The Exponential-Generalized Truncated Geometric (EGTG) Distribution: A New Lifetime Distribution

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1 Itratioal Joural of Statistics ad Probability; Vol. 7, No. 1; Jauary 018 ISSN E-ISSN Publishd by Caadia Ctr of Scic ad Educatio Th Epotial-Gralid Trucatd Gomtric (EGTG) Distributio: 1 Uivrsity of Tuis, ESSECT, Tuisia A Nw Liftim Distributio Mohiddi Rahmoui 1, & Ayma Orabi Kig Faisal Uivrsity, Commuity Collg i Abqaiq, Saudi Arabia Corrspodc: Mohiddi Rahmoui, Commuity Collg i Abqaiq, Kig Faisal Uivrsity, 3198, Al-ihsa, Eastr Rgio, Saudi Arabia. mohiddi.rahmoui@gmail.com Rcivd: Octobr 1, 017 Accptd: Octobr 4, 017 Oli Publishd: Novmbr 3, 017 doi: /ijsp.v71p1 URL: Abstract This papr itroducs a w two-paramtr liftim distributio, calld th potial-gralid trucatd gomtric (EGTG) distributio, by compoudig th potial with th gralid trucatd gomtric distributios. Th w distributio ivolvs two importat kow distributios, i.., th potial-gomtric (Adamidis ad Loukas, 1998) ad th tdd (complmtary) potial-gomtric distributios (Adamidis t al., 005; Louada t al., 011) i th miimum ad maimum liftim cass, rspctivly. Gral forms of th probability distributio, th survival ad th failur rat fuctios as wll as thir proprtis ar prstd for som spcial cass. Th applicatio study is illustratd basd o two ral data sts. Kywords: Liftim distributios; potial distributio; trucatd gomtric distributio; ordr statistics; failur rat; survival fuctio. 010 Mathmatics Subjct Classificatio: 60E05; 6E15; 6F Itroductio Liftim distributios ar widly usd i rliability thory, survival aalysis ad svral aras of studis, such as fiac, maufactur, biological scics, physics ad girig. Th potial distributio is th most commoly usd i rliability ad liftim tstig, assumig th failur rat is costat (Balakrisha ad Basu, 1995; Barlow ad Proscha, 1975). I rct yars, som of rsarch paprs hav b dvotd to tak ito accout data with icrasig or dcrasig failur rat fuctios. Th motivatio is to provid mor covit paramtric fit for ral data whr th udrlyig haard rats, arisig o a latt complmtary risk problm bas, prst mooto shaps (o-costat haard rats). Th gsis is statd o complmtary risk scarios with maskd causs of failur (Basu ad Kli, 198; Co ad Oaks, 1984). Svral familis of compoud liftim distributios ar itroducd as tsios of th potial distributio, followig Adamidis ad Loukas (1998), usig a mitur of discrt ad cotiuous distributios. Adamidis ad Loukas (1998) itroducd th potial-gomtric (EG) liftim distributio with dcrasig failur rat (DFR) arisig by miig powr-sris distributios. I th sam way, Adamidis t al. (005) ad Louada t al. (011) proposd th complmtary (or tdd) potial-gomtric (CEG) distributio with icrasig failur rat (IFR). Othr familis of liftim distributios hav b ivstigatd by svral authors. For ampl, Kus (007), Tahmasbi ad Rai (008), Chahkadi ad Gajali (009), Barrto-Soua ad Cribari-Nto (009), Silva t al. (010), Barrto-Soua t al. (011), Cacho t al. (011), Louada-Nto t al. (011), Morais ad Barrto-Soua (011), Hmmati t al. (011), Alkari ad Orabi (01), Nadarajah t al. (013), Bakouch t al. (014), ad othrs. Th proposd distributios com from th ida of modllig th rliability of sris ad paralll systms basd o th rliability of thir compots. Thy ar carrid out oly for th first (miimum) or th last (maimum) ordr statistics. 1

2 Th distributios of stochastic ordrd statistics 1 ar vry itrstig i may filds of statistics, particularly, th dtctio of outlirs, quality cotrol, auctio thory, rliability ad lif tstig modls. I this papr, w grali th distributios modllig th tim to th first (or th last) failur, to a distributio mor appropriat for modllig ay ordr statistic (scod, third, or ay k th liftim). For ampl, o may b itrstd i th priod of tim patits may spd i hospital durig a moth bfor thy lav. Lt's suppos that umbr of patits fill bds i a hospital ad lav durig a moth. W suppos that T i is th priod a patit spds bfor lavig ad Z th radom umbr of patits that lav th hospital. W may lt X (1) < X () < < X (Z) b th ordr statistics of idpdt obsrvatios of tim T = (T 1, T,, T ). Mayb prvious studis focusd o th miimum, X (1) = mi {T i } Z i=1, or th maimum, X (Z) = ma {T i } Z i=1, duratio of fillig bds. W ar itrstd hr i th duratio X (k) ad th w dtrmi th distributio for th obsrvd k th ordr statistic. Ordrd radom variabls ar alrady kow for thir ascdig ordr. Th cocpt of dual gralid ordrd statistics, itroducd by Burkschat t al. (003), abls a commo approach to th rvrs (or dscdig) ordr statistics. Rvrs ordr statistics hav also wid rag of applicatios i coomics such as providig divrs distributiv critria i assssig wlfar ad iqualitis of icoms ad walth (Wymark, 1981; Parrado-Gallardo t al., 014). W propos th w family of liftim distributios by compoudig th potial with th gralid trucatd gomtric distributios, amd th potial-gralid trucatd gomtric (EGTG) distributio. W show that th miimum liftim (Adamidis ad Loukas, 1998) ad th maimum liftim (Adamidis t al., 005; Louada t al., 011) ar spcial cass of our EGTG distributio. Fially, this papr is orgaid as follows: I sctio, w driv diffrt statistical propritis of our proposd distributio, icludig th probability dsity fuctio (pdf), th momt gratig fuctio, th rliability ad failur rat fuctios, th radom umbr gratio ad th tropy fuctio. Th stimatio of th paramtrs will b discussd i sctio 3. As a illustratio of th stimatio mthod, umrical computatios will b prformd i sctio 4. Th applicatio study is illustratd basd o two ral data sts i sctio 5. W coclud th papr i th last sctio.. Th Nw Liftim Distributio.1 Th Distributio Th drivatio of th w family of liftim distributios is obtaid by miig th potial with gralid trucatd gomtric distributios as follows: Lt T = (T 1, T,, T ) b iid potial radom variabls with scal paramtr θ > 0 ad pdf giv by: f(t) = θ θt, for t 0, whr Z is a discrt radom variabl (th radom umbr of uit i a systm) followig a trucatd at k 1 gomtric distributio with paramtr 0 < η < 1. Th probability mass fuctio (pmf), P k 1 (Z = ), is: whr th pmf of th gomtric distributio is giv by: P k 1 (Z = ) = (1 η)η k ; 0 < η < 1 ; k = 1,,, (1) P(Z = i) = (1 η)η i 1 ; 0 < η < 1 ad i = 1,, Z Z Lt X (1) = mi {T i } i=1 b th first ordr statistic (or th smallst ordr statistic) ad X (Z) = ma {T i } i=1 th last ordr statistic (or th largst ordr statistic). Th pdf of th k th ordr statistic X (k) (th k th -smallst valu of th liftim) is giv by th quatio () (David, 1981, p. 9; Balakrisha ad Coh, 1991, p. 1): ( 1) f ( k) ( k 1) Th joit probability dsity fuctio is obtaid from quatios (1) ad () as follows: ( k1) k1 k (, ) (1 ) ;, 0 () k k 1 (1 ) ( 1)(1 ) ( I) ; k 1,,..., ( k 1) ( k) ( k 1) g(,, ) (3) m1 m1 (1 ) ( 1)(1 ) ( II) ; k, 1,...,1 ; m k ( m1) ( m) ( m 1) 1 Ordr Statistics play a ky rol i liftim-distributio studis ad thy ar widly usd i statistical modls ad ifrc. O may rfr to Balakrisha ad Rao (1998, 1998a) for thory, mthods ad som applicatios o th ordr statistics of radom variabls.

3 whr, is th liftim of a systm, is th last ordr statistic ad m 0,1,.... Equatio (3) givs two dfiitios : th first is for th ascdig ordr statistics ad th scod is for th dscdig ordr. If w cosidr th ascdig ordr X (1) < X () < < X (Z), th first part (I) of this joit probability dsity i quatio (3) is rachd by compoudig a trucatd at l = k 1 gomtric distributio ad th pdf of th k th ordr statistic (k = 1,,, ). Th trucatd at k 1 gomtric distributio is motivatd by mathmatical itrst bcaus w ar itrstd i th k th ordr statistic. Thr is a lft-trucatio schm, whr oly ( k + 1) idividuals who surviv a sufficit tim ar icludd. I compariso with th formulatio of Adamidis ad Loukas (1998), w cosidr th k th -smallst valu of liftim istad of th miimum liftim (th first ordr statistic), X (1) = mi {T i } Z i=1. Howvr, wh w cosidr th rvrs (dscdig) ordr X (Z) > X (Z 1) > > X (1), w obtai th scod dfiitio (II) i quatio (3) by compoudig th pdf of th ( m) th ordr statistic ad a trucatd at m gomtric distributio. W iclud i th sampl oly ( m) idividuals who hav pricd th vt by th spcifid tim m = k. I compariso with th formulatio of Louada t al. (011), w cosidr th ( k) th -largst valu of liftim istad of th maimum liftim (th last ordr statistics), X (Z) = ma {T i } Z i=1. So, our proposd w family of liftim distributios (EGTG) is th margial dsity fuctio of giv by: k 1 k(1 ) (1 ) k 1,,..., k1 [1 ] g (, ) (4) ( m1) ( m1) (1 ) k, 1,...,1 ad m k m [1 (1 )] whr 0 1is th shap paramtr ad 0 is th scal paramtr. Our distributio is mor appropriat for modllig ay ordr statistic ( d, 3 rd or ay k th liftim). W show latr that th miimum liftim (Adamidis ad Loukas, 1998) is a spcial cas of th first part of th EGTG distributio, ad that th maimum liftim (Adamidis t al., 005; Louada t al., 011) is a spcial cas of th scod part of th EGTG distributio. It should b otd that w hav two sparat dfiitios. Th first cas is a graliatio of th miimum liftim distributio usig th ascdig ordr of liftim. Th scod cas is a graliatio of th maimum liftim distributio usig th dscdig ordr. Also, th cumulativ distributio fuctio (cdf) of corrspodig to th pdf i quatio (4) is giv by: k 1 k 1,,..., 1 G (, ) (5) m1 1 k, 1,...,1 ; m k 1 (1 ) I tabl (1), w prst th pdf i quatio (4) for som spcial cass at th first, scod ad third ordr statistics ad at th last, last-1 ad last- ordr statistics. Tabl (1) shows that th particular cas of our EGTG dsity fuctio, for k = 1, is th EG distributio modllig th tim to th first failur (Adamidis ad Loukas, 1998). Figur (1) illustrats som possibl shaps of th pdf for som slctd cass of th ordr statistics (k=1,; m=0,1) ad som slctd valus of (η, θ). Th pdf dcrass strictly i ad tds to ro as. Not that, for k = 1 th EGTG distributio is strictly dcrasig with modal valu θ(1 η) 1 at = 0. Th mdia is calculatd to b θ 1 l ( η) for k = 1. As η 0 ad k = 1, th EGTG distributio tds to a potial distributio with paramtr θ. Th graphs of th dsity rsmbl thos of th potial ad Parto II distributios. Th particular cas of th pdf for th last ordr statistic (m = 0) is th maimum liftim distributio (Adamidis t al., 005; Louada t al., 011). As η 0 ad m = 0, th EGTG distributio covrgs to th potial distributio with paramtr θ. Th EGTG dsity fuctio is dcrasig ad its mod is θ(1 η) at = 0 for η 1 ad ui-modal at 1 1 η log θ η. It is icrasig for η > 1. As otd by Louada t al. (011), th paramtrs η ad θ may b itrprtd Th proofs of all stps ad quatios ar availabl upo rqust. 3

4 dirctly i trms of complmtary risk problms i prsc of latt risks (s also, Louada-Nto, 1999). I fact, (1 η) ad θ rprst th ma of th umbr of complmtary risks ad th liftim failur rat, rspctivly. A dtaild prstatio of th comptig risks thory ca b foud i David ad Moschbrgr (1978) ad Crowdr (001). Tabl 1. Th probability dsity fuctio for som spcial cass Ordr statistics k, m Probability dsity fuctio First k=1 Scod k= Third k=3 Last m=0 Last -1 m=1 Last- m= (1 ) [1 ] (1 ) (1 ) 3 [1 ] 3 (1 ) (1 ) 4 [1 ] (1 ) [1 (1 )] (1 ) [1 (1 )] 3 3 (1 ) [1 (1 )] 3 4 4

5 .. Momt Gratig Fuctio ad r th Momt Figur 1. Dsity of th EGTG distributio for k=(1,) ad m=(0,1) Th momt gratig fuctio (mgf) is a altrativ spcificatio of th probability distributio of a radom variabl. It allows for th study of th charactristics ad th faturs of a distributio through its momts, such as th ma ad variac. If has th pdf i quatio (4), th th mgf is calculatd to b: k 1 1 k i k 1 j i t i j i0 j0 k(1 ) C C ( 1) i j 1 ; k 1,,..., M t E m C C m j k, 1,...,1; m k i 1 t m i 1 i j i t ( ) ( ) ( 1)(1 ) i j ( 1) 1 (6) i0 j0 Th mgf i quatio (6) grats th momts of X by diffrtiatio. I othr words, th r th momt ca b obtaid by valuatig th r th ordr drivativ of M () t at t = 0 as follows: d M dt r r ( r) X ( ) M X (0) r E Th ma is th first ordr drivativ, μ = E(X) = M X (1) (0), ad th scod ordr drivativ is E(X ) = M X () (0). Hc, th variac is σ = E(X ) μ. From quatio (6), th r th momt is giv by: k 1 ( r1) ( r1) k(1 ) k i k 1 j i ( 1) 1 1,,..., r Ci C j i j k r i0 j0 E ( ) (7) i ( r1) ( r 1)( m 1)(1 ) m i 1 i j i Ci C j( 1) m j 1 k, 1,...,1 ; m k r i0 j0 Tabl () provids th plicit prssios i trms of th polylogarithm fuctio for th first ad scod ordr momts of th radom variabl X. Li s (η) is th graliatio of Eulr's dilogarithm fuctio of η. It is kow as th polylogarithm fuctio dfid by th powr sris (Erdlyi t al., 1953, p. 31): () t t0 5

6 η Li s (η) = ηt t s t=1 = η + η η3 + ts t s + whr, Li 0 (η) = ad Li 1 η 1(η) = l (1 η) (Jodra, 008; Adamidis ad Loukas, 1998). Tabl. Th first ad scod ordr momts for som spcial cass Ordr statistics k, m E() E( ) First k=1 1 η ηθ Li 1(η) (1 η) ηθ Li (η) Scod k= 1 η η θ Li 1(η) 1 η ηθ (1 η ) η θ Li (η) (1 η) η θ Li 1 (η) Third k=3 1 η 3 η 3 θ Li 1(η) 3 (1 η) ηθ 1 η η θ (1 η 3 ) η 3 θ Li (η) 3(1 η) (1 + η) (1 η) η 3 θ Li 1 (η) + η θ Last m=0 1 ηθ Li 1(η) ηθ {Li (1) Li (1 η) + 1 [Li 1(η)] + l(η)li 1 (η)} Last -1 m=1 1 η θ Li 1(η) 1 ηθ η θ {Li (1) Li (1 η) + 1 [Li 1(η)] + [l(η) 1]Li 1 (η)} Last- m= 1 η 3 θ Li 1(η) 1. 1 ηθ 1 η θ η 3 θ {Li (1) Li (1 η) + 1 [Li 1(η)] + [l(η) 3 ] Li 1(η) + η }.3. Rliability ad Failur Rat Fuctios Th rliability (or th survival) fuctio is th probability of big aliv just bfor a duratio, giv by S() = Pr{X > } = 1 G() = f(t)dt which is th probability that th vt udr study has ot occurrd by duratio. So th rliability fuctio corrspodig to th pdf i quatio (4) is giv by: k 1 1 k 1,,..., 1 S(,, m) m1 k, 1,...,1 ad m k 1 (1 ) I othr words, th rliability, S(), is th probability that a systm oprats proprly i th itrval from tim 0 to tim, whr X is a radom variabl dotig th tim-to-failur or failur tim. O may rfr to th litratur o th thory ad applicatios of rliability (s Barlow ad Proscha, 1975, 1981; Basu ad Kli, 198). Th haard rat h(), kow as failur rat fuctio, is th istataous rat of occurrc of th vt of itrst at duratio (i.. th rat of vt occurrc pr uit of tim). Mathmatically, it quals th probability dsity g ( ) of 8 6

7 vts at, dividd by th probability, S( ) 1 G( ), of survivig to that duratio without pricig th vt. Thus, w dfi a failur rat fuctio as i Barlow ad Proscha (1965) by h( ) g( ) / S( ). Th failur rat fuctio corrspodig to th pdf i quatio (4) is giv by: 1 1 k k k(1 ) (1 ) 1 1 k 1,,..., k 1 [1 ] 1 h (, ) (9) ( m 1)(1 ) k, 1,...,1 ad m k 1 (1 ) Th haard rat fuctio is aalytically rlatd to th tim-to-failur probability distributio. It lads to th amiatio of th IFR or th DFR proprtis of lif-lgth distributios. G is a IFR distributio, if h() icrass for all X such that G(X) < 1. Th motivatio of th EGTG liftim distributio is th ralistic faturs of th haard rat i may ral-lif physical ad o-physical systms, which is ot a mootoically icrasig, dcrasig or costat failur rat. Not that for th first part of quatio (9), if k = 1, th haard rat fuctio is dcrasig (Adamidis ad Loukas, 1998). I fact, if 0 th h k (\η, θ) = θ(1 η) 1 ad if th h k (\η, θ) θ. If k > 1, thr is a IFR distributio. Idd, if 0 th h k (\η, θ) 0. If th h k (\η, θ) = θ. I th scod part of quatio (9), th EGTG is a IFR distributio. Th iitial rat valu is fiit ad giv by h m (0\ η, θ) = θ(m + 1)(1 η). Th log trm haard valu is fiit ad giv by θ(m + 1). Idd, w hav lim h (, ) lim h (, ). Figur () illustrats th shaps of th haard rat fuctio for som cass (k=1,; 0 m m=0,1) ad som slctd valus of (η, θ). m 1 (1 ) k 1 lim hk (, ) 0 0 k,3,..., lim h (, ) k 1,,3,..., k lim h (, ) ( m 1)(1 ) k, 1,...,1 ; m k 0 m lim h (, ) ( m 1) k, 1,...,1 ; m k m I tabl (3), w prst th rliability ad failur rat fuctios i quatios (8) ad (9) for som spcial cass at th first, scod ad third ordr statistics, ad at th last, last-1 ad last- ordr statistics. Tabl 3. Th rliability ad failur fuctios for som spcial cass Ordr statistics k, m Survival fuctio Haard rat First k= Scod k= (1 ) (1 ) [1 ] 1 1 7

8 Third k= (1 ) (1 ) [1 ] Last m=0 Last -1 m=1 1 (1 ) 1 (1 ) (1 ) 1 (1 ) (1 ) 1 (1 ) Last- m= 1 (1 ) 3 3 (1 ) 1 (1 ) Figur. Haard rat of th EGTG distributio for k=(1,) ad m=(0,1) 8

9 .4. Radom Numbr Gratio W ca grat a radom variabl from th cdf of i quatio (5) usig th followig stps: Grat a radom variabl U from th stadard uiform distributio. Calculat th valus of X such as: 1 k 1 1 l U k 1,,..., 1 k 1U X (10) 1 m1 1 1 (1 U ) l k, 1,...,1 ad m k 1 m1 (1 )(1 U ) W ca dtrmi th quatils by dividig th st of obsrvatios ito qual sid groups. Th mdia is computd by lttig U = 0.5. Th quatio (10) ca b usd for a simulatio study of th EGTG distributio..5. Etropy Etropy is a masur of th ucrtaity that was itroducd by Shao (1948) as a basic cocpt i iformatio ad commuicatio thory, masurig th avrag missig iformatio o a radom sourc (Ls, 014). Shao dfid th tropy as: 0 H b (X) = E [log b ( 1 )] = g() log g() b ( 1 ) d = g() [log g() b g()] (11) log b ( 1 ) is also calld a ucrtaity. Th probability distributio alrady dscribs th probability charactristics of a g() radom variabl. Howvr, if w hav two or mor probability distributios w do ot kow actly which variabl is mor radom tha th othr. I this cas, a compariso btw probability distributios is possibl thaks to tropy whras a probability distributio dscribs th radomss of o radom variabl. Th tropy appars as th avrag iformatio rquird to spcify th outcom X wh th distributio g() is kow. It quivaltly masurs th amout of ucrtaity rprstd by a probability distributio (Jays, 1957). Wh th tropy is larg this mas that th ucrtaity associatd to th radom variabl is larg, ad vic vrsa. Lt th tropy of a radom variabl X 1 b 0.8 ad that of aothr variabl X b 0.1. It is obsrvd that th first radom variabl is mor ucrtai tha th scod. From quatio (11), th tropy for th w family distributio is giv by: 0 j ( k j)! -log[ k(1- )] k( k -1)(1- ) B( i j 1, k) i j! k! i1 j0 i j ( k j)! - k( k 1)(1- ) B( i j 1, k) ; k 1,,..., i1 j0 i j! k! H( ) -log[ ( m 1)(1- )] ( m 1) - ( m 1)( m )(1- ) i j ( m j1)! B( i j 1, m 1) for, k, -1,...,1 ad m - k i1 j0 i j!( m 1)! (1) whr μ = E(X) Equatio (1) prsss th mathmatical pctatio of ucrtaity. Similar to th ma ad variac of a radom variabl, tropy is a drivd quatity from probability distributio, but it has a valu of its ow (Zog, 006). If w ar 9

10 skig a pdf subjct to crtai costraits (r th momt), w choos th dsity satisfyig th costraits ad havig tropy as larg as possibl. Th pricipl of maimum tropy stats that w should choos th probability distributio that maimis th ucrtaity subjct to som costrais about th radom variabl X. This distributio should b last surprisig i trms of th prdictios it maks. Etropy ca also b usd to produc a modl for data-gratig distributio i trms of iformatio costraits. Giv th prior iformatio about th pdf g(, η, θ), o may stimat th paramtrs by maimiig th tropy id H(X) or quivaltly, miimiig th log-loss. I th applicatio ampls, w will us th tropy as a statistic i assssmt of distributio, by tstig th hypothsis that th data follow a spcific distributio. Idd, it masurs how wll a probability distributio fits a st of obsrvd data (masur of th ihrt radomss). Giv th stimatd paramtrs η ad θ, w ca calculat th tropy H(X). 3. Estimatio I this sctio, w will dtrmi th stimatd paramtrs η ad θ for th EGTG w family of liftim distributios. Lt (X 1, X,, X ) b a radom sampl with obsrvd valus ( 1,,, ) from th EGTG distributio with th pdf i quatio (4). Th log liklihood fuctio giv th obsrvd valus, obs = ( 1,, ), is: i l( k) l(1 ) l( ) ( k 1) l(1 ) i1 i ( k 1) l(1 ) k 1,,..., i i1 i1 l L(, / obs ) l( m 1) l(1 ) l( ) ( m 1) i i1 i ( m ) l[1 (1 )] k, 1,...,1 ad m k i1 (13) W subsqutly driv th associatd gradits: i ( k 1) k 1,,..., (1 i ) i1 1 l L(, / ) (14) i 1 ( m ) ; k, 1,...,1 ad m k i (1 ) i1 1 (1 ) i i i ( k 1) ( k 1) i i i1 i1 1 i1 k 1,,..., l L(, / ) (15) i ( m 1) i ( m ) i i1 i1 (1 ) k, 1,...,1 ad m k W d th Fishr iformatio matri for itrval stimatio ad tsts of hypothss o th paramtrs. It ca b prssd i trms of th scod drivativs of th log-liklihood fuctio: 10

11 For th first dfiitio, For th scod dfiitio, ll θ I 11 = E ( ll η ) I 1 = E ( ll η θ ) I = I 1 = E ( ll ( θ η ) I = E ( ll θ ) ) ll η = θi (k + 1) (1 η) (1 η θ i ) = + (k 1) θ ll η θ i=1 i θ i i=1 ( θ (k + 1) i 1) = (k + 1) i θi (1 η θ i ) i=1 η i θ i ( θ i η) i=1 ll η = 1 θi (m + ) (1 η) [1 η(1 θ i)] ll θ ll η θ = (m + ) η(1 θ = (m + ) i=1 i=1 θ i η) i [ θ i(1 η) + η] i θi [ θ i(1 η) + η] i=1 Th maimum liklihood stimats (MLEs) η ad θ of th EGTG paramtrs η ad θ, rspctivly, ca b dtrmid umrically by solvig th oliar quatios (14) ad (15) of th associatd gradits, usig a statistical softwar (it ca b asily do usig R, Mathcad ad Matlab packags, amog othrs). Th choic of a good st of iitial valus is sstial. Th MLEs ca also b foud aalytically usig th itrativ EM algorithm to hadl th icomplt data problms (Dmpstr t al., 1977; McLachla ad Krisha, 1997). Th itrativ mthod cosists o rpatdly updatig th paramtr stimats by rplacig th "missig data" with th w stimatd valus. Th stadard mthod usd to dtrmi th MLEs is th Nwto-Raphso algorithm that rquirs scod drivativs of th log-liklihood fuctio for all itratios. Th mai drawback of th EM algorithm is its rathr slow covrgc, compard to th Nwto-Raphso mthod, wh th "missig data" cotai rlativly larg amout of iformatio (Littl ad Rubi, 1983). Rctly, svral rsarchrs hav usd th EM mthod such as Adamidis t al. (005), Karlis (003), Ng t al. (00), Adamidis ad Loukas (1998), Adamidis (1999), amog othrs. Nwto-Raphso is rquird for th M-stp of th EM algorithm. To start th EM algorithm, w dfi a hypothtical distributio of complt-data with th joit dsity fuctio i quatio (3). W driv th coditioal mass fuctio as: E-stp: M-stp: k ( k ) k 1 ( 1) (1 ) k 1,,..., ( k 1) ( k 1) p(,, ) m m1 ( 1)[1 (1 )] k, 1,...,1 ad m k m1 ( m) ( m )(1 ) (16) k k 1,,..., 1 E(,, ) 1 m k, 1,...,1 ad m k 1 (1 ) (17) 11

12 ( r1) ( r ) ( r) i i1 ( r ) ( r) i ( k 1) k 1,,..., (1 ) i1 ( r ) ( r) i (1 ) i1 k, 1,...,1 ad m k ( r ) ( r) i ( 1) (1 ) i1 (18) ( r1) k 1,,..., ( r ) ( r1) ( r) i k (1 k) i ( r1) i ( r) i ( r) i i 1 i 1 1 i 1 1 k, 1,...,1 ad m k ( r) ( r1) ( r) i ( r1) i (1 ) ( m1) i ( r) ( r1) ( r) i i i1 i1 1 (1 ) 1 4. Simulatio Study As a illustratio of th MLEs, umrical computatios hav b prformd usig th stps prstd i sctio.4 for radom umbr gratio. Th umrical study was basd o 1000 radom sampls of th sis 0, 50 ad 100 from th EGTG distributio for th ascdig ad dscdig ordr statistics with cass k = (1; ) ad m = (0; 1), rspctivly. W hav cosidrd th iitial valus of (η, θ) : (0.1, 0.5), (0.3, 0.5), (0.5,1), (0.1, 1.5), (0.7, 1.5). Aftr dtrmiig th paramtr stimats λ = (η, θ ) w comput th biass, variacs ad ma squar rrors (MSEs), whr MSE(λ ) = E(λ λ) = Bias (λ ) + var(λ ) ad Bias(λ ) = E(λ ) λ. A stimator λ is said to b fficit if its ma squar rror (MSE) is miimum amog all comptitors. I fact, λ 1 is mor fficit tha λ if MSE(λ 1) < MSE(λ ). Tabl 4 rports th rsults from th simulatd data whr th variacs ad th MSEs of th stimatd paramtrs ar giv. Th rsults show that, for ach cas, th covrgc has b achivd. Idd, th stimatd paramtrs λ = (η, θ ) approach to thir ral valus wh th si of th sampl icrass. th variacs ad th MSEs dcras ad covrg to ro wh th sampl si icrass, which may suggst that th MLEs ar prformd cosisttly. (19) 1

13 Tabl 4. Rsults from th simulatio study (η, θ) Ordr statistics Rvrs ordr statistics k η θ var(η ) var(θ ) MSE(η ) MSE(θ ) m η θ var(η ) var(θ ) MSE(η ) MSE(θ ) 0 (0.1, 0.5) m= (0.3, 0.5) k= (0.5, 1) (0.1, 1.5) (0.7, 1.5) (0.1, 0.5) k= m= (0.3, 0.5) (0.5, 1) (0.1, 1.5) (0.7, 1.5) (0.1, 0.5) k= m= (0.3, 0.5) (0.5, 1) (0.1, 1.5) (0.7, 1.5) (0.1, 0.5) k= m= (0.3, 0.5) (0.5, 1) (0.1, 1.5) (0.7, 1.5) (0.1, 0.5) k= m= (0.3, 0.5) (0.5, 1) (0.1, 1.5) (0.7, 1.5) (0.1, 0.5) k= m= (0.3, 0.5) (0.5, 1) (0.1, 1.5) (0.7, 1.5)

14 4. Applicatio Eampls I this sctio, w fit th EGTG distributio to two ral data sts usig th MLEs dtrmid umrically by dirct itgratio usig Mathcad Th first st (tabl A i appdi) cosists of "107 failur tims for right rar braks o D9G-66A catrpillar tractors", rproducd from Barlow ad Campo (1975) ad usd also by Chag ad Rao (1993). Ths data ar usd i may applicatios i rliability (Adamidis t al., 005; Tsokos, 01; Shahsaai t al., 01). Th scod st of data ivolvs 100 obsrvatios (tabl B i appdi) of th rsults from a primt cocrig "th tsil fatigu charactristics of a polystr/viscos yar". Ths data wr prstd by Picciotto (1970) to study th problm of warp brakag durig wavig. Th obsrvatios wr obtaid o th cycls to failur of a 100 cm yar sampl put to tst udr.3% strai lvl. Th sampl is usd i Qusbrry ad Kt (198) as a ampl to illustrat slctio procdur amog probability distributios usd i rliability. Th rliability fuctio of ths two data sts blogs to th icrasig failur rat class (Doksum ad Yadll, 1984; Adamidis t al., 005). W us diffrt statistical tsts to assss th agrmt btw th EGTG distributio ad th data sts. I additio to our class of distributios, th gamma ad Wibull distributios fittd ths data sts. Th rspctiv dsitis of gamma ad Wibull distributios ar: f 1 (, λ 1, β 1 ) = λ 1 β 1 β 1 1 p ( λ 1 )Γ(β 1 ) 1 ad f (, λ, β ) = β λ β β 1 p ( λ ) β. Tabls 5 ad 6 show th fittd paramtrs, calculatd valus of Kolmogorov Smirov (K-S) ad thir rspctiv p-valus for th two data sts. It should b otd that th K-S tst compars a mpirical distributio with a kow (ot stimatd) o. It is usd to dcid if a sampl coms from a populatio with a spcific distributio (H0: th data follow a spcifid distributio). W stimat som spcial cass of th EGTG distributio at 5% sigificat lvl. Th tabls rport also th AIC ad BIC iformatio critria ad th Shao s tropy (H) for modl slctio. Tabl (7) givs th mas ad th stadard rrors for som spcial cass of th EGTG distributio, compard to thir mpirical valus. I ordr to idtify mpirical bhaviors that th failur rat fuctio ca tak, w shall cosidr th graphical mthod basd o th total tst o tim (TTT-plot) proposd by Aarst (1987). I its mpirical vrsio th TTT-plot is costructd by valus r ad G(r ), whr G(r ) = r i=1 X i: + ( r)x r: i=1 X i: whr r = 1,, ad X i: rprsts th ordr statistics of th sampl. Th graphic TTT may hav various forms. It rsmbls to th Gii id ad it is usd as a crud idicativ of th shap of th failur rat fuctio. Idd, wh th curv approachs a straight diagoal fuctio, costat failur rat is adquat ad th data ar from a potial distributio. Wh th curv is approimatly cocav or cov, th data ar from IFR distributio or DFR distributio, rspctivly. Figur (3) shows cocav tdcis idicatig that th two data sts hibit IFR distributios. This rsult is i agrmt with th MLEs of th shap paramtrs ad th K-S tst. Th p-valus ar oly sigificat for th cas k = 1 for th two data sts. As show i sctio.3, If k = 1, th EGTG is DFR, but hr th data hibit a icrasig haard rat. Tabls (5) ad (6) show smallst valus of K-S statistics for th last ordr statistics (k = ) with largst associatd p-valus qual to ad , rspctivly. Th K-S distacs btw th mpirical distributio fuctio of th two sampls ad th cdf of th corrspodig distributio ar rspctivly ad W ca s that th w liftim family provids good fit to th data sts. Th K-S tst shows that th EGTG distributio is a attractiv altrativ to th popular gamma ad Wibull distributios. 14

15 Tabl 5. Th goodss of fit for som spcial cass, for th first data st (107 obs.) ML Estimats Distributios ˆ ˆ K-S p-valu Log-lik AIC BIC H EGTG: First ordr (k = 1) Scod ordr (k = ) Third ordr (k = 3) Fourth ordr (k = 4) Last ordr (k = ) Last ordr-1 (k = 1) Last ordr- (k = ) Last ordr-3 (k = 3) λ β Gamma Wibull Tabl 6. Th goodss of fit for som spcial cass, for th scod data st (100 obs). ML Estimats Distributios ˆ ˆ K-S p-valu Log-lik AIC BIC H EGTG: First ordr (k = 1) Scod ordr (k = ) Third ordr (k = 3) Fourth ordr (k = 4) Last ordr (k = ) Last ordr-1 (k = 1) Last ordr- (k = ) Last ordr-3 (k = 3) λ β Gamma Wibull

16 Barlow ad Campo (1975) data st Qusbrry ad Kt (198) data st Figur 3. Empirical TTT-plot For modls compariso, w comput th Akaik s iformatio critrio (AIC = ll + p) ad Schawar s Baysia iformatio critrio (BIC = ll + p log()), whr is th si of th sampl ad p is th umbr of paramtrs. Th rsults idicat that th last ordr statistic (k = ) has th smallst AIC ad BIC valus. Th, th maimum liftim distributio could b commoly chos as th prfrrd modl for dscribig th two data sts. Th tropy id shows that our distributio is a good altrativ i stimatig liftim data. Figur (4) illustrats th fittd modls ad th obsrvd histograms ad figur (5) shows th probability-probability plots for th two data sts. Th plots corroborat th prvious rsults ad cofirm th good prformac of our distributio. Th diagoal is th rfrc li i th PP-plot. Figur 4. Fittd modls ad obsrvd histograms 16

17 Figur 5. Th probability-probability plots for th data sts Tabl 7. Mas ad stadard rrors for som spcial cass Barlow & Campo (1975) data st Qusbrry & Kt (198) data st (107 obs.) (100 obs.) E(X) σ(x) E(X) σ(x) EGTG distributio First ordr (k = 1) Scod ordr (k = ) Third ordr (k = 3) Fourth ordr (k = 4) Last ordr (k = ) Last ordr-1 (k = 1) Last ordr- (k = ) Last ordr-3 (k = 3) Empirical valus

18 6. Coclusio I this papr w proposd th EGTG distributio, that gralis th potial-gomtric (Adamidis ad Loukas, 1998) ad th tdd (or complmtary) potial-gomtric distributio (Adamidis t al., 005; Louada t al., 011) i th miimum ad maimum cass, rspctivly. Th applicatio study was illustratd basd o two sts of ral data usd i may applicatios of rliability. W hav show that our proposd distributio is suitabl for modllig th liftim of ay ordr statistics. Futur rsarch, that should b cosidrd, icluds th Baysia approach with csord data. Ackowldgmt "Th authors would lik to thak th ditors ad th rfr for th valuabl commts ad suggstios." Rfrcs Aarst, M. V. (1987). How to idtify bathtub haard rat. IEEE Tras. Rliab. 36, Adamidis, K. (1999). A EM algorithm for stimatig gativ biomial paramtrs. Austral. Nw Zalad J. Statist. 41 (), Adamidis, K., Dimitrakopoulou, T., & Loukas, S. (005). O a tsio of th potial gomtric distributio. Statist. Probab. Ltt. 73, Adamidis, K., & Loukas, S. (1998). A liftim distributio with dcrasig failur rat. Statist. Probab. Ltt. 39, Alkari, S., Orabi, A. (01). A compoud class of Poisso ad liftim distributios. J. Stat. Appl. Pro. 1(1), Bakouch, H. S., Jai, M. A., Nadarajah, S., Dolati, A., & Rasool, R. (014). A liftim modl with icrasig failur rat. Applid Mathmatical Modllig. 38, Balakrisha, N., & Coh, A. C. (1991). Ordr Statistics ad Ifrc: Estimatio Mthods. Acadmic, Bosto. Balakrisha, N., & Basu, A. P. (1995). Th Epotial Distributio: Thory, Mthods ad Applicatios. Nwark, Nw Jrsy: Gordo ad Brach Publishrs. Balakrisha, N., & Rao, C. R. (1998). Ordr Statistics: Applicatios. Amstrdam: Elsvir. Balakrisha, N., & Rao, C. R. (1998a). Ordr Statistics: Thory ad Mthods. Amstrdam: Elsvir. Barlow, R. E., & Campo, R. (1975). Total tim o tst procsss ad applicatios to failur data aalysis. Rliability ad Fault Tr Aalysis. Socity for Idustrial ad Applid Mathmatics, Philadlphia. pp Barlow, R. E., & Proscha, F. (1981). Statistical Thory of Rliability ad Lif Tstig. Publishr: To bgi with, Silvr Sprig, Marylad. Barlow, R. E., & Proscha, F. (1965). Mathmatical thory of rliability. Joh Wily & Sos., Barrto-Soua, W., & Bakouch, H. S. (013). A w liftim modl with dcrasig failur rat. Statistics. 47(), Barrto-Soua, W., & Cribari-Nto, F. (009). A graliatio of th Epotial-Poisso distributio Statist. Probab. Ltt. 79, Barrto-Soua, W., Lmos-Morais, A., & Cordiro, G. M. (011). Th Wibull-gomtric distributio. J. Statist. computat. Simul. 81(5), Basu, A. P., & Kli, J. P. (198). Som rct rsults i comptig risks thory. I: Crowly JRA J, ditor. Survival Aalysis. Istitut of Mathmatical Statistics, Hayward., Burkschat, M., Cramr, E., & Kamps, U. (003). Dual gralid ordr statistics. Mtro. 61(1), Cacho, F., Louada-Nto, F., & Barriga, G. (011). Th Poisso-potial liftim distributio. Computat. Statist. Data Aal. 55, Chahkadi, M., & Gajali, M. (009). O som liftim distributios with dcrasig failur rat. Computat. Statist. Data Aal. 53, Chag, M. N., & Rao, P. V. (1993). Improvd stimatio of survival fuctios i th w-bttr-tha-usd class. Tchomtrics. 35, Co, D. R., & Oaks, D. (1984). Aalysis of Survival Data. Moographs o Statistics & Applid Probability. Lodo: 18

19 Chapma & Hall. Crowdr, M. J. (001). Classical Comptig Risks. Chapma ad Hall. David, H. A. (1981). Ordr Statistics. Scod d. Nw York, Wily. David, H. A., & Moschbrgr, M. L. (1978). Th Thory of Comptig Risks. Griffi. Dmpstr, A. P., Laird, N. M., & Rubi, D. B. (1977). Maimum liklihood from icomplt data via th EM algorithm (with discussio). J. Roy. Statist. Soc. Sr. B. 39, Doksum, K. A., & Yadll, B. S. (1984). Tsts for potiality. I: Hadbook of statistics. vol. 4. Elsvir. p Erdlyi, A., Magus, W., Obrhttigr, F., & Tricomi, F. G. (1953). Highr Trascdtal Fuctios. McGraw-Hill, Nw York. Hmmati, F., Khorram, E., & Rakhah, S. (011). A w thr-paramtr agig distributio. J. stat. pla. if. 141, Jays, E. T. (1957). Iformatio thory ad statistical mchaics. Physics Rviw. 106, Jodra, P. (008). O a coctio btw th polylogarithm fuctio ad th Bass diffusio modl. Procdigs of th Royal Socity. 464: Karlis, D. (003). A EM algorithm for multivariat Poisso distributio ad rlatd modls. J. Appl. Statist. 30(1), Kus, C. (007). A w liftim distributio. Computat. Statist. Data Aal. 51, Ls, A. (014). Shao tropy: a rigorous otio at th crossroads btw probability, iformatio thory, dyamical systms ad statistical physics. Mathmatical Structurs i Computr Scic. 4(03), Littl, R. J. A., & Rubi, D. B. (1983). Icomplt data. I: Kot, S., Johso, N. L. (Eds.), Ecyclopdia of Statistical Scics, vol. 4. Wily, Nw York. Louada-Nto, F. (1999). Modllig liftim data: a graphical approach. Applid Stochastic Modls i Busiss ad Idustry. 15, Louada, F., Roma, M., & Cacho, V. G. (011). Th complmtary potial gomtric distributio: Modl, proprtis, ad a compariso with its coutrpart. Computat. Statist. Data Aal. 55(8), Louada-Nto, F., Cacho, V. G., & Barrigac, G. D. C. (011). Th Poisso potial distributio: A Baysia approach. J. Appl. Statist. 38, McLachla, G. J., & Krisha, T. (1997). Th EM Algorithm ad Etsio. Wily, Nw York. Morais, A., & Barrto-Soua, W. (011). A compoud class of Wibull ad powr sris distributios. Computat. Statist. Data Aal. 55, Nadarajah, S., Cacho. V. G., & Ortga, E. M. M. (013). Th gomtric potial Poisso distributio. Stat. Mthods Appl., Ng, H. K. T., Cha, P. S., & Balakrisha, N. (00). Estimatio of paramtrs from progrssivly csord data usig EM algorithm. Comput. Statist. Data Aal. 39, Parrado-Gallardo, E. M., Barca-Martı, E., & Imdio-Olmdo, L. J. (014). Iquality, wlfar ad ordr statistics. I Ecoomic Wll-Big ad Iquality: Paprs from th Fifth ECINEQ Mtig (vol., p. 383). Emrald Group Publishig. Picciotto, R. (1970). Tsil Fatigu Charactristics of a Sid Polystr/Viscos Yar ad Thir Effct o Wavig Prformac. Mastr s thsis. North Caroli Stat Uivrsity. Qusbrry, C. P., & Kt, J. (198). Slctig amog probability distributios usd i rliability. Tchomtrics. 4, Shahsaai, F., Rai S., & Pak, A. (01). A Nw Two-Paramtr Liftim Distributio with Icrasig Failur Rat. Ecoomic Quality Cotrol. 7(1), Shao, C. E. (1948). A mathmatical thory of commuicatio. Bll Systm Tchical Joural. 7,

20 Silva, R. B., Barrto-Soua, W., & Cordiro, G. M. (010). A w distributio with dcrasig, icrasig ad upsid-dow bathtub failur rat. Computat. Statist. Data Aal. 54, Tahmasbi, R., & Rai, S. (008). A two-paramtr liftim distributio with dcrasig failur rat, Computat. Statist. Data Aal. 5, Tsokos, C. (01). Th thory ad applicatios of rliability with mphasis o Baysia ad oparamtric mthods. Elsvir. Wymark, J. A. (1981). Gralid Gii iquality idics. Mathmatical Social Scics. 1(4), Zog, Z. (006). Iformatio-Thortic Mthods for Estimatig of Complicatd Probability Distributios. Elsvir. Appdi Tabl (A): Ordrd Failur Tims (i hours) of 107 Right Rar Braks o D9G-66A Catrpillar Tractors (Barlow ad Campo, 1975; Chag ad Rao, 1993) Tabl (B): Rsults of Modl Slctio Program o Yar Data (Qusbrry ad Kt, 198) Copyrights Copyright for this articl is rtaid by th author(s), with first publicatio rights gratd to th joural. This is a op-accss articl distributd udr th trms ad coditios of th Crativ Commos Attributio lics ( 0