(PDF) Bayesian Estimation Procedures for Three-parameter Exponentiated-Weibull Distribution under Squared-Error Loss Function an

World Engineering & Applied Sciences Journal 6 (1): 45-58, 2015 ISSN 2079-2204 © IDOSI Publications, 2015 DOI: 10.5829/idosi.weasj.2015.6.1.22189 Bayesian Estimation Procedures for Three-parameter Exponentiated-Weibull Distribution under Squared-Error Loss Function and Type II Censoring Ajit Chaturvedi and Anupam Pathak Department of Statistics, University of Delhi-110007, India Abstract: The three-parameter exponentiated-Weibull distribution has been widely used especially in the modelling of life time event data. It provides a statistical model which has a wide variety of application in many areas and the main advantage is its ability in the context of life time event among other distributions. Bayesian estimation procedures are considered for estimating the reliability function R(t)=P(X>t) and P=P(X>Y) for three-parameter exponentiated-Weibull distribution under type II censoring. Considerations are given to squared-error loss. Approach: A new technique of obtaining Bayes estimators of these parametric functions is introduced in which major role is played by the estimators of the powers of the parameter and the functional forms of the parametric functions to be estimated are not needed. Simulation studied is performed. Key words: Three-parameter exponentiated-Weibull distribution Squared-error loss function (SELF) INTRODUCTION Type II Censoring Bayes estimators indicating a bathtub hazard rate. Non-monotone hazard rate is very common, for example, in the field of science and engineering, in the field of medical, in the field of ecological and space explorations. Due to these reasons, modeling lifetime data for non-monotonic hazard rates seems to be a growing interest. In this context, exponentiated-Weibull family can be considered as a suitable modal, which was initially introduced by [12], as a simple generalization of two-parameter Weibull family and is obtained by introducing one additional shape parameter. This family allows bathtub shaped as well as unimodal hazard rates. The probability density function (pdf), cumulative distribution function (cdf) and reliability function R(t) of exponentiated-Weibull family, respectively, are Reliability theory is mainly concerned with the determination of the probability that a system, consisting possibly of several components, will operate adequately for a given period of time in its intended application. The reliability function R(t) is defined as the probability of failure-free operation until time t. Thus, if the random variable (rv) X denotes the lifetime of an item, then R(t)=P(X>t). Another measure of reliability under stressstrength set-up is the probability P=P(X>Y), which represents the reliability of an item of random strength X subject to random stress Y[1-16]. Many researchers have considered the problems of estimation of R(t) and ‘P’ for various lifetime distributions and for a brief review, one may refer to and others. Some of these models (particularly, exponential and Weibull models) are widely discussed in the literature for the analysis of lifetime data. These models accommodate either constant or monotone type of the hazard rates. But non-monotonic hazard functions such as unimodal shaped and bathtub shaped also arises in practice. For example, data in reliability analysis specially life cycle of the product often involve high initial hazard rate (infant mortality) and eventual high hazard rates due to aging and wear-out in the end δ ( δ f ( x;α, λ, δ) = αλδxδ-1e-λx 1- e-λx ( δ F( x;α, λ,δ) = 1-e-λx and ( δ R(t) =1- 1-e-λx ) α-1 ; x, α, λ, δ > 0, ) ; x,α,λ,δ > 0 α ) ; x,α,λ,δ > 0. (1.2) α Corresponding Author: Anupam Pathak, Department of Statistics, University of Delhi-110007, India. 45 (1.1) (1.3) World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 Bayesian ideas were introduced for the first time in reliability and life testing by [1], who considered the problems of estimating the parameter and reliability function of one-parameter exponential distribution under type II censoring. Consideration was given to SELF. [11] considered a family of lifetime distribution and proposed Bayesian estimation procedures for the parameter and reliability function under SELF and type II censoring. For a brief review one may refer to the book by [10]. [6] derived Bayes estimator of ‘P’ under SELF when X and Y were assumed to follow exponential distributions. [15] discussed the classical and Bayesian methods of parameter estimation for complete sample case. [16] have discussed the classical and Bayesian methods of parameter estimation under type II censoring. This paper is an attempt in the direction of Bayesian estimation of R(t) and ‘P’ for three-parameter exponentiated-Weibull distribution under type II censoring. The purpose of the present paper is manifold. For the distribution (1.1), Bayes estimators are derived for the powers (positive as well as negative) of the parameter, reliability function R(t) and ‘P’ under SELF. Type II censoring is considered. Deviating from the conventional methods of obtaining Bayes estimators of R(t) and ‘P’, Bayes estimators of the powers of the parameter are utilized to obtain Bayes estimator of the pdf at a specified  r L ( α x ) = αr λr δr  x δ(i) x δ(r)  i=1 ∏ { } point. This estimator is subsequently used to obtain Bayes estimator of R(t) and ‘P’. Thus, in all the estimation problems, the major role is played by the estimators of the powers of the parameter. We have established an interrelationship between various estimation problems. Expressions for the risks, posterior risks and Bayes risks of various estimators are provided. In Section 2, we give the set-up of the estimation problems and introduce the notations and definitions. In section 3, we obtain Bayes estimators of the powers of , R(t) and ‘P’. In Section 4, simulation study is carried out to investigate the performance of estimators. In Section 5, discussion is made. Finally, in Section 6 conclusion are given. Set-up of the Estimation Problems, Notations and Definitions: Let the random variable (rv) X follows the three-parameter exponentiated-Weibull distribution whose pdf is given at (1.1). Throughout we assume that is unknown but and are known. Suppose n items are put on a test and the test is terminated after the first r ordered observations are recorded. Let 0 ≤ X(1) ≤ .... ≤ X( r ) ; 0 < r < n, be the lifetimes of the first r ordered observations. Obviously, (n-r) items survived until X(r). Denoting by x = ( x1, x2 , ..., x r ) , we have likelihood function as   r    exp -λ  xδ(i) + ( n - r ) xδ(r)        i =1 n-r  ∑ { } {   r -λxδ -λxδ ⋅ exp ( α -1)  ln 1- e (i) + ( n - r ) ln 1- e (r)  i =1  ∑ }    .   Therefore, (2.1) L ( α | x ) ∝ αr exp[ -αSr ] . Where,  r  -λxδ   -λxδ  = Sr -  ln 1- e (i)  + ( n - r ) ln 1- e ( r )  .     i =1  ∑ (2.2) Looking (2.1), we consider the natural conjugate prior distribution for ∏( α)= µν ν-1 α exp ( -αµ) ; α, µ > 0 and ν is positive int eger. Γν 46 to be gamma with pdf (2.3) World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 Combining (2.1) and (2.3) via Bayes theorem, the posterior density of h ( α | sr ) = comes out to ( sr + µ)r +ν αr +ν-1 exp - s + µ α . {(r ) } Γ ( r + ν) (2.4) In order to estimate ‘P’, let n items on X and m items on Y are put through a life test and r and s being their truncation numbers, respectively. The rv X has pdf f ( x; α1, λ1 , δ1 ) and rv Y has pdf f ( y; α2 , λ2 , δ2 ). Let us denote by  r  -λ1xδ1   -λ1xδ1  = Sr -  ln 1- e (i)  + ( n - r ) ln 1- e ( r)   i=1     ∑ and  s  -λ yδ2   -λ2 yδ2  2 = Ts -  ln 1- e ( j)  + ( m - s) ln 1- e (s)  .  j=1      ∑ Here, we assume that 1, 1, 2 and 2 are known but 1 and 2 are unknown. We consider the conjugate priors for 1 and 2 given at (2.3) with parameters (µ 1, 1) and (µ2, 2) respectively. Let us make the transformation U = -ln(1- e-λXδ ). It is easy to see that U follows exponential distribution with pdf f(u;a )=a exp(-au); u>0. If we consider the transformation then Zi’s are independent and identically distributed Zi = ( n-i+1) [U ( i ) -U( i-1) ]; i=1, 2, ..., r, rv’s each having exponential distribution. Moreover, since r ∑Zi = Sr , from the additive property of exponential i =1 distribution the pdf of Sr is h ( sr ; α ) = αr r-1 sr exp(-αsr ); sr > 0. Γ(r) (2.5) From (2.3) and (2.5) the marginal pdf of Sr is f ( sr ; α ) = ∞ µνsr-1 r αr +ν-1 exp{-α( sr + µ)} dα, Γ(r) Γ(ν) ∫ 0 or, µνsr-1 r f ( sr ; α) = Β( r, ν)( sr + µ) Denoting by by θˆ BS , r +ν ; sr > 0. ( θˆ BS and L θˆ BS , θ (2.6) ) the Bayes estimators of θ =Ψ( α) , under SELF and the loss resulting from estimating respectively, the associated risk is defined by ( ) { ( )} (2.7) R S θˆ BS = ESr |α L θˆ BS , θ . 47 World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 The posterior risk for estimating ( { ( ) R PS θˆ BS = E α|Sr L θˆ BS , θ is (2.8) by { ( ) θˆ BS )} and Bayes risk for estimating ( by θˆ BS is )} (2.9) ˆ R= ESr Eα|Sr L θˆ BS , θ  . BS θBS   It should be noted here that the risk of Bayes estimator of is a function of the modal parameter and is independent of the sample data, the posterior risk is a function of the sample data Sr and of the prior parameters and is independent of and Bayes risk is a function of the prior parameters only. We also note the following relationships ( { ( )} ) R BS θˆ = Eα  RS θˆ BS BS  and ( ) { ( )} . θˆ BS R BS = ESr  R PS θˆ BS  (*) In what follows we obtain θˆ BS and it’s various measures of performance under SELF. Bayes Estimators of the Powers of , R(t) and P Under SELF: The following theorem provides Bayes estimators of powers of . Theorem 1: For a positive integer p, under SELF, Bayes estimators of -p and p are given, respectively, by -p p aˆ and aˆ BS BS , where ˆ -p α BS = Γ( r +ν -p) Γ( r +ν) (Sr +µ)p ; ( p < r +ν) (3.1) and ˆ pBS = α Γ( r + ν+ p) Γ( r +ν) (Sr +µ)-p . (3.2) Proof: From (2.4) and using the result that Bayes estimator of any function of is it’s posterior mean, we have ∫ -p αˆ -p BS = α h ( α | Sr ) dα α = Γ( r +ν -p) Γ( r +ν) ( Sr +µ)p ; (p < r +ν) and the result (3.1) follows. Similarly, we can prove result (3.2). In the following theorem we derive expression for the risks, posterior risks and Bayes risks of Bayes estimators of the powers of . 48 World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 Theorem 2: 2 2p  Γ( r + ν - p)  = RS αˆ -p   BS  Γ( r + ν)  ( ) Γ( i + r )  -i α Γ( r )   2p  ∑ i  µ2p-1  i=0 p  Γ( r + ν - p)   p  p-1 Γ( i + r )  -i -2p - 2α-p    µ  α + α ; (p < r + ν),  Γ( r + ν)  i=0  i   Γ( r )  ∑  Γ r + ν - 2p Γ r +ν - p 2  ) - ( )   S +µ 2p ; 2p < r + ν ,  ( RPS αˆ -p = ) ( )   ( BS  Γ( r + ν)  Γ( r + ν)   r   ( ) ( ) ˆ -p RBS α BS 2   Γ ν - 2p Γ( r + ν - p)} ) µ2p; p < ν /2 , 1- {  ( ( )  Γ( r + ν) Γ( r + ν - 2p)  Γ(ν)   (3.3) (3.4) (3.5) ∞  Γ r + ν + p 2 ) 1 zr-1e-z  ( α2p  RS αˆ pBS = dz   Γ ( r + ν )  Γ ( r ) ( z + µα )2p 0  ( ) ∫ ∞  zr-1e-z  Γ ( r + ν + p )  1  dz, -2  1 +  p   Γ ( r + ν )  Γ ( r ) 0 ( z + µα )  ∫ ( ) RPS αˆ pBS Γ r + ν + 2p  Γ r + ν + p 2  )-  ( )   S + µ -2p  ( )    (  Γ( r + ν)   Γ( r + ν)   r   (3.6) (3.7) and 2   Γ ν + 2p Γ r + ν + p)} ) µ-2p. p 1- { (  ( = RBS αˆ BS  Γ( r + ν) Γ( r + ν + 2p)  Γν   ( ) Proof: From (3.1) the risk corresponding to ( )  ˆ -p - α-p  RS αˆ -p BS= ESr |α α BS  â -p BS (3.8) is 2 2 Γ( r + ν - p)  Γ( r + ν + p)  2p p -p  -2p   ESr |α ( Sr +µ) - 2α  ESr |α ( Sr +µ) + α  Γ( r + ν)   Γ( r + ν)  2  2p  2p    Γ ( r + ν - p)  2p-i i  E  µ Sr    Sr |α  i =0  i    Γ ( r + ν)  ∑  p p   Γ ( r + ν + p )  p-i i -2p - 2α-p  E  Sr |α   µ Sr  + α .  i =0  i    Γ ( r + ν )  ∑ (3.9) Now from (2.5) for q to be positive integer, we have 49 World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 ∞ r q +r-1 ( ) ∫ α ΓS(r r ) 0 E Sqr = = G( q+r ) G( r ) e-αSr dSr a -q . (3.10) Hence result (3.3) follows on combining (3.9) and (3.10). Now by definition ( ) ( ) -p -p  R PS αˆ= Eα|Sr  L αˆ -p BS BS , α   ( ) 2  ˆ -p = Eα|Sr α-2p - 2α-pαˆ -p BS + αBS    ( ) ( ) ( ) 2 = Eα|Sr α-2p - 2 Eα|Sr α-p  αˆ -p + αˆ -p BS .   BS (3.11) Hence result (3.4) follows from (2.4), (3.1) and (3.11). Using definition (*), (2.6) and (3.4), we get ( ) ( )  ˆ -p R BS α ESr  R PS αˆ -p = BS BS    Γ r + ν - 2p Γ r + ν - p 2 ) - ( )  E S +µ 2p .  (   ( r )  Γ( r + ν)  Γ( r + ν)     And hence (3.5) follows on using ∞ ν (Sr +µ)2p Sr-1 r µ ∫ B( r,ν)(Sr +µ)r+ν dSr 0 Γ r + ν) Γ( ν - 2p) ( ; (2p < r + ν). = µ2p Γ( ν) Γ( r + ν - 2p) ESr (Sr +µ) 2p = The proofs of the results (3.6), (3.7) and (3.8) are similar to those of (3.3), (3.4) and (3.5), respectively. In the following lemma we obtain Bayes estimator of the pdf (1.1) at a specified point ‘x’ with the help of Bayes estimators of powers of . Lemma 1: For f ( x; α, λ, δ) defined at (1.1) δ λδ(r + ν)xδ-1e-λx f̂BS ( x; α, λ, δ) = δ (Sr + µ)(1- e-λx )  ln(1- e-λxδ )  1 (Sr + µ)    -(r +ν+1) . Proof: We can write (1.1) as 50 World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 λδxδ-1e-λx f (= x; α, λ, δ) = -λxδ (1- e δ δ α(1- e-λx )α ) λδxδ-1e-λx δ δ ∞ [ln(1- e-λx )]i i +1 α δ i! (1- e-λx ) i=0 ∑ Utilizing lemma 1 of [4] and (3.2), we get ∞ (1- e ) ∑ fˆBS ( x; α, λ, δ) = = ( ) i ln 1- e-λxδ    i+1 αˆ BS i! i =0 δ λδxδ-1e-λx -λxδ δ-1 -λxδ λδx e ( ( -λxδ   r + ν + i   ln 1- e   i   (Sr + µ) i=0   ∞ (Sr + µ) 1- e-λx δ )∑ δ δ λδ(r + ν)xδ-1e-λx  ln(1- e-λx )    = 1δ (Sr + µ)  (Sr + µ)(1- e-λx )   )  i   -(r +ν+1) which is the required result In the following theorem, we obtain Bayes estimator of reliability function. Theorem 3: For the reliability function given at (1.3) (   -λt δ   ln 1- e R̂ BS ( t ) = 1- 1  (Sr + µ)    Proof: We have, ∞ ∫ -( r +ν)  )     .    ∞ ∫ Eα|S {f ( x; α,λ, δ)}dx f̂BS ( x;= α, λ, δ) dx r t t ∞ ∫ = Eα|Sr f ( x; α, λ, δ) dx t = Eα|Sr R ( t )  (3.12) = Rˆ BS ( t ) From above, we conclude that Bayes estimator of R(t) can be obtained with the help of Bayes estimator of f ( x;a,?,d ). Thus from Lemma 1 R̂BS (t) = δ δ ∞ λδ ( r + ν ) xδ-1e-λx  ln(1- e-λx )    1(Sr + µ) t 1- e-λxδ  (Sr + µ)  ∫ 1- = ( r + ν) ( ) -(r +ν+1) dx - λxδ ln(1-e ) (Sr +µ) ∫ z-(r +ν+1)dz 1 51 World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 and the theorem follows. The following theorem gives the expression for risk, posterior risk and Bayes risk of Theorem 4: For R̂ BS (t). 1 RS Rˆ BS ( t ) = Γ( r ) ( ) ∞ ∫ 0 -2 R PS ( R̂ BS (t). obtained in Theorem 3, { ( )} -2( r +ν )  -λt δ  zr-1e-z z + α µ - ln 1- e  dz -2 r +ν z + αµ ( ) ( ( 1- e-λt δ Γ( r ) ) 2α ∞ ∫ 0 ( δ  2ln 1- e-λt  ˆ ( t ) = 1R  BS ( Sr + µ)   ) ) { ( )} -( r +ν)  -λt δ  z ln 1e zr-1e-z + α µ   dz + 1- e-λtδ -( r +ν ) z + αµ )  ( -( r +ν)   { ) (  -λt δ  ln 1- e - 1(Sr + µ)   )  } 2α , (3.13) -2( r +ν)   (3.14) and RBS (  µ ˆR ( t ) =   BS δ  µ -2ln 1- e-λt  ) ( ν )   r +ν  r + ν  µ 1    δ  Β( r, ν) i =0  i   µ - ln 1- e-λt   ⋅ Β( r + i,r + 2ν - i) . ∑ ( )      r +2ν-i (3.15) Proof: By definition ( ) ˆ (t) = E Rˆ (t) - R(t) RS R BS Sr |α  BS  (  -λt δ  ln 1- e = ESr |α 1(Sr + µ)   2 )  -2( r +ν)   (  ln 1- e-λtδ  ⋅ ESr |α 1(Sr + µ)   ( - 2 1- e-λt )  -( r +ν )   ( δ + 1-e-λt ) δ α ) 2α . (3.16) For q to be positive integer, from (2.5),   Sr + µ ESr |α  δ  ( Sr + µ) - ln 1- e-λt  ( q )  ∞ -αSr  (Sr + µ)q Sr-1 αr r e dSr  = q Γ( r )  δ  0 ( S + µ) - ln 1- e-λt    r  ∫ ∞ = ( ) q ( z + αµ) zr-1e-zdz . 1 q Γ( r )  δ  0 z + α µ - ln 1- e-λt     ∫ { ( )} (3.17) 52 World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 Result (3.13) follows from (3.16) and (3.17). Again by definition ( ) R PS Rˆ BS(t) = Eα|Sr Rˆ BS (t) - R(t)    ln(1- e-λtδ )   = 1Sr + µ)   (   2 -2( r +ν ) ( ⋅ Eα|Sr 1- e-λt δ ) α  ln(1- e-λtδ )   - 2 1Sr + µ)   (   ( + Eα|Sr 1- e-λt ) δ 2α -( r +ν) . (3.18) Using (2.4), we have ( Eα|Sr 1- e-λt ) δ α ∞ = (Sr + µ)r+ν αr +ν-1 exp -α S + µ 1- e-λtδ { ( r )} ∫ Γ ( r + ν) 0 ( (  -λt δ  ln 1- e = 1(Sr + µ)   Similarly, ( Eα|Sr 1- e-λt δ ) 2α ( )  ) dα α -( r +ν) .    -λtδ  2ln 1- e = 1(Sr + µ)   (3.19) )  -( r +ν) .   (3.20) Result (3.14) follows from (3.18), (3.19) and (3.20). From (3.14), R BS ( (  t  2ln 1- e-λt  Rˆ BS (t) = ESr 1 (Sr + µ)   ) -( r +ν ) )    (  -λt t  ln 1- e - 1(Sr + µ)   -2( r +ν)  )   .      (3.21) Using (2.6), we have (  -λt t  2ln 1- e ESr 1(Sr + µ)   and (  -λt t  ln 1 - e ESr 1( Sr + µ)   ) )  -( r +ν)        -2( r +ν )   µ = δ  µ -2ln 1- e-λt  ( )      ν (3.22) r + 2ν-i     r + ν µ 1 =   δ  B(r, ν) i=0  i   µ - ln 1- e-λt    ⋅ Β( r + i, r + 2ν - i ) . r +ν ∑ ( ) Result (3.15) follows from (3.21), (3.22) and (3.23). 53 (3.23) World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 In what follows, we obtain Bayes estimator of P. Theorem 5: For C =1- (Sr + µ1) (Ts +µ2) and 1 = 2 = say and 1 = 2 = , say,  (r +ν1 ) (1-C)(r+ν1) 2 F1 (r +ν1 +s +ν2 ,r +ν1 +1;r +ν1 +s+ν2 +1;C); C <1  (r +ν1 +s +ν2 ) P̂BS =   (r +ν1 ) (1-C)-(s+ν2 ) 2 F1 (r +ν1 +s +ν2 ,s +ν2 ;r +ν1 +s+ν2 +1; C ); C ≤-1, C-1  (r +ν1 +s +ν2 ) where = 2 F1(a,b,c;z) ∞ a zj b ∑ (j)c(j)(j) . (j)! ,c ≠ 0,-1,-2,.... , (3.24) is Gauss hyper geometric series and I(k) = j=0 Γ(1 + k) ; k 1, 2, ..... = Γ(1) Moreover, for α1 ≠ α 2 , λ1 ≠ λ 2 and δ1 ≠ δ2 λ δ (s + ν 2 ) P̂BS = 1- 2 2 (Ts + µ2 ) ∞  ln(1- e-λ1yδ1 )  1 (Sr + µ1 )    y =0  -(r +ν1 ) ∫  ln(1- e-λ2 yδ2 )  1 ⋅ δ2 (Ts + µ2 )  (1- e-λ 2 y )   yδ2 -1e-λ2 y Proof: For λ1 =λ2 =λ, say δ2 and -(s +ν 2 +1) dy. (3.25) δ1 =δ2 =δ, say, = P P(X > Y) ∞ = ∫ R( y;α1, λ, δ) f ( y;α2, λ, δ) dy. y=0 Putting δ , 1- e-λY = Z = h ( α1, α2 ) we get P= α1 . α1 +α2 From (2.6), the joint posterior density of (α1, α2 ) (Sr + µ1 )r +ν1 ( Ts + µ2 )s+ν2 α r +ν1 -1α s+ν2 -1 1 2 Γ ( r + ν1 ) Γ ( s + ν2 ) ⋅ exp{- ( Sr + µ1 ) α1 - ( Ts + µ2 ) α2 }. In (3.26), let us make the transformation P= α1 and W =α1 +α2. α1 + α2 is (3.26) The Jacobian of transformation is w. From (3.26), the joint pdf of P and W is thus h ( P, w ) = (Sr + µ1 )r +ν1 ( Ts + µ2 )s+ν2 wr+s+ν1 +ν2 -1Pr+ν1 -1(1- P)s+ν2-1 Γ ( r + ν1 ) Γ ( s + ν2 ) × exp -w{(Sr + µ1 ) P + ( Ts + µ2 ) (1- P)} .   Integrating out w from (3.27), the marginal posterior pdf of ‘P’ comes out to be 54 (3.27) World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 h ( P) = (Sr + µ1 )r+ν1 ( Ts + µ2 )s+ν2 ∞ wr +s+ν1 +ν2 -1Pr+ν1 -1(1- P)s+ν2-1 ∫ Γ ( r + ν1 ) Γ ( s + ν2 ) 0 { } × exp -w ( Sr + µ1 ) P + ( Ts + µ2 ) (1- P)  dw   = (Sr + µ1 )r+ν1 ( Ts + µ2 )s+ν2 Pr +ν1 -1(1- P) s+ν2-1 B(n + ν1, m + ν2 ) { } ⋅ (Sr + µ1 ) P + ( Ts + µ2 ) (1- P) Denoting by h ( P) = C =1- (Sr + µ1) (Ts +µ2) (1-C)r +ν1 B(r + ν1,s + ν2 ) r +s+ν1 +ν2 (3.28) . , we can write (3.28) as Pr +ν1-1(1- P) s+ν2-1 (1- CP)-(r+s+ν1 +ν2 ) . Result (3.24) now fallows on applying a result of [7]. From the argument similar to used in Theorem 3, for ∞ α1 ≠ α 2 , λ1 ≠ λ 2 and δ1 ≠ δ2 ∞ ∫ ∫ fˆBS ( x; α1, λ1, δ1) fˆBS ( y; α2, λ2 , δ2 ) dxdy = P̂BS = y 0= x y ∞ = ∫ Rˆ BS ( y; α1, λ1, δ1) fˆBS ( y; α2 , λ2 , δ2 ) dy y=0 λ δ (s + ν 2 ) = 1- 2 2 (Ts + µ2 ) ∞  ln(1 - e-λ1yδ1 )  1 (Sr + µ1)    y=0  -(r +ν1 ) ∫ δ2 δ2 yδ2 -1e-λ 2 y  ln(1- e-λ2 y )    1⋅ δ2 (Ts + µ2 )  (1 - e-λ2 y )   -(s +ν 2 +1) dy and result (3.24) follows. Corollary 1: For δ1 =δ2 =δ, say λ1     (Ts +µ2 )(1- 1z )  λ2  1   1- e  (s+ν2 -1)  1-   P̂= BS 1- (s + ν2 ) z (Sr + µ1)   0     -(r +ν1 ) ∫ Proof: Corollary directly follows from (3.25), For dz. δ1 =δ2 = δ. Remarks 1: The complete sample case results can be obtained on putting r = n and s = m. 55 World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 From (3.1), ˆ -1 = α BS Γ( r +ν -1) S +µ = (Sr +µ) r Γ( r +ν) r +ν -1 Now 1 E(χ2 ) +µ r +µ 2α 2r = α r + ν -1 r + ν -1 E(Sr ) +µ = r + ν -1 ˆ -1 = E(α BS) → a -1 as r → ∞. Moreover, V( aˆ -1 BS )= V(Sr ) 2 4a (r+?-1) 2 = r 2 a (r+?-1) 2 → 0 as n → ∞. We can write (3.12) as δ λδ(r + ν)xδ-1e-λx f̂BS ( x; α, λ, δ) = δ (Sr +µ)(1- e-λx )  ln(1- e-λxδ )  1 (Sr +µ)     -(r +ν+1)  ln(1- e-λxδ )  1 f̂BS ( x; α, λ, δ) = -λxδ  (r + ν -1)α-1  (r + ν -1)α-1 ) BS  BS(1- e δ λδ(r + ν)xδ-1e-λx δ → αλδxδ-1e-λx δ (1- e-λx ) ⇒ f̂ ( x; α, λ, δ) Hence, -(r+ν+1) -λxδ ) eαln(1-e = f ( x;α, λ, δ) is a consistent estimator of f ( x; α, λ, δ) . R̂BS (t) and P̂BS are also consistent estimators of R(t )and ‘P’, respectively. If we look at remark (2), we observe that the estimators of negative powers of of f̂ ( x; α, λ, δ) . This justifies the estimation of negative powers of . are used to prove the consistency In order to obtain Bayes estimators of R(t) and ‘P’, in literature, authors first obtain the expression of these quantities and then their Bayes estimators, say, posterior mean under SELF. In the present approach to obtain Bayes estimators of R(t) and ‘P’ we made use of Bayes estimator of the pdf and one does not need their expression. Moreover, we have established an interrelationship between these two estimation problems. Simulation Studies: In order to validate the performance of the estimators under SELF with respect to the actual reliability estimates obtained through numerical integration, we have simulated a sample of size r = n = 50 from (1.1) with = 2, = 2 and =1. 0.4173003, 0.9460239, 0.7186531, 0.9736477, 1.6545909, 1.1489930, 1.0387765, 0.8689265, 1.1750468, 0.9411219, 1.8013401, 1.1333229, 2.2251164, 0.8826638, 1.8679742, 1.2884515, 1.0713660, 1.4829736, 1.5310753, 0.7714249, 0.7771927, 1.0573936, 1.5314517, 0.8741688, 2.1261284, 1.1641412, 0.9074558, 1.3315983, 0.6891152, 1.5015916, 1.9824928, 1.2762304, 1.5224234, 0.7976915, 0.7174940, 1.0339409, 1.2862068, 0.9775802, 0.8620062, 1.1182395, 1.0568100, 0.9023865, 0.7227187, 1.9491991, 1.1022730, 1.6380632, 1.1811405, 1.7408185, 1.6126990, 0.9343715. 56 World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 Table 1: p=1 â BS p=2 â -1 0.4562096 BS â 2 5.070009 BS â -2 0.2119814 BS ) 0.08894753 R PS (aˆ -1 ) 0.003854207 BS R PS (aˆ 2 ) 1.852177 BS R BS (aˆ -2 ) 0.003489233 BS ) 0.02947368 R BS (aˆ -1 ) 0.02272727 BS R BS (aˆ 2 ) 0.3251993 BS R BS (aˆ -2 ) 0.3752806 BS 2.231829 R PS (aˆ BS R BS (aˆ BS For computing the value of P̂BS we have simulated two samples from (1.1) of sizes r = n = 30 and s = m = 35, respectively. First is with parameters ( α1 = 3, δ1= 2, λ1 =1) and second is with ( α2 = 2.5, λ2 = 1, δ2 = 2) , respectively. X-Population: 1.1725308, 1.8200127, 1.7052567, 1.4018941, 1.2583337, 1.6616181, 0.9813064, 1.5846732, 0.5500817, 0.8015553, 1.4781880, 1.3155080, 0.9924352, 0.4397273, 1.0000568, 1.1561893, 0.7403056, 0.9018083, 0.5021047, 1.9817868, 1.4721579, 1.3098427, 1.4935193, 2.1591953, 0.8969304, 0.8221974, 1.5739064, 1.0102813, 0.5673100, 1.3246853. Fig. 1: Prior and Posterior densities Fig. 2: Graph of f(x; , , ) and Y-Population 1.1403903, 0.7555690, 1.2309668, 2.3932917, 1.2066033, 1.8399822, 1.3407853, 0.6030329, 1.1180712, 1.5327907, 1.5622690, 1.4313309, 1.6976184, 2.2411743, 0.7800085, 1.3023954, 1.0793686, 0.9517087, 1.4332955, 0.8087409, 0.7456061, 0.4338839, 1.3554415, 0.8024363, 1.0191814, 0.5005716, 0.6418640, 0.7757467, 0.6331233, 1.4079004, 1.1508947, 1.3903010, 1.4176388, 0.9870763, 1.2192468. f̂BS (x; α, λ, δ). Table 2: t R(t) R̂ BS (t) 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 0.993 0.978 0.951 0.909 0.850 0.777 0.692 0.600 0.507 0.995 0.984 0.962 0.926 0.875 0.808 0.727 0.637 0.544 In the case when, prior parameters are (µ2 = 7, ν2 = 5) ,  µ =5, ? =6   1 1    and respectively. For q=1, we get Sr =13.50066, Ts=16.5051, = P = 0.5 and PˆBS 0.5326872. We have shown under Remark (1), that consistent estimator of f(x;α, λ,δ) . f̂ (x; α, λ, δ) is a In order to verify these results, we have drawn a sample of sizes n = r = 30 from (1.1), with α= 2, λ=1, δ=3.5 and µ=ν=5.InFig.2,wehaveplotted f̂ (x; α, λ, δ) and Here, we have Sr = 20.09153. For the case when prior parameters are µ = 5 and = 6, we have plotted prior and posterior densities in Fig. 1 and constructed Table 1. In order to compute values of R(t) and R̂ (t) , we f(x;α, λ, δ), simultaneously. It is clear from Fig. 2 that both the curves overlap. This justifies the consistency property of the estimator. BS DISCUSSION assume that the data represents life spans of items in hours. With respect to above sample and for same prior parameters as above, computed values of R(t) and R̂ (t) , [15] obtained the classical and Bayesian methods of parameter estimation for complete sample case. [16] obtained the classical and Bayesian methods of parameter BS over t=0.30(0.10)1.10 are given in Table 2. 57 World Eng. & Appl. Sci. J., 6 (1): 45-58, 2015 estimation under type II censoring. Here, we obtained Bayes estimators of the powers of the , R(t) and ‘P’ under type II censoring. Consideration was given to SELF. We also obtained expressions for the risks, posterior risks and Bayes risks of the powers of and R(t). Therefore we have extended previous research. In the literature, the researchers have dealt with the estimation of R(t) and ‘P’, separately. If we look at the proofs of Theorems 3 and 5, we observe that the Bayes estimator of the sampled pdf is used to obtain the Bayes estimators of R(t) and ‘P’, respectively. With the help of Fig. 2, we justified the consistency property of the estimators. Table 1 and 2 Shows that estimated values are very close to actual values. 5. 6. 7. 8. 9. 10. 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