Estimating of $P(Y<X)$ in the Exponential case Based on Censored Samples

Estimating of P ( Y < X )in the Exp onen tial case Based on Censored Samples Ab d Elfattah, A. M. Departmen t of mathematical Stat istics , Institute of Statistical Studies a nd Researc h Cairo Univ ersity , Cairo, Egypt. Marw a O. Mohamed Departmen t of mathematics, Z agazig Univ ersit y , Cairo, Egypt. Abstract In this article, the estimation of reliability of a system is discussed p ( y < x )when strength, X , and stress, Y , a r e tw o indep enden t exp onential distribution with differ- ent scale para meter s when the av ailable data are type I I Censored sample. Different metho ds for estimating the reliability ar e applied . The point estimato rs obtained are ma x im um likeliho od estimator , unifor mly minimum v ariance unbiased estimator, and Bay esia n estimators based on conjugate and non informative prior distributions. A compariso n of the estimates obtained is p erformed. Interv al estimato rs of the r eliabilit y are also discusse d. Key W ords: Maxim um lik eliho o d estimator; Unbiasedness; Consistency ; Uniform mini- m u m v ariance unbiased estimato r ; Ba y esian estimator; Piv otal quan tit y; Fisher inf ormation . 1 1 In tro du ction In life testing, it is often the case that items dr a wn from a p opulation are put on test and their times of f ailure are recorded. F or a num b er of reasons, such as budget or limited time, it is often necessary to terminate th e test b efore all th e f ailur e times ha ve b een observed; In th is case th e data b ecome av ailable in some ord ered manner. Th e test is usu ally terminated after a fixed time or a fixed num b er of f ailur es are observ ed giving a censored sample see E pstein and Sob el (1953) . In s tr ess-strength mo del, the stress Y and the strength X are treated as random v ariables and the reliabilit y of a comp onen t du ring a give n p erio d (0 , T )is tak en to b e the p robabilit y that the strength exceeds the stress during the entire in terv al. the reliabilit y of a comp onen t is P ( Y < X ). Sev eral authors ha ve considered differen t studies for stress and strength with exp onen tial d istribution with complete sample. T ong(1974) discussed the estimatio n of P ( Y < X ) in the exp onen tial case. T ong(1977) had a lo ok at the estimation of P ( Y < X ) for exp onen tial families. A go o d review of the literature can b e fou n d in Johnson (1988). Beg(19 80a,b and c)e s timated the exp onentia l family ,t wo parameter exp onentia l distribution and truncation parameter distributions . Basu(19 81) considered maximum lik eliho o d estimators (MLE) for P ( Y ≤ X ) in case of gamma and exp onentia l distributions. Sathe and Shah (198 1) stud ied estimation of P ( Y > X ) for the e xp onen tial distr ibution. C h ao (19 82) provi ded simple approxi mations for b ias and mean square error of the maxim u m lik eliho o d estimators of reliabilit y when stress and strength are indep en d en t exp onenti ally distributed random v ariables. Aw ad and Charraf (1986) stud ied three differen t estimators for r elia bilit y has a b iv a r iate exp onential distribution. Dinh, et al (199 1) obtained the MVUE of R when X and Y ha ve th e biv ariate normal d istribution. Moreo v er, they considered the case when X and Y ha ve th e biv ariate exp onen tial distribution. Bai and Hong (1992 ) estimated P ( Y ≤ X ) in the exp onentia l case with common location parameter. Kunch ur and Mounoli (1993 ) o btained UMVUE of stress- strength m odel for multi comp onen t surviv al mo del b ased on exp onentia l distrib ution for parallel system. Siu-Keun g Tse and Geoffrey Tso (1996) studied the sh rink age estimation of reliabilit y f or exp onen tial distributed lifetime. A note on the UMVUE on P ( Y ≤ X ) in the exp onential case discussed in Cramer and Kamps (1997). S elv av el, et al (2000) studied 2 reliabilit y R when ( X , Y ) jointly follo ws a tr uncated biv ariate exp onential distribu tion with a common parameters. Kha y ar (2001) discussed the reliabilit y of time d ep en den t stress- strength mo dels for exp onentia l and Ra yleigh d istributions. Shrink age estimation of P ( Y ≤ X ) in the exp onential case discussed by Ayman and W alid (2003).T ac hen (2005)dea ls with the empirical Bay es testing the reliabilit y of an exp onen tial d istr ibution . In the present articl e, the reliabilit y , R , is studied when X and Y tw o indep end ent exp onen tial distribution with different s cale parameters. Whic h can b e represen ted as f ( x ; α ) = 1 α e − x α , x > 0 , α > 0 . Differen t estimators of R are deriv ed, namely , maxim um lik eliho o d estimator (MLE), un i- form minimum v ariance unbiased estimator, (UMVUE), and Ba y esian estimators with mean square error loss f u nctions corresp ondin g to conjugate and non informativ e pr iors. A com- prehensive comparison of the v arious p oint estimators (MLE,UMVUE, and Bay es) is p er- formed on the basis of the mean squared err or. Interv al estimators of R are also discus s ed. A numerical comparison of the int erv als obtained. 2 Reliabilit y Let X b e the strength of a comp onent and Y b e the stress acting on it. Let X and Y b e exp onen tial indep endent rand om v ariables with p arameters α and β , resp ectiv ely . That is , the prob ab ility densit y functions (p dfs) of X and Y are, resp ectiv ely , f ( x ; α ) = 1 α e − x α , x > 0 , α > 0 (2 . 1) , and f ( y ; β ) = 1 β e − y β , y > 0 , β > 0 (2 . 2) , where α and β are un kno wn parameters . The reliabilit y of the comp onen t will b e R = P ( Y < X ) = 1 α R 0 ∞ (1 − e − x β ) e − x α dx = f r acαα + β . (2 . 3) if α and β are kn o wn then R is simply calculated u sing Eq .(2.3). 3 3. P oin t Estimation of R 3.1 Maxim um Likelih o o d Estimator of R If α and β are unk n o wn the MLE of, ˆ R 1 , of R is giv en b y ˆ R 1 = f r ac ˆ α ˆ α + ˆ β , (3 . 1) where ˆ α and ˆ β are the MLEs of α and β , resp ectiv ely . F or obtaining ˆ α and ˆ β we argue as follo ws: Supp ose th at r 1 comp onen ts wh ere r 1 ≤ n with strengths X i ; i = 1 , ..., r 1 , eac h of whic h ha ving exp onenti al distrib u tion with p aramete r α as in Eq. (2.1) are s u b jected, resp ectiv ely , the lik eliho od function will b e L = n ! ( n − r 1 )! Q i =1 r 1 1 α e − x i α [ e − x r 1 α ] n − r 1 L = n ! ( n − r 1 )! ( 1 α ) r 1 e − P r 1 i =1 x i α − − x r 1 ( n − r 1 ) α , eq no (3 . 2) T aking the logarithm of Eq. (3.2) and fi nd th e deriv ativ e w ith resp ect to α dlnL α = − r 1 α + P r 1 i =1 x i α 2 + ( n − r 1 ) x r 1 α 2 = 0 − r 1 α = P r 1 i =1 x i + ( n − r 1 ) x r 1 , ˆ α = P r 1 i =1 x i + ( n − r 1 ) x r 1 r 1 , (3 . 3) T o stress Y j ; j = 1 , ..., r 2 , ha ving exp onent ial distribution w ith p aramete r β as in E q. (2.2), where r 2 ≤ m . Assuming that X 1 and Y j ; i = 1 , ..., r 1 and j = 1 , ..., r 2 , are in d ep enden t , the lik eliho od function will b e L = m ! ( m − r 2 )! Q j =1 r 2 1 β e − y j β [ e − y r 2 β ] m − r 2 L = m ! ( m − r 2 )! ( 1 β ) r 2 e − P r 2 j =1 y j β − − y r 2 ( m − r 2 ) β , (3 . 4) T aking the logarithm of Eq. (3.4) and fi nd th e deriv ativ e w ith resp ect to β dlnL β = − r 2 β + P r 2 j =1 y j β 2 + ( m − r 2 ) y r 2 β 2 = 0 − r 2 β = P r 2 j =1 y j + ( m − r 2 ) y r 2 , The MLE of β will b e ˆ β = P r 2 j =1 y j + ( m − r 2 ) y r 2 r 2 , (3 . 5) 4 No w w e shall study some p rop erties of ˆ R 1 . First, if r 1 = r 2 = r : 1] Un biasedness E ( ˆ R 1 ) = α α + r β r +1 [1 − (2 r − 1) r ( r − 2) (1 − α α + r β r +1 ) 2 ] lim r →∞ E ( ˆ R 1 ) = R − R lim r →∞ (1 − R ) 2 ( r − 1) then lim r →∞ E ( ˆ R 1 ) = R then, ˆ R 1 asymptotically unbiased estimator of R . 2] Consistency V ar ( ˆ R 1 ) = (2 r − 1) r ( r − 2) [ r β ( r − 1) α ( r +1) αβ ( r − 1) β ] 2 [ 1 1+ r β ( r − 1) α ] 2 lim r →∞ V ar ( ˆ R 1 ) = R 2 ( β α lim r →∞ 1 r then lim r →∞ V ar ( ˆ R 1 ) = 0 then, ˆ R 1 is a consistent estimator for R . Second, if r 1 6 = r 2 : 1] Un biasedness E ( ˆ R 1 ) = α α + r 1 β r 1 +1 [1 − ( r 1 + r 2 − 1) r 2 ( r 1 − 2) (1 − α α + r 1 β r 1 +1 ) 2 ] F or fixed r 2 , lim r 1 →∞ E ( ˆ R 1 ) = lim r 1 →∞ α α + r 1 β r 1 +1 [1 − ( r 1 + r 2 − 1) r 2 ( r 1 − 2) (1 − α α + r 1 β r 1 +1 ) 2 ] lim r 1 →∞ E ( ˆ R 1 ) = R [1 − 1 r 2 (1 − R ) 2 ] then lim r 1 ,r 2 →∞ E ( ˆ R 1 ) = R then, ˆ R 1 asymptotically unbiased estimator of R . 2] Consistency V ar ( ˆ R 1 ) = ( r 1 + r 2 − 1) r 2 ( r 1 − 2) [ r 1 β ( r 2 − 1) α ( r 1 +1) αβ ( r 2 − 1) β ] 2 [ 1 1+ r 1 β ( r 1 − 1) α ] 2 F or fixed r 2 , 5 lim r 1 →∞ V ar ( ˆ R 1 ) = lim r 1 →∞ ( r 1 + r 2 − 1) r 2 ( r 1 − 2) lim r 1 → ∞ [ r 1 β ( r 2 − 1) α ( r 1 +1) αβ ( r 2 − 1) β ] 2 lim r 1 →∞ [ 1 1+ r 1 β ( r 1 − 1) α ] 2 and, lim r 1 ,r 2 →∞ V ar ( ˆ R 1 ) = R 2 [ β α ] 4 lim r 2 →∞ 1 r 2 then lim r 1 ,r 2 →∞ V ar ( ˆ R 1 ) = 0 then, ˆ R 1 is a consistent estimator for R . 3.2 Uniform Minim um V ariance Un biased Est imator of R Let X 1 , ..., X r 1 and Y 1 , ..., Y r 2 b e tw o indep endent random s amp les , of size r 1 and r 2 , resp ec- tiv ely , dr a wn from exp onen tial distribu tions with parameters α and β , resp ectiv ely , Define z i = l ne x i , v j = l ne y j , i = 1 , ..., r 1 and j = 1 , ..., r 2 , Z = P i = 1 r 1 z i ,and V = P j = 1 r 2 v j Clearly fr om Eq. (3.2) and (3.4) we see that Z , V is a complete sufficient statisti c for α , β . No w, w e ha ve E ( W ) = 1 .P ( v 1 < z 1 ) + 0 .P ( v 1 ≥ Z 1 ) E ( W ) = P ( l ne y j < l ne x i ) E ( W ) = P ( y < x ) = R Let W b e the indicator v ariable I [0 ,z 1 ) ( W ).it could b e seen that W b e unbiased estimator for R , by u sing Rao-Blac k W ell and Lehmann-S c heff ´ e we h a v e ˆ R 2 is UMVUE for R .(see Mo od et al (1974)). ˆ R 2 = E ( W / Z, V ) ˆ R 2 = R z 1 R v 1 wf ( z 1 , v 1 / Z, V ) dv 1 dz 1 where f ( z 1 , v 1 / Z, V ) is the conditional p df of z 1 , v 1 giv en Z , V . Notice that z 1 and v 1 are indep endent exp onenti al random v ariables with parameters α and β , resp ectiv ely , and that Z and V are indep en d en t gamma random v ariables with parameters ( n , α ) and ( m, β ) , resp ectiv ely . W e see that Z − z 1 and V − v 1 are imp endent gamma random with parameters ( n − 1 , α ) and ( m − 1 , β ) , resp ectiv ely . Moreo ver Z − z 1 and z 1 ,as w ell as V − v 1 and v 1 are also 6 indep endent. W e see that ˆ R 2 = R z 1 R v 1 w Γ( r 1 )( Z − z 1 ) ( r 1 − 2) Γ( r 2 )( V − v 1 ) ( r 2 − 2) Γ( r 1 − 1)( z 1 ) ( r 1 − 1) Γ( r 2 )( v 1 ) ( r 2 − 1) dv 1 dz 1 ˆ R 2 = Γ( r 1 )Γ( r 2 ) Γ( r 1 − 1)Γ( r 2 − 1) z ( r 1 − 1) v ( r 2 − 1)      R v 0 R z v 1 ( V − v 1 ) ( r 2 − 2) ( Z − z 1 ) ( r 1 − 2) dz 1 dv 1 , v 1 < z 1 , R z 0 R z 1 0 ( Z − z 1 ) ( r 1 − 2) ( V − v 1 ) ( r 2 − 2) dv 1 dz 1 , v 1 ≥ z 1 ; (3 . 6) The computation of the UMVUE ˆ R 2 is v ery complicated as it can s een from equation (3.6).so , w e will u se the MA T HCAD program to ev aluate the v alue of ˆ R 2 . 3.3. Ba yes Est imator of R W e obtain Bay es Estimator of R w ith resp ect to the mean s quare error loss fu nction with resp ect to conjugate and n on informativ e prior distributions. 3.3.1. Conjugate gamma prior distribution Let X 1 , ..., X r 1 and Y 1 , ..., Y r 2 b e the first r 1 and r 2 failure observ ations from X 1 , ..., X n and Y 1 , ..., Y m resp ectiv ely , where b oth of them h a v e exp onen tial d istribution with parameters α and β r esp ectiv ely . Assume that the p rior distrib ution of α is give n by π 01 = f ( α ) = v u 1 1 Γ( u 1 ) ( 1 α ) u 1 − 1 e − v 1 α , u 1 , v 1 , α > 0 the lik eliho od function with t yp e I I censored sample is , resp ectiv ely , f ( x 1 , ..., x r 1 | α ) = n ! ( n − r 1 )! ( 1 α ) r 1 e − 1 α ( P i =1 r 1 x i + x r 1 ( n − r 1 )) , (3 . 7) and f ( y 1 , ..., y r 2 | β ) = m ! ( m − r 2 )! ( 1 β ) r 2 e − 1 β ( P j =1 r 2 y j + y r 2 ( m − r 2 )) , (3 . 8) Assuming that α and β are indep end en t ha ving p rior gamma distribu tions, the p osterior distributions of α and β will b e gamma distrib ution also, π 2 = f ( α | x 1 , ..., x r 1 ) = ( v 1 + P r 1 i =1 x i + x r 1 ( n − r 1 )) u 1 + r 1 ( 1 α ) r 1 + u 1 e − ( P r 1 i =1 x i + x r 1 ( n − r 1 )) α Γ( r 1 + u 1 + 1) , (3 . 9) 7 and π 3 = f ( β | y 1 , ..., y r 2 ) = ( v 2 + P r 2 j =1 y j + y r 2 ( m − r 2 )) u 2 + r 2 ( 1 β ) r 2 + u 2 e − ( P r 2 j =1 y j + y r 2 ( m − r 2 )) β Γ( r 2 + u 2 + 1) , (3 . 10) the joint p osterior fu nction, put, ζ = ( v 1 + r 1 X i =1 x i + x r 1 ( n − r 1 )) , τ = ( v 2 + r 2 X j =1 y j + y r 2 ( m − r 2 )) and k = 1 Γ( r 1 + u 1 +1)Γ( r 2 + u 2 +1) π ( α, β | x, y ) = kζ ( u 1 + r 1 ) τ ( u 2 + r 2 ) ( 1 α ) r 1 + u 1 ( 1 β ) r 2 + u 2 e − ζ α e − τ β , (3 . 11) Hence Ba yes estimator ˆ R 3 of R will b e ˆ R 3 = E ( R | x, y ) = k ζ ( u 1 + r 1 ) τ ( u 2 + r 2 ) Γ( r 1 + u 1 + r 2 + u 2 + 1) Z 1 0 R ( r 1 + u 1 ) (1 − R ) ( r 2 + u 2 +1) ((1 − R ) ζ + Rτ ) u 1 + r 1 + u 2 + r 2 dR, (3 . 12) F r om equation (3.12) there is no explicit form of ˆ R 3 so, The compu tatio n of the Bay es estimator ˆ R 3 ,is very complicated ,we will u se the MA THCAD program to ev aluate the v alue of ˆ R 3 . 3.3.2. Non Informativ e P rior Distributions Let X 1 , ..., X r 1 b e a rand om sample from exp onent ial distribution with parameter α . The prior distribution of α is prop ortional to p I ( α ) , where I ( α ) is Fisher’s information of the sample ab out α , an d is giv en by I ( α ) = 1 α 2 , (3 . 13) from that the prior distribution π 1 ∝ 1 α , (3 . 14) Similarly , if Y 1 , ..., Y r 2 is a r andom sample from exp onen tial distrib u tion w ith parameter β , the prior d istribution of β will b e giv en by: π 2 ∝ 1 β , (3 . 15) 8 if w e h a v e α and β are indep endent then the p osterior joint distribution of α and β ,will b e π ( α, β | x 1 , ..., x r 1 , y 1 , ..., y r 2 ) ∝ L ( x 1 , ..., x r 1 | α ) L ( y 1 , ..., y r 2 | β ) π 1 ( α ) π 2 ( β ) , (3 . 16) then π ( α, β | x 1 , ..., x r 1 , y 1 , ..., y r 2 ) = n ! ( n − r 1 )! ( 1 α ) r 1 e − 1 α ( P i =1 r 1 x i + x r 1 ( n − r 1 )) m ! ( m − r 2 )! ( 1 β ) r 2 e − 1 β ( P j =1 r 2 y j + y r 2 ( m − r 2 )) , α, β > 0 put δ = ( X i =1 r 1 x i + x r 1 ( n − r 1 )) , ε = ( X j =1 r 2 y j + y r 2 ( m − r 2 )) π ( α, β | x 1 , ..., x r 1 , y 1 , ..., y r 2 ) = n ! m ! α − (1+ r 1 ) β − (1+ r 2 ) e − δ α e − ε β ( n − r 1 )!( m − r 2 )! , α, β > 0 , (3 . 17) under the m ean squ are error, Bay es estimator ˆ R 4 of R will b e ˆ R 4 = E ( R | x, y ) = n ! m ! ( n − r 1 )!( m − r 2 )!( r 1 + r 2 + 3)! Z 1 0 R ( r 2 +2) (1 − R ) ( r 1 +1) ((1 − R ) δ + Rε ) r 1 + r 2 +2 dR, (3 . 18) F r om equation (3.18) there is no explicit form of ˆ R 4 so, The compu tatio n of the Bay es estimator ˆ R 4 ,is very complicated ,we will u se the MA THCAD program to ev aluate the v alue of ˆ R 4 . 4. In terv al Estimation of R 4.1 Appro ximate confidence interv al Let X 1 , ..., X r 1 and Y 1 , ..., Y r 2 b e a r andom samples with size r 1 , r 2 from exp onentia l d istri- bution with parameter α, β resp ectiv ely ,w e can show that the maxim u m likelihoo d function of R , ˆ R is asymptotically normal distr ib ution with mean R and v ariance-co v ariance matrices I − 1 ( α, β ) where I ( α, β ) is the Fisher information matrix and giv en by: I ( α, β ) = r 1 α 2 0 0 r 2 β 2 ! 9 The v ariance-co v ariance matrix is obtained by in ve r ting the information matrix with elemen ts that are negat ives of the exp ected v al ues of the second order d eriv at ives of log- arithms of the lik eliho od functions, and th e asymptotic v ariance-co v ariance matrix is ob- tained by replacing v alues by their m axim um like liho od estimators. Hence, the asymptotic v ariance-co v a r iance matrix will b e I − 1 ( α, β ) = β 2 r 2 0 0 α 2 r 1 ! F r om (Su rles and P adj ett (2001)) that the maxim um lik elihoo d fu nction of R , ˆ R 1 is asymp - totical ly normal d istribution with mean R an d v ariance σ 2 ˆ R 1 = r 1 r 2 ( α + β ) 2 α 2 β 2 Hence,(1 − α )10 0 an appro ximate confidence inte r v al for R would b e ( L 1 , U 1 ) ,and the v alue of L 1 , U 1 is giv en as: L 1 = ˆ R 1 − z 1 − α 2 σ ˆ R 1 , (4 . 1) and U 1 = ˆ R 1 + z 1 − α 2 σ ˆ R 1 , (4 . 2) where z 1 − α 2 quan tile of the stand ard norm al distribution and ˆ R 1 is giv en by Eq.(3.1). 4.2 Exact confidence in terv al Let X 1 , ..., X r 1 and Y 1 , ..., Y r 2 b e rand om samples with size r 1 , r 2 from exp onen tial d istri- bution with parameter α, β resp ectiv ely . Since z i = lne x i and v j = lne y j are in dep enden t with gamma distrib ution with p arame- ters ( r 1 , α ) and ( r 2 , β ) resp ectiv ely ,2 αl ne P r 1 i =1 x i and 2 β lne P r 2 j =1 y j are indep endent with chi square distribu tions with degree of freedom 2 r 1 and 2 r 2 hence ˆ r R 1 = (1 + β α ) − 1 w e kno w that F 1 = f r acV αZ β has F-distribution with (2 r 1 , 2 r 2 )degrees of freedom , ˆ r R 1 = (1 + β α F 1 ) − 1 ,this equation can b e written as follo w F 1 = (1 − ˆ R 1 ) ˆ R 1 R (1 − R ) using F 1 as a pivo tal quanti t y ,we obtain a (1 − α ) 100% confidence interv al for R as L 2 = F 1 − α 2 (2 r 2 , 2 r 1 )( F 1 − α 2 (2 r 2 , 2 r 1 ) + V Z ) − 1 10 and U 2 = F α 2 (2 r 2 , 2 r 1 )( F α 2 (2 r 2 , 2 r 1 ) + V Z ) − 1 where F α is (1 − α )th of an F distribu tion r andom v ariables with (2 r 2 , 2 r 1 ) degrees of fr eedom. 5. Numerical illustrations in this section we w ill compare the d ifferen t p oint estimat ors of R , namely ˆ R 1 , ˆ R 2 , ˆ R 3 and ˆ R 4 and in the cases of R = 0 . 25 , R = 0 . 4 , R = 0 . 5 and R = 0 . 538 ,and differen t v alues of their p a- rameters α , β . 299 9 samples are generated of v arious size of n = 5 , 10 , 15 , 20 , n = 25 and n = 50 f rom exp onential d istribution with parameters α , β resp ectiv ely . 5.1. Numerical illustrations in the case r 1 = r 2 in tables(1-4) 1-MLE has the s mallest mean square err ors exp ect at some p oin ts of the UMVUE is the smallest. and f or α = 2 , β = 6 .b oth Ba yes estimator and Non-Informativ e estimator are the smallest. 2-F or some v alues of n, m, r 1 , r 2 , α and β Ba y esian estimator has adv an tage ov er UMVUE comparing the mean square errors, such th at: ∗ In table (1) in case α = 2 , β = 3 at n = m = 5 , r 1 = r 2 = 3. ∗ In table (2) in case α = 2 , β = 6 for all v alues of n, m except at n = m = 5 , r 1 = r 2 = 4. ∗ In table (4) in case α = β = 7 for the v alues of n = m = 15 , r 1 = r 2 = 12 . 3-F or all v alues of n, m, r 1 and r 2 Ba y esian estimator equal to Non-Informativ e estimator. 5.2. Numerical illustrations in the case r 1 6 = r 2 in tables(5-7) 1-MLE has the s mallest mean square err ors exp ect at some p oin ts of the UMVUE is the smallest .and for α = 2 , β = 6 .b oth Ba ye s estimator and Non-Informative estimator are the smallest. 2-F or some v alues of n, m, r 1 , r 2 , α an d β UMVUE has adv an tage ov er MLE comparing the mean square err ors, su c h that: 11 ∗ In table (5) in case α = 2 , β = 3 f or the v alues of n = 5 , m = 4 , r 1 = 3 , r 2 = 2 and in the case n = 10 , m = 5 , r 1 = 6 , r 2 = 5, also in the case n = 25 , m = 10 , r 1 = 20 , r 2 = 9 . ∗ In table (6) in case α = 2 , β = 6 th e UMVUE is the smallest in the all ca s es exp ect for the v alues of n = 10 , m = 25 , r 1 = 10 , r 2 = 20 and in the case n = 15 , m = 25 , r 1 = 10 , r 2 = 20 ,also in the case and n = 50 , m = 25 , r 1 = 4 , r 2 = 24. ∗ In table (7) in case α = β = 7 the UMVUE is th e smallest in the all cases exp ect for th e v alues of n = 5 , m = 4 , r 1 = 3 , r 2 = 2 and n = 5 , m = 5 , r 1 = 4 , r 2 = 3. 3-F or some v alues of n , m, r 1 , r 2 , α and β Ba y esian estimator is equal to UMVUE comparing the mean square err ors. 4-all mean square errors (MSE1, MSE2, MSE3 gamma and MSE4) increases as α or β increases when we compare the case of r 1 = r 2 and r 1 6 = r 2 w e ha v e that, in case r 1 6 = r 2 the mean square errors of all p oints estimators are s maller than the mean square errors in r 1 = r 2 case, exp ect some p oin ts. 12 Comparisons b et w een differen t estimators for R = P ( Y < X ) in censored exp onen tial case T abl e (1) m n r 1 r 2 R 1 MSE1 R 2 MSE2 R 3 MSE3 R 4 MSE4 5 5 3 3 0.276 0.0150 0. 536 1.39 0.0017 7 0.1 59 0.0 000432 0. 160 5 5 4 4 0.631 0.0530 1.90 0.7 02 0.00 0.1 60 0.00 0.160 10 10 6 6 0.62 0. 05 0.66 0.25 0.00 0.16 0.00 0.16 10 10 7 7 0.70 0. 09 2.14 0.00 0.00 0.16 0.00 0.16 10 10 8 8 0.44 0. 00 0.58 0.02 0.00 0.16 0.00 0.16 10 10 9 9 0.76 0. 13 0.63 0.68 0.00 0.16 0.00 0.16 15 15 12 12 0.70 0.09 0.73 0.59 0.00 0.16 0.00 0.16 15 15 13 13 0.59 0.04 0.58 0.52 0.00 0.16 0.00 0.16 15 15 14 14 0.53 0.02 0.59 0.57 0.00 0.16 0.00 0.16 20 20 15 15 0.66 0.07 0.56 0.53 0.00 0.16 0.00 0.16 20 20 16 16 0.54 0.02 0.64 0.75 0.00 0.16 0.00 0.16 20 20 17 17 0.73 0.11 0.70 0.00 0.00 0.16 0.00 0.16 25 25 23 23 0.60 0.04 0.00 0.61 0.00 0.16 0.00 0.16 25 25 24 24 0.48 0.01 0.67 0.73 0.00 0.16 0.00 0.16 50 50 4 4 0.60 0.0 4 0.73 0.73 0.00 0.16 0.00 0.16 50 50 6 6 0.37 0.0 0 0.68 0.76 0.00 0.16 0.00 0.16 50 50 9 9 0.67 0.0 8 0.61 0.45 0.00 0.16 0.00 0.16 α = 2 , β = 3 andR = 0 . 4 13 T abl e (2) m n r 1 r 2 R 1 MSE1 R 2 MSE2 R 3 MSE3 R 4 MSE4 5 5 3 3 0.725 0.226 0.526 0.07 6 0.0003457 0.062 0.051 0.04 5 5 4 4 0.789 0.291 2.244 3.97 8 8.594E- 11 0.062 0.000667 8 0.062 10 10 6 6 0.773 0.27 3 0.668 0.175 0 0.0 63 0 0.0 63 10 10 7 7 0.657 0.16 5 0.789 0.29 0 0.063 0 0.063 10 10 8 8 0.671 0.177 0.656 0.164 0 0.063 0 0.063 10 10 9 9 0.708 0.209 0.619 0.136 0 0.063 0 0.063 15 15 12 12 0.657 0.165 0.645 0.156 0 0.063 0 0.063 15 15 13 13 0.741 0.241 0.783 0.284 0 0.063 0 0.063 15 15 14 14 0.812 0.316 0.673 0.179 0 0.063 0 0.063 20 20 15 15 0.662 0.17 0.779 0.28 0 0.063 0 0.063 20 20 16 16 0.71 0.212 0.679 0.184 0 0.063 0 0.063 20 20 17 17 0.727 0.228 0.719 0.22 0 0.063 0 0.063 25 25 23 23 0.799 0.301 0.734 0.234 0 0.063 0 0.063 25 25 24 24 0.643 0.154 0.78 0.281 0 0.063 0 0.063 50 50 4 4 0.518 0.072 0.771 0.272 0 0.063 0 0.063 50 50 6 6 0.688 0.192 0.571 0.103 0 0.063 0 0.063 50 50 9 9 0.794 0.296 0.734 0.234 0 0.063 0 0.063 α = 2 , β = 6 andR = 0 . 25 14 T abl e (3) m n r 1 r 2 R 1 MSE1 R 2 MSE2 R 3 MSE3 R 4 MSE4 5 5 3 3 0.486 0.0027 21 0.896 0.128 0.0029 0.287 0.00001 6 0.29 5 5 4 4 0.546 0.0000 6 0.91 0.138 6.109E-08 0.29 1.652 1.239 10 10 6 6 0.516 0.00 05 0.532 0.00004 5 0 0.29 0 0.29 10 10 7 7 0.607 0.00 46 0.639 0.01 0 0.29 0 0.29 10 10 8 8 0.563 0.00059 0.569 0.00 095 0 0.29 0 0.29 10 10 9 9 0.556 0.00031 1.703 1.357 0 0.29 0 0.29 15 15 12 12 0.434 0.011 0.578 0.0016 0 0.29 0 0.29 15 15 13 13 0.353 0.035 0.545 0.00 004 0 0.29 0 0.29 15 15 14 14 0.413 0.016 0.616 0.006 0 0.29 0 0.29 20 20 15 15 0.429 0.012 0.532 0.0000417 0 0.29 0 0.29 20 20 16 16 0.543 0.000023 0.0004 5 0.289 0 0.29 0 0.29 20 20 17 17 0.482 0.00 32 0.639 0.01 0 0.29 0 0.29 25 25 23 23 0.526 0.00016 0.0000053 0.29 0 0.29 0 0.29 25 25 24 24 0.46 0.006 215 4.86E-12 0.29 0 0.29 0 0.29 50 50 4 4 0.484 0.002912 0.663 0.016 0 0.29 0 0.29 50 50 6 6 0.443 0.009077 0.72 0.033 0 0.29 0 0.29 50 50 9 9 0.453 0.007321 0.412 0.016 0 0.29 0 0.29 α = 7 , β = 6 andR = 0 . 538 15 T abl e (4) m n r 1 r 2 R 1 MSE1 R 2 MSE2 R 3 MSE3 R 4 MSE4 5 5 3 3 0.535 0.0012 38 0.549 0.002414 0.00715 7 0.243 0 .000089 0.25 5 5 4 4 0.286 0.046 0.592 0.008417 0.29 0.044 0.00000 24 0. 25 10 10 6 6 0.681 0.033 0.5 62 0.0038 45 0.0008 5 0.249 0 0.2 5 10 10 7 7 0.519 0.00 035 0.7 0.04 0 0.25 0 0.25 10 10 8 8 0.426 0.00 546 0.713 0.046 0 0.25 0 0.25 10 10 9 9 0.412 0.007753 0.628 0.017 0 0.25 0 0.25 15 15 12 12 0.525 0.0006029 1.619 1.252 0 0.25 0 0.25 15 15 13 13 0.628 0.016 0.0 55 0.198 0 0.25 0 0.25 15 15 14 14 0.682 0.033 0.6 0.01 0 0.25 0 0.25 20 20 15 15 0.43 0.0048 5 0.687 0.035 0 0.25 0 0.25 20 20 16 16 0.487 0.0001711 0.0040 66 0.246 0 0.25 0 0.25 20 20 17 17 0.426 0.00 552 0.000001 057 0.2 5 0 0.25 0 0.25 25 25 23 23 0.366 0.018 0.5 91 0.0083 15 0 0.25 0 0.25 25 25 24 24 0.46 0.006 215 8.178E-14 0.25 0 0.25 0 0.25 50 50 4 4 0.516 0.0002437 0.647 0.022 0 0.25 0 0.25 50 50 6 6 0.487 0.0001691 0.659 0.025 0 0.25 0 0.25 50 50 9 9 0.443 0.003232 0.206 0.086 0 0.25 0 0.25 α = 7 , β = 7 andR = 0 . 5 16 T abl e (5) m n r 1 r 2 R 1 MSE1 R 2 MSE2 R 3 MSE3 R 4 MSE4 5 4 3 2 0.677 0.077 0.362 0.001 465 8.027E- 09 0.16 0.00002 938 0.1 6 5 5 4 3 0.521 0.015 0.759 0.129 2.095E-1 1 0.16 6.352E- 08 0.16 10 5 6 5 0.78 0.144 0.457 0.003292 0 0.16 0 0.16 10 10 7 6 0.56 0.026 0.716 0.1 0 0.16 0 0.16 10 15 8 7 0.638 0.057 0.62 0.049 0 0.16 0 0.16 10 20 9 10 0.611 0.044 0.0001 796 0.16 0 0.16 0 0.16 10 25 10 20 0.524 0.015 0.000001 812 0.16 0 0.16 0 0.16 15 5 6 5 0.796 0.157 0.567 0.028 0 0.16 0 0.16 15 10 7 6 0.533 0.018 0.8 0.16 0 0.16 0 0.16 15 15 8 7 0.546 0.021 0.621 0.049 0 0.16 0 0.16 15 20 9 10 0.518 0.014 0.601 0.04 0 0.16 0 0.16 15 25 10 20 0.589 0.036 0.0 01212 0.159 0 0.16 0 0.16 20 5 15 4 0.325 0.00569 6 0.65 0.063 0 0.16 0 0.16 20 10 16 9 0.614 0.046 0.59 0.036 0 0.16 0 0.16 20 15 17 14 0.551 0.023 0.531 0.017 0 0.16 0 0.16 25 10 20 9 0.724 0.105 0.689 0.083 0 0.16 0 0.16 25 15 22 14 0.657 0.066 0.644 0.06 0 0.16 0 0.16 25 20 24 19 0.569 0.029 0.574 0.03 0 0.16 0 0.16 50 25 4 24 0.806 0.165 0.226 0.03 0 0.16 0 0.16 50 30 6 29 0.727 0.107 0.018 0.146 0 0.16 0 0.16 50 40 9 39 0.759 0.129 0.0001 917 0.16 0 0.16 0 0.16 α = 2 , β = 3 andR = 0 . 4 17 T abl e (6) m n r 1 r 2 R 1 MSE1 R 2 MSE2 R 3 MSE3 R 4 MSE4 5 4 3 2 0.911 0.436 0.90 7 0.432 0.0003433 0.062 0.00157 1 0.062 5 5 4 3 0.715 0.216 0.81 1 0.314 1.118E-1 3 0.062 5.941E-07 0.062 10 5 6 5 0.805 0.308 0.875 0.39 0 0.062 0 0.063 10 10 7 6 0.837 0.344 0.825 0.33 0 0.062 0 0.063 10 15 8 7 0.727 0.228 0.739 0.24 0 0.062 0 0.063 10 20 9 10 0.71 0.212 0.73 0.231 0 0.062 0 0.063 10 25 10 20 0.762 0.262 0.0001 145 0.062 0 0.062 0 0.063 15 5 6 5 0.865 0.378 0.789 0.291 0 0.062 0 0.063 15 10 7 6 0.702 0.204 0.8 0.302 0 0.062 0 0.063 15 15 8 7 0.801 0.304 0.59 0.116 0 0.062 0 0.063 15 20 9 10 0.792 0.294 0.77 6 0.27 7 0 0.062 0 0.063 15 25 10 20 0.729 0.23 0.0034 33 0.06 1 0 0.062 0 0.063 20 5 15 4 0.583 0.111 0.718 0.219 0 0.062 0 0.063 20 10 16 9 0.587 0.114 0.72 0.221 0 0.062 0 0.063 20 15 17 14 0.781 0.282 0.862 0.374 0 0.062 0 0.063 25 10 20 9 0.738 0.238 0.758 0.258 0 0.062 0 0.063 25 15 22 14 0.71 0.211 0.66 3 0.17 0 0.062 0 0.063 25 20 24 19 0.78 0.281 0.81 0.314 0 0.062 0 0.063 50 25 4 24 0.841 0.35 0.395 0.021 0 0.062 0 0.063 50 30 6 29 0.818 0.323 0.88 9 0.40 8 0 0.062 0 0.063 50 40 9 39 0.859 0.371 0.00000394 0.062 0 0.062 0 0.063 α = 2 , β = 6 andR = 0 . 25 18 T abl e (7) m n r 1 r 2 R 1 MSE1 R 2 MSE2 R 3 MSE3 R 4 MSE4 5 4 3 2 0.61 0.012 0.436 0.004035 0.01 3 0.2 37 0.022 0.229 5 5 4 3 0.259 0.058 0.509 0.00 007698 0.00 0102 0.25 0.0095 86 0.241 10 5 6 5 0.427 0.005336 0.602 0. 01 0.00 06316 0.249 0 0.25 10 10 7 6 0.381 0.014 0.66 0.026 0 0.25 0 0.25 10 15 8 7 0.389 0.012 0.74 5 0.06 0 0.25 0 0.25 10 20 9 10 0.012 0.0014 02 0.369 0.017 0 0.25 0 0.25 10 25 10 20 0.363 0.019 0.06 0.194 0 0.25 0 0.25 15 5 6 5 0.439 0.003772 0.728 0.052 0 0.25 0 0.25 15 10 7 6 0.723 0 .05 0.715 0.046 0 0.25 0 0.25 15 15 8 7 0.535 0.001225 0.608 0.012 0 0.25 0 0.25 15 20 9 10 0.66 0.025 0.628 0.016 0 0.25 0 0.25 15 25 10 20 0.639 0.019 0.17 8 0.104 0 0.25 0 0.25 20 5 15 4 0.505 0.00002 566 0.406 0.008777 0 0.25 0 0.25 20 10 16 9 0.541 0.001673 0.609 0.012 0 0.25 0 0.25 20 15 17 14 0.472 0.0007671 0.002 26 0.248 0 0.25 0 0.25 25 10 20 9 0.487 0.00016 27 0.551 0.002556 0 0.25 0 0.25 25 15 22 14 0.509 0.00008313 0.487 0.000164 1 0 0.25 0 0.25 25 20 24 19 0.528 0.0007997 0.576 0.00584 0 0.25 0 0.25 50 25 4 24 0.791 0.085 1.68 1.392 0 0.25 0 0.25 50 30 6 29 0.752 0.063 1.144 0.415 0 0.25 0 0.25 50 40 9 39 0.755 0.065 0.023 0.227 0 0.25 0 0.25 α = 7 , β = 7 andR = 0 . 5 Where b oth α and β are the scale p aramete rs for X and Y resp ective ly . 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