Research Article Estimating the Reliability Function for a Family of Exponentiated Distributions

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1 Probabiliy ad Saisics, Aricle ID 56393, 1 pages hp://dx.doi.org/1.1155/214/56393 Research Aricle Esimaig he Reliabiliy Fucio for a Family of Expoeiaed Disribuios Aji Chaurvedi ad Aupam Pahak Deparme of Saisics, Uiversiy of Delhi, Delhi 117, Idia Correspodece should be addressed o Aupam Pahak; pahakaupam24@gmail.com Received 4 Jue 213; Revised 17 December 213; Acceped 22 March 214; Published 29 April 214 Academic Edior: Dejia Lai Copyrigh 214 A. Chaurvedi ad A. Pahak. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial work is properly cied. A family of expoeiaed disribuios is proposed. The problems of esimaig he reliabiliy fucio are cosidered. Uiformly miimum variace ubiased esimaors ad maximum likelihood esimaors are derived. A comparaive sudy of he wo mehods of esimaio is doe. Simulaio sudy is preformed. 1. Iroducio The reliabiliy fucio R() is defied as he probabiliy of failure-free operaio uil ime. Thus,if he radom variable (rv) X deoes he lifeime of a iem, he R() = P(X > ). Aoher measure of reliabiliy uder sresssregh seup is he probabiliy P = P(X > Y), which represes he reliabiliy of a iem of radom sregh X subjec o radom sress Y. A lo of work has bee doe i he lieraure o deal wih ifereial problems relaed o various expoeiaed disribuios. I paricular, Mudholkar ad Srivasava [1] cosidered expoeiaed Weibull family for aalyzig bahub failure-real daa. The expoeiaed Weibull family was also applied o he bus-moor-failure daa i Davis [2], o head-ad-eck cacer cliical rial daa i Efro [3] [see[4]], ad i aalyzig flood daa [see [5]]. Jiag ad Murhy [6] cosidered expoeiaed Weibull family ad illusraed some of is properies by usig graphical approach. Nassar ad Eissa [7, 8] sudied a wo-parameer expoeiaed Weibull disribuio ad hey gave some of is properies ad esimaed he parameers by usig he maximum likelihood ad Bayes mehods based o ype II cesored daa. They used he squared-error ad liear i expoeial (LINEX) loss fucios ad a iformaive prior o obai he Bayes esimaes. Pal e al. [9] showedha he failure rae of expoeiaed Weibull behaves more like he failure rae of he Weibull disribuio ha ha of he Gamma disribuio. They also obaied maximum likelihood esimaors, Fisher s iformaio marix, ad cofidece iervals for relaed parameers. Gupa e al. [1] iroduced expoeiaed expoeial disribuio i a series of papers. Gupa ad Kudu [11 17], Gupa e al. [18], ad Kudu e al. [19] coceraed o he sudy of he expoeiaed expoeial disribuio. Kudu ad Gupa [2] saed ha expoeiaed expoeial disribuio is a aleraive o he well-kow wo-parameer Gamma, wo-parameer Weibull or wo-parameer logormal disribuios. Kudu ad Gupa [2] esimaed he reliabiliy of he sress-sregh model P(X < Y),whe X ad Y are idepede expoeiaed expoeial radom variables. Raqab ad Ahsaullah [21] esimaed he parameers of he expoeiaed expoeial based o ordered saisics. Iferece for expoeiaed Weibull disribuio based o record values was made by Raqab [22]. Abdel-Hamid ad AL-Hussaii [23] obaied he maximum likelihood esimaes of he parameers whe sep-sress acceleraed life esig is applied for expoeiaed expoeial disribuio. Al-Hussaii ad Hussei [24, 25] fied he expoeiaed Burr model o he daa of breakig sregh of sigle carbo fibers of paricular legh (daa i [26])ihecompleesamplecaseadwheypeIIcesorig isimposedodaa.selfadlinexlossfuciosare used i he Bayes esimaio. Gupa e al. [1] iroduced a ew disribuio, called he expoeiaed Pareo disribuio.shawkyadabu-ziadah[27] obaiedmaximum likelihood esimaors of he differe parameers of he

2 2 Probabiliy ad Saisics expoeiaed Pareo disribuio. They also cosidered five oher esimaio procedures ad compared heir performaces hrough umerical simulaios. The purpose of he prese paper is mayfold. We propose a family of expoeiaed disribuios, which covers as may as e expoeiaed disribuios as specific cases. We cosider he problems of esimaig R() ad P. Uiformly miimum variace ubiased esimaors (UMVUES) ad maximum likelihood esimaors (MLES) are derived. I order o obai hese esimaors, he major role is played by he esimaor of he probabiliy desiy fucio (pdf) a a specified poi, ad he fucioal forms of he parameric fucios o be esimaed are o eeded. I is worh meioig here ha, i order o esimae P, i he lieraure, he auhors have cosidered he case whe X ad Y follow he same disribuios, may be wih differe parameers. We have geeralizedheresulohecasewhex ad Y may follow ay disribuio from he proposed expoeiaed family of disribuios. Simulaio sudy is carried ou o ivesigae he performace of he esimaors. I Secio 2, we iroduce he family of expoeiaed disribuios. I Secio 3, we derive he UMVUES of R() ad P. I Secio 4,weobaiheMLESofR() ad P. I Secio 5, aalysis of a simulaed daa has bee preseed for illusraive purposes. Fially, i Secio 6, coclusios have bee provided. 2. The Family of Expoeiaed Disribuios Le he rv X follow he disribuio havig he pdf f (x; a, α, λ, θ) =αλg (x;a,θ)e λg(x;a,θ) (1 e λg(x;a,θ) ) α 1 ; x>a, α,λ>. (1) Here, g(x; a, θ) is a fucio of X ad may also deped o (may be vecor-valued) parameer θ. Moreover,g(x; a, θ) is a real-valued, moooically icreasig fucio of X wih g(a; a, θ) =, g(; a, θ) =,adg (x;a,θ) deoes he derivaive of g(x; a, θ) wih respec o x. Ourcaseisa special model of he bea exeded Weibull family proposed by Cordeiro e al. [28] ad ca be obaied from equaio (7) whe b =.Weoeha(1) represesafamily of expoeiaed disribuios, as i covers he followig expoeiaed disribuios as special cases. (i) For g(x; a, θ) =xad a=, we ge he expoeiaed expoeial disribuio [see [1]]. (ii) For g(x; a, θ) =x 2 ad a=, i gives expoeiaed Rayleigh disribuio [see [29]]. (iii) For g(x; a, θ) = x δ, δ > ad a =,wegehe expoeiaed Weibull disribuio [see [1]]. (iv) For g(x; a, θ) =log(1+x δ ), δ>ad a=,ileadsus o pdf of expoeiaed Burr disribuio [see [24]]. (v) For g(x; a, θ) =log(x/a),iursouobeexpoeiaed Pareo disribuio [see [1]]. (vi) For g(x; a, θ) = (1 + (x/ξ)), ξ>ad a=,igives us expoeiaed Lomax disribuio [see [3]]. (vii) For g(x; a, θ) = (1 + (x ξ /])), ξ>, ] >ad a=,i urs ou o be expoeiaed Burr disribuio wih scale parameer ] [see [31]]. (viii) For g(x; a, θ) =x γ exp(]x), γ>, ] >ad a=, i gives he expoeiaed form of modified Weibull disribuio of Lai e al. [32]. (ix) For g(x; a, θ) =γexp[(x/γ) ] 1], γ>, ] >ad a=, we ge he expoeiaed form of a modificaio of Weibull disribuio [see [33]]. (x) For g(x; a, θ) = (x a)+(]/λ) log((x + ])/(a + ])), ] >, i gives expoeiaed form of he geeralized Pareo disribuio [see [34]]. 3. UMVUES of R() ad P Throughou his secio, we assume ha α is ukow, bu a, λ, adθ are kow. Le X 1,X 2,...,X be a radom sample of size from (1). Lemma 1. Le T = log(1 e λg(x i,a,θ) ).The,T is complee ad sufficie for he family of disribuios give i (1).Moreover,hepdfofT is g (; α, λ, θ) = α Γ () ( ) 1 e α ; < <. (2) Proof. Deoig by h(x 1,x 2,...,x 2 ;a,α,λ,θ) he joi pdf of X 1,X 2,...,X,wehave h(x 1,x 2,...,x 2 ;a,α,λ,θ) = (αλ) g (x i ;a,θ)e λ g(x i;a,θ) e (α 1) log(1 e λg(x i ;a,θ)). I follows from (3) ad Fishers-Neyma facorizaio heorem [see [35, p. 341]] ha T is sufficie for he family of disribuios f(x; a, α, λ, θ). I is easy o see ha he rv V = 2αlog(1 e λg(x;a,θ) ) follows χ 2 (2) disribuio. Thus, from he well-kow addiive propery of Chi-square disribuio [see Johso ad Koz, [36, p. 17]], 2αT = 2α log(1 e λg(x i;a,θ) ) is a χ 2 (2) rv ad he resul follows. Sice he disribuio of T belogs o expoeial family, i is also complee [see [35, p. 341]]. The followig lemma provides he UMVUES of he powers of α. Lemma 2. For q (, ),heumvueofα q is α q = Γ () Γ( q) ( T) q ; q <, ; oherwise. (3) (4)

3 Probabiliy ad Saisics 3 Proof. The resul follows from Lemma 1, Lehma-Scheffé heorem [see [35, p. 357]], ad he fac ha E( 2αT) q = Γ( q) 2 q ; q <. (5) Γ () I he followig lemma, we provide he UMVUE of he sampled pdf (1) a a specified poi x. Lemma 3. The UMVUE of f(x; a, α, λ, θ) a a specified poi x is f (x; a, α, λ, θ) ( 1) λt 1 (1 e λg(x;a,θ) ) 1 g (x; a, θ)e λg(x;a,θ) = [1 T 1 log (1 e λg(x;a,θ) )] 2 ; T < log 1 e λg(x;a,θ) }, ; oherwise. (6) Proof. Sice T is complee ad sufficie for he family of disribuios f(x; a, α, λ, θ), ay fucio H(T) of T saisfyig E[H(T)] = f(x; a, α, λ, θ) will be he UMVUE of f(x; a, α, λ, θ).tohised,from(1)adlemma 1, α H () Γ () ( ) 1 e α d (7) =αλg (x; a, θ) e λg(x;a,θ) (1 e λg(x;a,θ) ) α 1, or α 1 Γ () log(1 e λg(x;a,θ) ) H[ u+log 1 e λg(x;a,θ) }] [ u+log (1 e λg(x;a,θ) )] 1 e αu du =λ(1 e λg(x;a,θ) ) 1 g (x; a, θ) e λg(x;a,θ). Equaio (8) is saisfied if we choose H[ u+log (1 e λg(x;a,θ) )] ( ( 1) λg (x;a,θ) e λg(x;a,θ) ( u) 2 ) = ((1 e λg(x;a,θ) ) [ u + log (1 e λg(x;a,θ) )] 1 ) 1 ; u >, ; oherwise, (9) ad he lemma holds. Remark 4. We ca wrie (1)as f (x; a, α, λ, θ) = λ(1 e λg(x;a,θ) ) 1 (8) Usig Lemma 1 of Chaurvedi ad Tomer [37]ad Lemma 2, he UMVUE of f(x; a, α, λ, θ) a a specified poi x is or f (x; a, α, λ, θ) = λ(1 e λg(x;a,θ) ) 1 f (x; a, α, λ, θ) g (x; a, θ)e λg(x;a,θ) i= [log (1 e λg(x;a,θ) )] i i! α i+1 =λ(1 e λg(x;a,θ) ) 1 g (x;a,θ)e λg(x;a,θ) ( T) 1 2 Γ ( 1) Γ ( i 1) ( T)i i= [log (1 e λg(x;a,θ) )] i ( 1) λt 1 (1 e λg(x;a,θ) ) 1 i!, (11) g (x; a, θ)e λg(x;a,θ) = [1 T 1 log (1 e λg(x;a,θ) )] 2 ; T < log 1 e λg(x;a,θ) }, ; oherwise, (12) which coicides wih Lemma 3. Thus, he UMVUE of he power of α ca be used o derive he UMVUE of f(x; a, α, λ, θ) a a specified poi. I he followig heorem, we obai he UMVUE of R(). Theorem 5. The UMVUE of R() is give by 1 [1 T 1 log 1 e λg(;a,θ) }] 1 ; R () = T<log (1 e λg(;a,θ) ), ; oherwise. (13) Proof. Le us cosider he expeced value of he iegral f(x; a, α, λ, θ)dx wih respec o T;hais, f (x; a, α, λ, θ)dx}h(;α,λ,θ)d g (x;a,θ)e λg(x;a,θ) α i= [log (1 e λg(x;a,θ) )] i α i. i! (1) = [E T f (x; a, α, λ, θ)}] dx = f(x;a,α,λ,θ)} dx = R (). (14)

4 4 Probabiliy ad Saisics We coclude from (14) ha he UMVUE of R() ca be obaied simply iegraig f(x; a, α, λ, θ) from o.thus, deoig by g 1 ( ), he iverse fucio of g(x; a, θ), from Lemma 3,weobai R () = ( 1) T 1 g 1 λ 1 log(1 e T )} λ(1 e λg(x;a,θ) ) 1 g (x; a, θ)e λg(x;a,θ) [1 T 1 log (1 e λg(x; a,θ) )] 2 dx; (15) T<log 1 e λg(x;a,θ) 1 }= ( 1) T 1 log(1 e λg(;a,θ) ) (1 y) 2 dy, ad he heorem follows. Le X ad Y be wo idepede rvs followig he families of disribuios f 1 (x; a 1,α 1,λ 1,θ 1 ) ad f 2 (y; a 2,α 2,λ 2,θ 2 ), respecively. We assume ha α 1 ad α 2 are ukow bu a 1,a 2,λ 1,λ 2,θ 1,adθ 2 are kow. Le X 1,X 2,...,X be a radom sample of size from f 1 (x; a 1,α 1,λ 1,θ 1 ) ad le Y 1,Y 2,...,Y m be a radom sample of size m from f 2 (y; a 2,α 2,λ 2,θ 2 ). Le us deoe by T = log(1 e λ 1g(x i ;a 1,θ 1 ) ) ad S= m j=1 log(1 e λ 2h(y j ;a 2,θ 2 ) ). I wha follows, we obai he UMVUE of P. Theorem 6. The UMVUE of P isgiveby P = 1+(m 1) 1+ (m 1) [1 T 1 log 1 e λ 1g(h 1 ( (1/λ 2 ) log(1 e s] ))) }] 1 (1 ]) m 2 d]; 1 h 1 λ 1 2 log (1 es )}>g 1 λ 1 1 log (1 et )}, log1 S 1 e λ 2 h(g 1 ( (1/λ 1 ) log(1 e T ))) } [1 T 1 log 1 e λ 1g(h 1 ( (1/λ 2 ) log(1 e s] ))) }] 1 (1 ]) m 2 d]; h 1 λ 1 2 log (1 es )}<g 1 λ 1 1 log (1 et )}. (16) Proof. From he argumes similar o hose adoped i provig Theorem 5,i ca be show ha he UMVUE of P is give by P = y=a 2 x=y f 1 (x; a 1,α 1,λ 1,θ 1 ) f 2 (y; a 2,α 2,λ 2,θ 2 )dxdy. (17) Thus, usig Lemma 3, P = [ y=a 2 x=y ( 1) λ 1 (1 e λ 1g(x;a 1,θ 1 ) ) 1 g (x; a 1,θ 1 )e λ 1g(x;a 1,θ 1 ) (1 T 1 log (1 e λ 1g(x;a 1,θ ) 1 )) 2 dx] f2 (y; a 2,α 2,λ 2,θ 2 )dy =1+(m 1) λ 2 S 1 y=max[h 1 λ 1 2 log(1 es )},g 1 λ 1 1 log(1 et )}] 1 T 1 log (1 e λ 1g(y;a 1,θ 1) )} 1 (18) (1 e λ 2h(y;a 2,θ 2 ) ) 1 h (y; a 2,θ 2 )e λ 2h(y;a 2,θ 2 ) [1 S 1 log (1 e λ 2h(y;a 2,θ 2 ) )] m 2 dy. Le us firs cosider he case whe h 1 λ 1 2 log(1 es )} > g 1 λ 1 1 log(1 et )}.Ihiscase,from(18), P =1+(m 1) [1 T 1 log 1 e λ 1g(h 1 ( (1/λ 2 ) log(1 e S] ))) }] 1 (1 ]) m 2 d]. (19) 1

5 Probabiliy ad Saisics 5 Nowwecosiderhecasewheh 1 λ 1 2 log(1 es )} < g 1 λ 1 1 log(1 et )}.Ihiscase,from(18), P =1+(m 1) [1 T 1 log 1 e λ 1g(h 1 ( (1/λ 2 ) log(1 e s] ))) }] 1 (1 ]) m 2 d]. (2) log1 S 1 e λ 2 h(g 1 ( (1/λ 1 ) log(1 e T ))) } The heorem ow follows o combiig (19)ad(2). Corollary 7. For he case whe a 1 =a 2 =a,say,λ 1 =λ 2 =λ, say, θ 1 =θ 2 =θ,say,adg(x; a, θ) =h(y;a,θ), 1 1 ( 1) i ( 1)!(m 1)! i= ( i 1)!(m+i 1)! ( S i T ) ; T > S, P = m 2 1 ( 1) i i+1 ( 1)!(m 1)! i= (+i)!(m i 2)! (T S ), T < S. Remarks 1. (21) (i) I he lieraure, he researchers have deal wih he esimaio of R() ad P, separaely.ifwelooka he proofs of Theorem 5 ad Theorem 6, weobserve ha he UMVUE of he sampled pdf is used o obai he UMVUES of R() ad P. Thus, we have esablished ierrelaioship bewee he wo esimaio problems. (ii) I he lieraure, he researchers have derived he UMVUES of P forhecasewhex ad Y follow he same disribuio (may be wih differe parameers). We have obaied UMVUES of P forallhehree siuaios, whe X ad Y followhesamedisribuio havig all he parameers same oher ha α s, whe X ad Y havehesamedisribuiowihdiffere parameers ad whe X ad Y have differe disribuios. (iii) I follows from Lemma 2 ha Var( α) = α 2 /( 1) as.thus, α is a cosise esimaor of α. Sice f(x; a, α, λ, θ), R(), ad P are coiuous fucios of cosise esimaors, hey are also cosise esimaors of f(x; a, α, λ, θ), R(),ad P, respecively. 4. MLES of R() ad P Whe All he Parameers Are Ukow Lookig a he disribuios belogig o he family (1), we oice ha, excep for expoeiaed Pareo ad geeralized expoeiaed Pareo disribuios [(V) ad (X)], a=;ha is, a is kow. For hese wo disribuios θ coais a. Irrespecive of he disribuio, he MLE of a is a =X (1) = mi 1 i X i.from(3), he log-likelihood fucio is L (a, α, λ, θ x)=log α+log λ+ λg(x i ;a,θ)+(α 1) log (1 e λg(x i;a,θ) ). log g (x i ;a,θ) (22) Firsofall,wereplace a byx (1) i (22); he we differeiae (22) wih respec o all he ukow parameers ad equae hese differeial equaios o zero. The MLES of ukow parameers are obaied o solvig hese equaios simulaeously. Le α, λ, ad θ be he maximum likelihood esimaors of α, λ, adθ, respecively. The followig lemma provides The MLE of f(x; a, α, λ, θ) a a specified poi x. Lemma 8. The MLE of f(x; a, α, λ, θ) a a specified poi x is f (x; a, α, λ, θ) = α λg (x; a, θ)e λg(x; a, θ) (1 e λg(x; a, θ) ) α 1. (23) Proof. The proof follows from (1) ad he oe-o-oe propery of he MLE. I he followig heorem, we derive he MLE of R(). Theorem 9. The MLE of R() is give by R () =1 (1 e λg(; a, θ) ) α. (24) Proof. Usig Lemma 8 ad ivariace propery of he MLES, R () = f (x; a, α, λ, θ)dx = α λ g (x; a, θ)e λg(x; a, θ) (1 e λg(x; a, θ) ) α 1 dx = α 1 (1 e λg(; a, θ) ) ad he heorem follows. (1 u) α 1 du, (25) I he followig heorem, we obai he maximum likelihood esimaor of P.

6 6 Probabiliy ad Saisics Theorem 1. The MLE of P isgiveby 1 P = α 1 [1 exp λ 2 h (g 1 λ 1 1 log (1 u)})}] α 2 u α 1 1 du. (26) Proof. Usig Lemma 8 ad he oe-o-oe propery of he MLES P = y=y (1) x=y f 1 (x; a 1,α 1,λ 1,θ 1 ) f 2 (y; a 2,α 2,λ 2,θ 2 )dxdy = [1 R 2 (x)] f 1 (x; a 1,α 1,λ 1,θ 1 )dx y=y (1) = [1 exp λ 2 h(x; a 2, θ 2 2 )}] α y=y (1) ad he resul follows. α 1 λ1 g (x; a 1, θ 1 )e λ 1 g (x; a 1, θ 1 ) (1 e λ 1 g (x; a 1, θ 1 ) ) α 1 dx, (27) Corollary 11. For he case whe a 1 =a 2 =a,say,λ 1 =λ 2 =λ, say, θ 1 =θ 2 =θ,say,adg(x; a, θ) =h(y;a,θ), Remarks 2. P = α 1 α 1 + α 2. (28) (i) All he commes made uder Remarks 1 for UMVUES are eable for MLES. (ii) I he prese approaches of obaiig UMVUES ad MLES, oe does o eed he expressios of R() ad P. (iii) Sice he UMVUES ad MLES of powers of α are obaied uder he same codiios, we compare heir performaces. For q = 1 he UMVUE ad MLE of α are, respecively, α = ( 1)( T) 1 ad α = ()( T) 1. For hese esimaors V ( α) = Hece, α 2 ( 2), V( α) = 2 ( 1) 2 ( 2). (29) V ( α) V( α) = (2 1) ( 1)( 2) α2 >. (3) Thus, he UMVUE of α is more efficie ha is MLE. Similarly, we ca compare he performaces of hese esimaors for oher powers of α f3 =3 =2 =15 =1 =5 Figure 1: Cures of f(x; α, λ) ad f(x; α, λ). 5. Simulaio Sudies I order o verify he cosisecy of esimaors, we have draw samples of sizes = 5,2ad3from(1) wih g(x; a, θ) = x, a =, α = 3,adλ = 1.IFigure 1, we have ploed f(x; α, λ) correspodig o hese samples. Wecocludehaashesamplesizeicreases,hecurves of f(x; α, λ) come closer o he curve of f(x; α, λ), whichis ploedihesamefigureolyfor=3.for=3,he curves overlap. A similar paer follows for he curves of f(x; α, λ). For he case whe α is ukow bu oher parameers are kow, we have coduced simulaio experimes usig boosrap resamplig echique for sample sizes = 5, 1, 2, ad 5. The samples are geeraed from (1), wih, g(x; a, θ) = x, a =, α = 3,adλ = 1.Fordiffere values of, we have compued R(), R(), heir correspodig bias, variace, 95% cofidece legh, ad correspodig coverage perceage. All he compuaios are based o 5 boosrap replicaios ad resuls are repored i Table 1. I order o esimae P, for he case whe α 1 ad α 2 areukowbuoherparameersarekow,wehavecoduced simulaio experimes usig boosrap resamplig echique for sample sizes (, m) =(5,5),(1,1),(15,15), (25, 25), ad (3, 3). The samples are geeraed from (1), wih g(x; a 1,θ 1 ) = x,h(y;a 2,θ 2 ) = y, a 1 = a 2 =, λ 1 = λ 2 =1, α 1 =2,adα 2 =.5(.25)1.25. The compuaios are based o 5 boosrap replicaios. We have compued P, P, bias, variace, 95% cofidece legh, ad correspodig coverage perceage. The resuls are preseed i Table 2. I order o demosrae he applicaio of he heory developed i Secio 4, we cosider he followig example. f f

7 Probabiliy ad Saisics 7 Table 1: Simulaio resuls for R(). R() R() =5 =1 =2 =5 R() R() R() R() R() R() R() e 4 2e e e 4 5e e e e e 4.5 6e e Here, he firs row idicaes he esimae, he secod row idicaes he bias, he hird row idicaes variace, he fourh row idicaes 95% boosrap cofidece legh, ad he fifh row idicaes he coverage perceage. Le he rv X follow expoeiaed expoeial disribuio wih pdf f (x; α, λ, δ) = αλδx δ 1 e λxδ (1 e λxδ ) α 1 ; x,α,λ,δ>, (31) where α, λ,adδ are ukow. Deoig, L 1 (α, λ, δ x),he likelihood, he log-likelihood is give by log L 1 (α, λ, δx) =log α+log λ+log δ + (δ 1) + (α 1) log x i λ x δ i log (1 e λxδ i ). (32) Cosiderig egaive log-likelihood, he differeiaig i wih respec o all ukow parameers, ad equaig hese differeial coefficies o zero, from (32), α log (1 e λxδ i )=, λ + x δ x δ i i (α 1) e λxδ i (1 e λxδ i ) =, δ log x i +λ x δ i e λxδ i log x i (1 e λxδ i ) =. x δ i log x i (δ 1) λ (33)

8 8 Probabiliy ad Saisics Table 2: Simulaio resuls for P. (α 1,α 2 ) (1,.5) (1,.75) (1, 1.) (1, 1.25) P (, m) P P P P P P P P e 4 (5, 5) e 4 (1, 1) (15, 15) e 4 6e (25, 25) e 4 1e (3, 3) Here, he firs row idicaes he esimae, he secod row idicaes he bias, he hird row idicaes variace, he fourh row idicaes 95% boosrap cofidece legh, ad he fifh row idicaes he coverage perceage. The followig sample of size 5 is geeraed from (31), for α = 3.5, λ=1,adδ=1as follows. Sample ,.588, ,.9756, 2.362, , 1.433, ,.9241, 2.591, 1.221, 1.322, ,.8146, 1.783,.3126, ,.9462,.4557, 2.12, , ,.973, 1.35,.4618,.7465, , , ,.9742, 2.741, , , , , , , , 1.191,.8248, 3.91,.9511, 1.338, , 2.216, 1.674, 1.562, 2.339,.6882, Assumig ha he daa represes life spas of iems i hours, R(.35) =.986, adsolvig(33) simulaeously, we ge α = , λ = , δ = ,ad R(.35) = I order o obai he maximum likelihood esimaor of P, we have geeraed oe more sample of size 5 from (31), for α=3, λ=1,adδ=1as follows. Sample ,.9467, 1.98,.6613,.4526, 1.571, 1.414, 1.98, , , 1.412,.564, 1.674, 2.615, 1.436, ,.7443,.9568, 1.636,.2327, 1.24,.8178,.774, ,.846, , 1.72, , , , 3.41, ,.6874, , , ,.4313, , , 1.575, , 1.437, , , ,.829,.6693, , 3.265, For his sample, we have α = , λ = , ad δ = Usig hese resuls (obaied from Samples 1 ad 2) ad Theorem 1, wegep = ad P = Coclusios We propose a class of disribuios, which covers as may as e expoeiaed disribuios as special cases. The problems of esimaig R() ad P are cosidered. UMVUES ad MLES are derived. A comparaive sudy of he wo mehods of esimaio is doe. The esimaors of P are derived, which cover he cases whe X ad Y may follow he same, as well as, differe disribuios. By esimaig he sampled pdf o obai he esimaors of R() ad P, a ierrelaioship bewee he wo esimaio problems is esablished. Simulaio sudy is performed, ad a real-daa example is preseed.

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11 Advaces i Operaios Research Advaces i Decisio Scieces Applied Mahemaics Algebra Probabiliy ad Saisics The Scieific World Joural Ieraioal Differeial Equaios Submi your mauscrips a Ieraioal Advaces i Combiaorics Mahemaical Physics Complex Aalysis Ieraioal Mahemaics ad Mahemaical Scieces Mahemaical Problems i Egieerig Mahemaics Discree Mahemaics Discree Dyamics i Naure ad Sociey Fucio Spaces Absrac ad Applied Aalysis Ieraioal Sochasic Aalysis Opimizaio