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Burr-XII Distribution Parametric Estimation and Estimation of Reliability of Multicomponent Stress-Strength

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Burr-XII Distribution Parametric Estimation and Estimation of Reliability of Multicomponent Stress-Strength Burr-XII Distribution Parametric Estimation and Estimation of Reliability of Multicomponent Stress-Strength G. Srinivasa Rao 1 and Muhammad Aslam 2 and Debasis Kundu 3 1Department of Statistics, School of M & C S, Dilla University, Dilla, Po. Box: 419, Ethiopia. 2Department of Statistics, Forman Christian College University, Lahore, Pakistan. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin 208016, India 1 E-mail: [email protected]; 2Email: [email protected] ; 3Email: [email protected] Abstract: In this paper, we estimate the multicomponent stress-strength reliability by assuming the Burr-XII distribution. The research methodology adopted here is to estimate the parameters by using maximum likelihood estimation. The reliability is estimated using the maximum likelihood method of estimation and results are compared using the Monte-Carlo simulation for small samples. By using real data sets we well illustrate the procedure. Key Words : Burr-XII distribution, reliability estimation, stress-strength, ML estimation, confidence intervals. 1. Introduction The Burr-XII distribution, which was originally derived by Burr (1942) and received more attention by the researchers due to its broad applications in different fields including the area of reliability, failure time modeling and acceptance sampling plan. Reader can find the applications in various fields from Ali and Jaheen (2002) and Burr (1942). The two parameters Burr-XII distribution has the following density function −(α + 1) fx(;,)αβ= αβ xxβ−1 ( 1 + β ) ;for x > 0 (1) and the distribution function −α Fx(;,)1α β =−+( 1 xβ ) ; for x > 0. (2) Here α>0and β > 0 are the shape parameters respectively. It is important to note that when α =1, the Burr-XII reduces to the log-logistic distribution. Wu and Yu (2005) show that the shape parameter β plays a vital role for the Burr-XII. Now onwards two parameters Burr-XII distribution with shape parameters αand β will be denoted by BD (α, β ) . Let the random samples YX,1 , X 2 ,... X k be independent, G(y) be the continuous distribution function of Y and F(x) be the common continuous distribution function of X1, X 2 ,... X k . The reliability in a multicomponent stress-strength model developed by Bhattacharyya and Johnson (1974) is given by 1 [ ] Rs, k =P at least s of the( XXX 1 , 2 ,...k )exceed Y k ∞ k − =∑ ∫ [1 − Fy ( )]i [ Fy ( )] k i dGy ( ), (3) i= s i −∞ where X1, X 2 ,... X k are identically independently distributed (iid) with common distribution function F( x ) and subjected to the common random stress Y. The probability in (3) is called reliability in a multicomponent stress-strength model [Bhattacharyya and Johnson (1974)]. The survival probabilities of a single component stress-strength version have been considered by several authors for diffent distributions. Enis and Geisser (1971), Downtown (1973), Awad and Gharraf (1986), McCool (1991), Nandi and Aich (1994), Surles and Padgett (1998), Raqab and Kundu (2005), Kundu and Gupta (2005& 2006), Raqab et al. (2008), Kundu and Raqab (2009). The reliability in a multicomponent stress-strength was developed by Bhattacharyya and Johnson (1974), Pandey and Borhan (1985). Pandey and Uddin (1991) assumed the parameters not involved in reliability as known using the Bayesian estimation. Recently Rao and Kantam (2010) studied estimation of reliability in multicomponent stress-strength for the log-logistic distribution and Rao (2012) developed an estimation of reliability in multicomponent stress- strength based on generalized exponential distribution. The aim of this paper is to study the reliability in a multicomponent stress-strength based on X, α β α β Y being two independent random variables, where X~ BD (,)1 and Y~ BD (2 ,) . We will use the parametric estimation and estimation reliability. Suppose a system, with k identical components, functions if at least s(1≤ s ≤ k ) components simultaneously operate. In its operating environment, the system is subjected to a stress Y which is a random variable with distribution function G(.) . The strengths of the components, that is the minimum stresses to cause failure, are independently and identically distributed random variables with distribution function F(.) . The reliability of system can be obtained by equation (3). The estimation of survival probability when the stress and strength variates follow two parameters Burr-XII distribution is not paid much attention. Therefore, an attempt is made here to study the estimation of reliability in multicomponent stress-strength model with reference to two parameters Burr-XII distribution. The rest of the paper is organized as follows. In Section 2, we discussed the research methodology and procedure for expression of Rs, k . The asymptotic distribution and confidence 2 interval of equation (3) is obtained using the MLE. The results of small sample comparisons made through Monte Carlo simulations are in Section 3. Some findings are discussed in Section 4. 2. Research Methodology for Maximum Likelihood Estimator of Rs, k α β α β Let X~ BD (,)1 and Y~ BD (2 ,) are independently distributed with unknown shape α α β parameters 1and 2 and common shape parameter respectively. The reliability in multicomponent stress-strength for two parameter Burr-XII distribution using (3) we get ∞ k −αi − α k− i −+ α = +β1 −+ β 1 α β ββ−1 + (2 1) Rkys, k ∑( ) ()()()1 11 y2 yydy 1 i ∫ i= s 0 1 k − −α α =∑(k) ν ti+ν − 1 ()1 − tk i dt , where t=()1 + y β 1 , ν = 2 i ∫ α i= s 0 1 k =∑(k)ν Β+( ν iki , −+ 1). i i= s After the simplification we get −1 k k =νk! ν + Rs, k ∑ ∏( j ) since ki and are integers. (4) i= s i! j= i The probability in (4) is called reliability in a multicomponent stress-strength model. It is important to note that the MLE (maximum likelihood estimation) of Rs, k depends on the MLE of α α 1and 2 . Therefore, we need to find the MLE of the later before the MLE of first one. α α β Similarly, to derive the MLE of 1and 2 we need the MLE of . Let us assume that < α β < < X1 X 2<....<X n is random sample from BD (1 , ) and Y<Y1 2 .... Y m is a random sample α β from BD (2 , ) , then the log-likelihood function (LLF) of these observed sample is n m ααβ=+ β + α + αβ +−() + L(1 , 2 , ) ( mnnm )ln ln1 ln 2 1∑ ln xyi ∑ ln j i=1 j = 1 n m −+α +−+β α + β (1 1)∑ ln()() 1xi ( 2 1) ∑ ln 1 y j (5) i=1 j = 1 βα α βαˆ α The MLE of ,12 and ; say ,ˆ 1 and ˆ 2 respectively, can be obtained as the solution of β nm nβ m ∂L m + n xln x yjln y j = 0 ⇒ +lnx + ln y −+ (α 1)i i −+ ( α 1) = 0 (6) ∂β β ∑∑i j 1 ∑+β 2 ∑ + β ij==11 i = 1()1xi j = 1 () 1 y j 3 n ∂ L n β = 0 ⇒ −ln() 1 +x = 0 , (7) ∂α α ∑ i 1 1 i=1 m ∂ L m β = 0 ⇒ −ln() 1 +y = 0 (8) ∂α α ∑ j 2 2 j=1 From (7) and (8), we obtain n m αβˆ⇒ αβ ˆ ⇒ 1( ) n and 2 ( ) m (9) +β + β ∑ln() 1xi ∑ ln() 1 y j i=1 j = 1 αβ αβ Putting the values of ˆ1( ) and ˆ 2 ( ) into (6), we obtain β + n m n xβ ln x m yln y mn++− ni i − m j j ∑∑lnxi ln y j n ∑β m ∑ β β = =β = ()+β = () + i1 j 1 + i 1 1xi + j 1 1 y j ∑ln() 1xk ∑ ln() 1 y k k=1 k = 1 β nxβ ln x m yln y −i i −j j = 0 (10) ∑+β ∑ + β i=1()1xi j = 1 () 1 y j Therefore, βˆ can be obtained using the following non-linear equation (11) h(β ) = β (11) β nln xβ m ln y + +i + j (m n ) ∑β ∑ β =()1+x = () 1 + y β = i1i j 1 j Where h( ) β (12) n xβ ln x m yln y ni i + m j j n ∑β m ∑ β β = ()+β = () + +i 1 1xi + j 1 1 y j ∑ln() 1xk ∑ ln() 1 y k k=1 k = 1 Because βˆ is a fixed point solution of the non-linear (11), therefore, it can be obtained by using a simple iterative procedure as β= β h((j ) ) ( j + 1) , (13) βth β ˆ where (j ) is the j iteration of . It should be noted that during the simulation process when β β the difference between (j ) and ( j + 1) is sufficiently small then stop the iterative process. Once βˆ α α αβˆ αβ ˆ we obtain , the parameters ˆ1and ˆ 2 can be obtained from (9) as ˆ1( ) and ˆ 2 ( ) respectively. The MLE of Rs, k becomes −1 k k!k αˆ Rˆ =νˆ∏( ν ˆ + j ) where ν ˆ =2 . (14) s, k ∑ α i= s i! j= i ˆ1 4 To obtain the asymptotic confidence interval for Rs, k , we proceed as follows: The asymptotic variance of the MLE is given by −1 −1 ∂2 L α 2 ∂2 L α 2 V(αˆ ) =E − = 1 and V(αˆ ) =E − = 2 (15) 1 ∂α 2 2 ∂α 2 1 n 2 m The asymptotic variance (AV) of an estimate of Rs, k which is a function of two independent α α statistics (say) 1, 2 is given by Rao (1973). 2 2 ∂R ∂ R AV(Rˆ )=V(αˆ )sk,+ V( α ˆ ) sk , . (16) s, k 1∂α 2 ∂ α 1 2 ˆ Thus from Equation (16), the asymptotic variance of R s, k can be obtained.
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