as b obtaid by fidig t distributioof t aiu of idpdtly,idtically ad potially distributd rado variabls wr is radofollowig,zro trucatd Poisso distributio. tis study, w assu tat t liftis av Poisso- Epotial distributio. T otivatio for tis faily of distributio ca also b tracd i t study of copltary risk(cr probls i prsc of latt risks i., for tos situatios w oly lif ti valus ar obsrvd but o iforatio is availabl about t factors rsposibl for copot failurs. For otr datails rgardig CR ad rlatd odls,t radrs ay rfr Basu ad Kli [4] ad

2 4 Josp Ndritu Gitai t al.: Estiatio of t Paratrs of Poisso-Epotial Distributio Basd o Progrssivly Typ Csorig Usig t Epctatio Maiizatio (E Algorit Adaidis ad Loukas [5]. ral lif, sotis it is ard to gt a coplt data st; oft t data ar csord. Scitific prits igt av to stop bfor all its fail bcaus of t liit of ti or lack of oy. Tis rsults to availability of csord data. Typ- ad Typ- csorig ar t ost basic aog t diffrt csorig scs. Typ- csorig apps w t prital ti T is fid, but t ubr of failurs is rado. Typ- csorig occurs w t ubr of failurs r is fid, t prital ti is rado. Vast litratur is availabl o ts two csorig scs ad o ay rfr to Bai ad Eglardt [6] for dtaild discussio o various aspcts of ts scs. Ufortuatly,ts tods do ot allow t roval of uits bfor t copltio of t prit. Howvr, i dical ad grig survival aalysis,roval of its ay occur at itrdiat stps also du to various rasos wic ar byod t cotrol of t pritr. For suc a situatio,progrssiv csorig is a appropriat csorig sc as it allow t roval of survig its bfor t triatio poit of t tst. Trfor, i tis study, w will focus o progrssiv csorig du to its flibility tat allows t pritr to rov activ uits durig t prit. May autors av discussd ifrc udr progrssiv csorig usig diffrt lifti distributios, icludig Co, [7], Aggarwala [8] ad Aal t al [9]. For a coprsiv rct rviw of progrssiv csorig, radrs ay rfr to Balakrisia [0]. Lt X b a o-gativ rado variabl dotig t lif ti of a copot/syst. T rado variabl X is said to av a PE distributio wit paratrs ad, if its probability dsity fuctio (pdf is giv by, f (,, =, > 0 > 0, > 0 ( T corrspodig cuulativ distributio fuctio (cdf is giv by, F( =, >0 > 0, > 0 ( Wr is t scal paratr, wil is sap paratr of t distributio. Louzada-Nto t al. [], poitd out tat t paratrs ad of t distributio av dirct itrprtatio i trs of copltary risk. fact rprsts t a of t ubr of copltary risk wras dots t lifti failur rat. frtial issus for t Poisso-Epotial distributio basd o coplt data av b addrssd by Louzada- Nto t al wo studid t statistical proprtis of PE distributio ad discussd about t Bays stiators udr squard rror loss fuctio (SELF.Sig t al. [] obtaid t aiu likliood stiators ad Bays stiators of t paratrs udr sytric ad asytric loss fuctio for Poisso-potial distributio ad copard t proposd stiators wit aiu likliood stiators i trs of tir risks. Raqab ad Madi [3] discussd t classical ad Baysia ifrtial procdur for progrssivly typ csord data fro t gralizd Raylig distributio. T rsults sowd tat t aiu likliood stiators of t scal ad sap paratrs ca b obtaid via EM algorit basd o progrssiv csorig. Krisa ad Kuar [4] discussd t ifrc probls i Lidly distributio ad t rsults sows tat Lidly distributio provid good paratric fit udr progrssiv csorig sc for so ral lif situatios. Also, so of t rct work o progrssiv csorig iclud but ot liitd to Kuar t al. [5], Pak t al. [6] ad Rastogi ad Tripati [7]. As far as w kow, o o as dscribd t EM algorit for dtriig t MLEs of t paratrs of t Poisso-Epotial distributio basd o progrssiv typ- csorig sc. tis study, w propos to us EM algorit for coputig MLEs. Tis is bcaus t EM algorit is rlativly robust agaist t iitial valus copard to t traditioal Nwto-Rapso (NR tod as sow by Wataab ad Yaaguci [8] ad Ng t al. [9]. t guarats a sigl uifor o-dcrasig likliood trial fro t iitial valu to t covrgc valu. Morovr, wit t EM algorit, tr is o d to valuat t first ad scod drivativs of t log-likliood fuctio, wic lps sav t ctral procssig uit (CPU ti of ac itratio. T Epctatio aiizatio algorit is coputatioal stabl, asy to iplt ad asyptotic variacs ad covariac ar also obtaid. For or rct rlvat rfrcs o EM algorit ad csorig iclud [0-]. T purpos of tis study is to stiat t sap ad scal paratrs of t Poisso-Epotial distributio udr progrssiv typ- csorig usig t EM algorit ad to copar t rsults udr diffrt csorig scs. T rst of tis papr is orgaizd as follows:sctio,provids a brif dscriptio of Progrssiv typ csorig sc. Furtror, t asyptotic variac ad covariac of t aiu likliood stiats wic ar gratd troug EM algorit ar giv. Siulatio study is coductd i sctio 3. Fially, coclusio ad rcodatio ar prstd i sctio 4.. Paratr Estiatio.. Progrssiv Typ- Csorig Sc Suppos tat uits ar placd o a lif tst at ti 0. Prior to t prit, a ubr (< is fid ad t csorig sc R = R, R,..., Rar prdtrid wit R 0 ad R + = is spcifid. At t first failur = ti X : :, R uits, cos at rado, ar rovd fro t survivig uits. At t scod failur ti X : :,

3 Arica Joural of Tortical ad Applid Statistics 07; 6(3: R radoly cos uits fro t raiig R t uits ar rovd. T tst cotius util t failur ti X : :. At tis ti, all raiig uits ar rovd; tr ar R = R of ts. T st of obsrvd = lifti : : : : : : X = X, X,..., X is a progrssivly Typ rigt csord sapl as rffrd by Balakrisa ad Aggarwala [3]... Maiu Likliood Estiatio Basd o Progrssiv Typ- Csorig Suppos idtical uits ar placd o a lifti tst. At t t ti of t i failur, R i survivig uits ar radoly witdraw fro t prit, i. Tus, if failurs ar obsrvd t R + R R uits ar progrssivly csord; c = + R + R R, R R R : : : :... : : X X X dscrib t progrssivly csord failur tis, wr R ( R R R =,..., dots t csorig sc. f t failur tis of t its origially o tst ar fro a cotiuous populatio wit pd.. f f ad cdf F( giv by quatio ( ad ( ( rspctivly, t t oit probability dsity fuctio for R R R X X... X is giv by, : : : : : : R R R R R (... = ( ( ( f A f F,,..., : : : : : : = : : : : R R R Wr : : : :... : : R (3 < < ad A = ( R ( R R...( R R... R + Fro quatio ( ad (, t likliood fuctio basd o progrssivly Typ csord sapl is giv by; ( ( ( L(, = A (4 = T log-likliood fuctio of quatio (4 ca b writt as follows ( ( ll, = cost + l l( ( + Rl = = = Diffrtiatig (5 w. r. t. (wit rspct to to ad ad quatig t drivativs to zro, w gt t followig oral quatios: R (5 0 ( R = (6 = + = ( + R = 0 = = = (7 T oral quatios (6 ad (7 ar iplicit syst of quatios i ad. Ty caot b solvd aalytically. Trfor, w propos to us EM algorit for solvig ts quatios urically, for aiu likliood stiat of ad..3. Epctatio-Maiizatio (EM Algorit T E M algorit was itroducd by Dpstr t al. [4] to adl ay issig or icoplt data situatio. McLacla ad Krisa[5] discussd EM algorit ad its applicatios. T progrssiv typ- csorig ca b viwd as a icoplt data st, ad trfor, t EM algorit is a good altrativ to t NR tod for urically fidig t MLEs. X = X, X,..., X wit Lt : : : : : : X: : < X: : <...< X: : dots t progrssiv typ- rigt-csord data fro a populatio wit pdf ad cdf giv i Equatios ( ad (, rspctivly. For otatio siplicity, w will writ X for X : :. =,,..., wit Z ( Z, Z,..., Z R Lt Z ( Z Z Z =, =,,..., b t csord data. W cosidr t csord data as issig data. T cobiatio of X, Z = Y fors t coplt data st. T Likliood ( fuctio basd o Y is L( Y,, = z R k zk (8 = k = T log-likliood fuctio basd o Y is ( R ( ( k z k (9 ll( Y,, = l( l + z + = = k = T MLEs of t paratrs ad for coplt sapl Y ca b obtaid by drivig t log-likliood fuctio i Equatio (9 wit rspct to ad ad quatig t oral quatios to 0 as follows: ll( Y,, zk = = 0 (0 R = = k = R R zk zk zk = = = k = = k = ll( Y,, = + + = 0 ( To start t algorit, t oit distributio of ad z is giv by, ( z z f(, z = p( z( z = z > 0, z =,, 3... z! ( ( Wr > 0ad > 0ar paratrs. t is straigtforward to vrify tat t coputatio of t coditioal pctatio

4 44 Josp Ndritu Gitai t al.: Estiatio of t Paratrs of Poisso-Epotial Distributio Basd o Progrssivly Typ Csorig Usig t Epctatio Maiizatio (E Algorit of (Z X usig t pdf is giv by ( z z f(, z p( z = = f( Siplifyig (3, w gt ( ( z ( z! z z + f(, z p( z = = f(! z (3 (4 Tus it is straigtforward to vrify tat t E-stp of a EM cycl rquirs t coputatio of t coditioal pctatio (, Z X, wr (, stiats of (,. z= is t currt E( z,, = zp( z,, (5 Usig quatio (5, w gt, z= ( z z + E( z,, = z (6! ( z Siplifyig (6, w gt ( E( z,, = + s Sadg ad Rasool [6](7 T EM cycl is copltd wit M-stp, wic is,, wit issig coplt data aiu likliood ovr ( Z s rplacd by tir coditioal pctatios (,, Tus a EM itratio, takig ( + +, ito (, giv by R + = = k = Z X. ll( Y,, = = 0 = k = ( ( ( ( R = = = k = ll( Y,, = R + + = 0 W obtai t itrativ procdur of t EM-algorit as ad + = R = = is (8 (9 (0 + = T ( + ( + (, + + R R + + = = = = is t usd as a w valu of (, i t subsqut itratio. T MLEs of (, ca b obtaid by rpatig t E-stp ad M-stp util covrgc. Eac itratio is guaratd to icras t loglikliood ad t algorit is guaratd to covrg to a local aiu of t likliood fuctio, i.. startig fro a arbitrary poit i t paratr spac, t EM algorit will always covrg to a local aiu. tis work, t MLEs of ad basd o coplt sapl ar usd as iitial valus for ad i t EM algorit..4. Asyptotic Variacs ad Covariac T variac covariac atri is usd to provid a asur of prcisio for paratr stiators by utilizig t log-likliood fuctio. Applyig t usual larg sapl β =, ca b tratd as approiatio, t MLE of ( big approiatly bivariat oral wit a β ad variac-covariac atri, wic is t ivrs of t J β = E, β, wr pctd iforatio atri ( ( ( β : obs = is t obsrvd iforatio atri wit lts i i ( l = wit i, =, ad t pctatio is to b tak wit rspct to t distributio of X. For a coplt data st fro t Poisso-potial distributio, t variac covariac atri of paratrs β =, basd ad is giv by t likliood fuctio of ( o t obsrvd sapl of siz, ( t PE distributio is giv by, L ( β = =,,...,, fro ( ( ( log log = = Tor So Crar-Rao rgularity coditios old ad β =, blogs to a op itrval of t ral li. f t ( variac of a ubiasd stiator attais t Crar s-rao Lowr Boud, t likliood quatio as a uiqu solutio ˆβ tat aiizs t likliood fuctio. t is kow tat udr suc rgularity coditios, as t sapl siz icrass, t distributio of t MLE tds to t bivariat oral distributio wit a (, ad

5 Arica Joural of Tortical ad Applid Statistics 07; 6(3: covariac atri qual to t ivrs of t Fisr iforatio atri, s Co & Hikly [7]. T Fisr iforatio atri is giv by, (,, = = ll Y ll( Y,, ll( Y,, E ˆ ˆ ˆ E var( cov(, = ˆ ˆ ˆ cov(, var( ll( Y,, ll( Y,, E E Wr ll( Y,, = = ll( Y,, = + ( ll( Y,, = = = + (3 ll( Y,, = = ll( Y,, = T pctatios ar giv by, ( E ( E ( = = F, 4 [,],[3,3], ( ( = F 3,3 4 atri for, w gt, ([,,],[3,3,3], (4 (5 F, ([,],[3,3], ˆ ˆ ˆ var( cov(, ( 4 ( = cov( ˆ, ˆ var( ˆ F, ([,],[3,3], + F 3,3 ([,,],[3,3,3], 4 ( 4 ( (6 T ivrs of ( J β, valuatd at ˆβ provids t asyptotic variac-covariac atri of t MLEs. tis study, t procdur dvlopd by Louis ad Tar [8] is usd to driv t asyptotic variac covariac atri for t MLEs basd o t EM algorit. T ida of tis procdur is giv by ( η ( η ( η = (7 obs c iss Wr obs ( η, c ( η ad iss ( η dot t coplt, obsrvd, ad issig (pctd iforatio, rspctivly, η =,. T Fisr iforatio atri for a sigl ad ( obsrvatio wic is csord at t ti of t giv by ( iss ( η ( ; η lfz y zk z k > = E η t failur is (8 Giv X =, t coditioal distributio of Z k follows a trucatd Poisso-Epotial distributio wit lft trucatio at. Tat is, ( k X ( fx z f ( z z > ; η =, z > F Hc, z k k k z z z z ( ; f z z > η = =, z > k k Takig t logarit of bot sids, w gt, (9 (30 ( ( ( z lfz/ zk zk > ; η = l + l z l Diffrtiatig (30 wit rspct to β (, =, w gt ( η lf / ; z zk zk > z = (3

6 46 Josp Ndritu Gitai t al.: Estiatio of t Paratrs of Poisso-Epotial Distributio Basd o Progrssivly Typ Csorig Usig t Epctatio Maiizatio (E Algorit ( η z k k lf z z > ; = + ( η lfz zk zk > ; z = z + z + lfz/ ( zk zk > ; η + + z = z + lf / ( ; z zk zk > η z = z T pctatios ar giv by, z ( E z z ( E z = F, 4 [,],[3,3], ( ( ( = F 3,3 4 ([,,],[3,3,3], (3 (33 T pctd valus of t scod partial of t loglikliood fuctio of Z giv X ar calculatd as, ( > η E = + = lf ; z zk zk ( ( fz/ ( z k zk > ; η E = F, ([,],[3,3], = 4 ( ( ; η F3,3 ([,,],[3,3,3], + 4 ( lfz/ zk zk > + + E = = Wr Not tat ( iss ( η ( iss ( = (34 η iss is a fuctio of ad η, sic t pctatio is tak wit rspct z ; trfor, t pctd iforatio atri is siply iss ( η ( = R η (35 = iss ( Trfor, t variac covariac atri of paratr η ca b obtaid by ( η ( η ( η obs [ cop iss ] = (36 A approiat ( α 00% cofidc itrval for ad is obtaid as ˆ ˆ ± z α / var( ad z α / ˆ ± var( ˆ wr zα/ is t (α/00 t prctil of stadard oral distributio. 3. Rsults ad Discussios tis sctio, a siulatio study is coductd to ivstigat ow t proposd stiators prfor i stiatig t paratrs of Poisso-Epotial distributio basd o progrssiv typ csord data. T sapls wr gratd basd o t algorits of Balakrisa ad Sadu [6]. tis study, sapls of sizs 0,, ad 00 wr usd ad t csorig scs cosidrd ar giv i Tabl, ad 3 blow. Tabl. Csorig sc R = ( r, r,..., r for β = ( =.5, =.5. Csorig sc 0.5.5, 0,,, 0,, 0,, 0, , 0, 0, 0, 0,, 0,, 0,, 0, 0, 0, 0, ,, 3, 3, 0, 3,,,, 0, 3, 0,,, 0,,, 0,,,0,0,0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0,,, 0, 0, 0,, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0,, 0, 0,, 0,, 0,,,,, 0, 0,,, 0,, 0, 4, 0,, 0,, 3, 0,,, 0,, 0, 0, 0, 0, 3,,, 0,, 0,,, 0,,,, 0,, 0,, 0,, 0, 5 0, 0,, 0,, 0, 0, 0, 0, 0,, 0, 0,, 0, 0, 0, 0,, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0,, 0, 0, 0, 0, 0,, 0,, 0, 0, 0,, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0,

7 Arica Joural of Tortical ad Applid Statistics 07; 6(3: Tabl rprsts t progrssiv csorig sc for diffrt sapls siz ad diffrt ubrs of failurs for paratrs β = ( =.5, =.5. Tabl. Csorig sc R = ( r, r,..., r for β = ( =.5, =. Csorig sc 0.5, 0,,, 0,, 0,, 0, , 0, 0, 0, 0,, 0,, 0,, 0, 0, 0, 0, ,, 3, 3, 0, 3,,,, 0, 3, 0,,, 0,,, 0,,, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0,,, 0, 0, 0,, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0,, 0, 0,, 0,, 0,,,,, 0, 0,,, 0,, 0, 4, 0,, 0,, 3, 0,,, 0,, 0, 0, 0, 0, 3,,, 0,, 0,,, 0,,,, 0,, 0,, 0,, 0, 5 0, 0,, 0,, 0, 0, 0, 0, 0,, 0, 0,, 0, 0, 0, 0,, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0,, 0, 0, 0, 0, 0,, 0,, 0, 0, 0,, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0, Tabl rprsts t progrssiv csorig sc for diffrt sapls siz ad diffrt ubrs of failurs for paratrs β = ( =.5, =. Csorig sc Tabl 3. Csorig sc R = ( r, r,..., r for β = ( =.3, = , 0, 0,,,, 0, 0,, 0, 0, 0, 0, 0, , 0,,, 0,,, 0,,,, 0,, 0, , 0,,, 0,,, 5, 0,,, 3,, 0, 5 Tabl 3 rprsts t progrssiv csorig sc for icrasig sapls siz but fid ubrs of failurs for paratrs β( =.3, =. No rstrictio as b iposd o t aiu ubr of itratios ad covrgc is assud w t absolut diffrcs btw succssiv stiats ar lss ta 5 0. All coputatioal rsults wr coputd usig R softwar. Tabl 4. T MLEs, Variacs, covariac ad 95% cofidc liits of t MLEs for t paratrs of Poisso-potial distributio udr progrssiv typ csord sapl w β = ( =.5, =.5. ˆ ˆ var( ˆ var( ˆ cov( ˆ, ˆ CL( CL( LCL UCL LCL UCL Fro t abov tabl, it is obsrvd tat irrspctiv of t csorig rat ad at wic poit t csord uits ar rovd fro t sapl, for icrasig sapl siz; a t stiatd valu of t paratr covrg to t tru valu, b t variacs ad covariac of t MLEs dcras. t is also obsrvd tat, for t cas w is fid (i.. = 0, w ot tat as icrass (i.. fro 0 to 5 t variacs ad t covariac valus dcras (also s tabl 5. Tabl 5. T MLEs, Variacs, covariac ad 95% cofidc liits of t MLEs for t paratrs of Poisso-potial distributio udr progrssiv typ csord sapl w β = ( =.5, =. ˆ ˆ var( ˆ var( ˆ cov( ˆ, ˆ CL( CL( LCL UCL LCL UCL

8 48 Josp Ndritu Gitai t al.: Estiatio of t Paratrs of Poisso-Epotial Distributio Basd o Progrssivly Typ Csorig Usig t Epctatio Maiizatio (E Algorit Fro t abov tabl, it is obsrvd tat irrspctiv of t csorig rat ad at wic poit t csord uits ar rovd fro t sapl, for icrasig sapl siz; a t stiatd valu of t paratr covrg to t tru valu. b t variacs ad covariac of t MLEs dcras. Tabl 6. Cofidc itrvals of β = ( =.5, =.5. Widt of C. ( Widt of C. ( Fro t abov tabl, it is obsrvd tat t widts of 95% cofidc itrvals td to b arrowr for a icrasig sapl siz. Tabl 7. Cofidc itrvals of β = ( =.5, =. Widt of C. ( Widt of C. ( Fro t abov tabl, it is also obsrvd tat t widts of 95% cofidc itrvals td to b arrowr for a icrasig sapl siz. Tabl 8. T MLEs, Variacs, covariac ad 95% cofidc liits of t MLEs for t paratrs of Poisso-potial distributio udr progrssiv typ csord sapl w β( =.3, = wit diffrt sapl siz but fid ubr of failurs copltly obsrvd. 0 ˆ ˆ var( ˆ var( ˆ cov( ˆ, ˆ CL( CL( LCL UCL Widt ( LCL UCL Widt ( Fro t abov tabl, it is obsrvd tat irrspctiv of t csorig rat ad at wic poit t csord uits ar rovd fro t sapl wit fid ubr of failurs copltly obsrvd for icrasig sapl siz; a t stiatd valu of t paratr covrg to t tru valu. b t variacs ad covariac of t MLEs dcrass. 4. Coclusios Tis study as addrssd t probl of stiatio of paratrs of t Poisso-potial distributio basd o progrssiv Typ- csord data. T aiu likliood stiators of t scal ad sap paratrs wr obtaid by usig EM algorit. A copariso of t MLEs ad tir variacs as wll as tir cofidc itrvals was ad by siulatio for diffrt csorig scs. t was obsrvd tat: (i for a icrasig sapl siz, t stiatd valu of t paratr bcos closr to t tru valu, t variacs ad covariac of t MLEs dcras ad t widts of t cofidc itrvals bco arrowr. (ii rducig t ubr of uits to b rovd i t csorig sc, lads to bttr stiats for a fid sapl siz. T rsults provid t EM algorit tat is rlativly robust agaist t iitial valus. t guarats a sigl uifor o-dcrasig likliood trial fro t iitial valu to t covrgc valu. Morovr, wit t EM algorit, tr is o d to valuat t first ad scod drivativs of t log-likliood fuctio, wic lps sav t ctral procssig uit (CPU ti of ac itratio. T Epctatio Maiizatio algorit is coputatioal stabl, asy to iplt ad asyptotic variacs ad covariac of stiats ar also obtaid. Rfrcs [] Kus, C. (007. A w lifti distributio. Coputatioal Statistics ad Data Aalysis 5, [] Barrto, S. W. ad Cribari, N. F. (009. A gralizatio of Epotial-Poisso distributio. Statistics ad Probability Lttrs 79, [3] Caco, V. G. Louzada-Nto, F. ad Barriga, G. D. C. (0. T Poisso-Epotial lifti distributio. Coputatioal Statistics ad Data Aalysis 55, [4] Basu,A.,Kli, L. (98. So Rct Dvlopt i Coptig Risks Tory. Survival Aalysis,MS, Hayward. [5] Adaidis,K.,Loukas, S. (998. A Lifti distributio wit dcrasig failur rat. Statistics ad Probability Lttrs 39(, [6] Bai, L. T ad Eglardt, M. (99. Statistical Aalysis of Rliability ad Lif Tstig Modl, Marcl Dkkr; Nw York. [7] Co. A. C. (976. Progrssivly csord saplig i t tr paratr logoral distributio, Tcotrics 8, [8] Aggarwala, R. (00. Progrssivly itrval csorig: So atatical rsults wit applicatio to ifrc. Couicatios i Statistics-Tory ad Mtods 30,

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