Inferences on stress-strength reliability from weighted Lindley distributions

Transcription

1 Inferences on stress-strength reliability from weighted Lindley distributions D.K. Al-Mutairi, M.E. Ghitany & Debasis Kundu Abstract This paper deals with the estimation of the stress-strength parameter R = P (Y < X), when X and Y are two independent weighted Lindley random variables with a common shape parameter. The stress strength parameter cannot be obtained in closed form and it has been expressed in terms of hyper-geometric functions. When the common shape parameter is known, the minimum variance unbiased estimator of R exists. It is quite complicated, and it is not pursued any further here. In this case the maximum likelihood estimators (MLEs) can be obtained explicitly. When the common shape parameter is unknown, the MLEs of the unknown parameters cannot be obtained in explicit forms. The MLEs can be obtained by maximizing the profile log-likelihood function in one dimension. The asymptotic distribution of the MLEs are also obtained, and they have been used to construct the asymptotic confidence interval of R. Bootstrap confidence intervals are also proposed. Monte Carlo simulations are performed to verify the effectiveness of the different methods, and data analysis has been performed for illustrative purposes. Key Words and Phrases: Lindley distribution; Maximum likelihood estimator; Asymptotic distribution; Uniformly minimum variance unbiased estimator; Bootstrap confidence interval. Department of Statistics and Operations Research, Faculty of Science, Kuwait University, P.O. Box 5969 Safat, Kuwait Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin , INDIA. Corresponding author, Phone: , Fax: ; e- mail: kundu@iitk.ac.in. 1

2 2 1 Introduction Lindley (1958) originally proposed a lifetime distribution which has the following probability density function (PDF) f(x; θ) = θ2 1 + θ (1 + x)e θx, x > 0, θ > 0. (1) The distribution function which has the PDF (1) is well known as the Lindley distribution. This distribution was proposed in the context of Bayesian statistics, and it is used as a counter example of fudicial statistics. Ghitany et al. (2008) discussed different properties of the Lindley distribution. It is known that the shape of the PDF of the Lindley distribution is either unimodal or a decreasing function, and the hazard function is always an increasing function. Although, Lindley distribution has some interesting properties, due to presence of only one parameter, it may not very useful in practice. Due to this reason, Ghitany et al. (2011) proposed a two-parameter extension of the Lindley distribution, and named it as the weighted Lindley (WL) distribution. The WL distribution is more flexible than the Lindley distribution, and Lindley distribution can be obtained as a special case of the WL distribution. The WL distribution can be defined as follows. A random variable X is said to have a WL distribution if it has the PDF f(x) = θ c+1 (θ + c) Γ(c) xc 1 (1 + x)e θx, x > 0, c, θ > 0. (2) From now on a WL distribution with the parameters c and θ, and PDF (2) will be denoted by WL(c, θ). The WL distribution has several interesting properties. A brief description of the WL distribution, and some of its new properties are discussed in Section 2. The main aim of this paper is to consider the inference on the stress-strength parameter R = P (Y < X), where X WL(c, θ 1 ), Y WL(c, θ 2 ) and they are independently distributed. Here the notation means follows or has the distribution. The estimation of the

3 3 stress-strength parameter arises quite naturally in the area of system reliability. For example, if X is the strength of a component which is subject to stress Y, then R is a measure of system performance and arises in the context of mechanical reliability of a system. The system fails if and only if at any time the applied stress is greater than its strength. In this paper first we consider the case when all the three parameters are unknown. The MLEs of the unknown parameters cannot be obtained in explicit forms. For a given c, the MLEs of θ 1 and θ 2 can be obtained explicitly, and finally the MLE of c can be obtained by maximizing the profile likelihood function of c. The asymptotic distribution of the MLEs of the unknown parameters and also of R are obtained. The asymptotic distribution has been used to construct asymptotic confidence interval of the stress strength parameter. Since the asymptotic variance of the MLE of R is quite involved, we have proposed to use parametric and non-parametric Bootstrap confidence intervals also. Different methods are compared using Monte Carlo simulations and data analysis has been performed for illustrative purposes. We further consider a special case when the common shape parameter c is known. In this case it is observed that the uniformly minimum variance unbiased estimator (UMVUE) of R exists, although the exact form is very complicated and it is not not pursued any further. When c is known, the MLEs of θ 1 and θ 2 can be obtained in explicit forms. The asymptotic distribution of the MLE of R also can be obtained quite conveniently, and which can be used to construct asymptotic confidence interval of R. It may be mentioned that the estimation of the stress-strength parameter has received considerable attention in the statistical literature starting with the pioneering work of Birnbaum (1956), where the author has provided an interesting connection between the classical Mann-Whitney statistic and the stress-strength model. Since then extensive work has been done on developing the inference procedure on R for different distributions, both from the Bayesian and frequentist points of view. The monograph by Kotz et al. (2003) provided

4 4 an excellent review on this topic till that time. For some of the recent references the readers are referred to the articles by Kundu and Gupta (2005, 2006), Kim and Chung (2006), Krishnomoorthy et al. (2007), Saraçoğlu and Kaya (2007), Jiang and Wong (2008), Raqab et al. (2008), Saraçoğlu et al. (2009), Kundu and Raqab (2009), Saraçoğlu et al. (2012), Al-Mutairi et al. (2013), and see the references cited therein. The rest of the paper is organized as follows. In Section 2, we briefly discuss about the WL distribution, and discuss some of its properties. In Section 3, we discuss the MLE of R and derive its asymptotic properties when all the parameters are unknown. Different confidence intervals are discussed in Section 4. The estimation of R when the common shape parameter c is known is discussed in Section 5. Monte Carlo simulation results are provided in Section 6, and the data analysis has been presented in Section 7. Finally, we conclude the paper in Section 8. 2 Weighted Lindley Distribution A random variable X is said to have a WL distribution with parameter c and θ, if it has the PDF (2). The cumulative distribution function (CDF) and the survival function (SF) take the following forms; F (x) = 1 (θ + c)γ(c, θx) + (θx)c e θx, x > 0, c, θ > 0 (3) (θ + c) Γ(c) S(x) = (θ + c)γ(c, θx) + (θx)c e θx, x > 0, c, θ > 0, (4) (θ + c) Γ(c) where Γ(a, z) = z y a 1 e y dy, a > 0, z 0. It has been observed by Ghitany et al. (2011) that the PDF of the WL distribution is either decreasing, unimodal or decreasing-increasing-decreasing shape depending on the values of c

5 5 and θ. The hazard function of the WL distribution is either bathtub shaped or an increasing function depending on the value of c. It is clear from (3) and (4) that the inversion of the CDF or the SF cannot be performed analytically. The following mixture representation is useful in generating random sample and also deriving some other properties for the WL distribution. The PDF of WL distribution can be written as Here p = θ/(θ + c), and f GA (x; c + j 1) = f(x) = pf GA (x; c, θ) + (1 p)f GA (x; c + 1, θ). (5) θ c+j 1 Γ(c + j 1) xc+j 2 e θx ; x > 0, c, θ > 0, j = 1, 2, is the PDF of the gamma distribution with the shape parameter c+j 1 and scale parameter θ. From now on it will be denoted by Gamma(c+j 1, θ), j = 1, 2. Since the generation from a gamma distribution can be performed quite efficiently, see for example Kundu and Gupta (2007), the generation from a WL distribution becomes quite straight forward. Using (5), it immediately follows that if X WL(c, θ), then the moment generating function (MGF) of X is M X (t) = E(e tx ) = θ ( 1 t ) c + c ( 1 t ) (c+1). (6) θ + c θ θ + c θ The following result will be useful to derive the UMVUE of R, and it may have some independent interest also. Theorem 1: If X 1,..., X n are independent identically distributed WL(c, θ) random variables, then the PDF of Z = X 1, X n n f Z (t) = p n,k (c, θ)f GA (t; nc + n k, θ), k=0 where p n,k (c, θ) = ( ) n θ k c n k k (c + θ). n

6 6 Proof: The proof mainly follows using the MGF of Z. The MGF of Z is M Z (t) = E(e tz ) = = [ θ θ + c ( n n k k=0 ( 1 t ) c + c θ ) ( θ θ + c Now the result follows by using the fact that θ + c ) k ( c θ + c ( 1 t ) ] (c+1) n θ ) n k ( 1 t θ ) (nc+n k). ( 1 t θ ) a is the MGF of Gamma(a, θ). It may be mentioned that the WL distribution belongs to the two-parameter exponential n n family. If x 1,..., x n, is a random sample of size n from WL(c, θ), then ( x i, x i ) is jointly a complete sufficient statistic for (c, θ). We further have the following result regarding likelihood ratio ordering. i=1 i=1 Theorem 2: Let X WL(c 1, θ 1 ) and Y WL(c 2, θ 2 ). If c 1 c 2 and θ 1 θ 2, then X is larger than Y in the sense of likelihood ratio order, written as X lr Y. Proof: From the definition of the likelihood ratio, f(x; c 1, θ 1 ) f(x; c 2, θ 2 ) = (c 2 + θ 2 ) Γ(c 2 ) θ1 c1 1 (c 1 + θ 1 ) Γ(c 1 ) θ c x c 1 c 2 e (θ 1 θ 2 ) x 2 1 increases in x, since c 1 c 2 and θ 1 θ 2. Therefore the result follows. 2 Remarks. (i) If c 1 = c 2, then X lr Y when θ 1 θ 2. It further implies the stochastic ordering also between X and Y. 3 Maximum Likelihood Estimator of R In this section first we provide the expression of R, and then provide the maximum likelihood estimator of R when all the parameters are unknown.

7 7 3.1 Expression of R To derive the expression of R, we will be using the following notation: 0 where Γ(a, x) x s 1 Γ(a + s) β e x dx = 2F s(1 + β) a+s 1 (a + s, 1; s + 1; ), s > 0, β > β 2F 1 (a, b; c; z) = 2 F 1 (b, a; c; z) = n=0 (a) n (b) n (c) n z n n 1 n!, z < 1, (a) 0 = 1, (a) n = (a + k), k=0 is the hyper-geometric function. We assume c 0, 1, 2,... to prevent the denominators from vanishing, see Erdélyi et al. (1954), p. 325, formula (16). Let X WL(c, θ 1 ) and Y WL(c, θ 2 ) be independent random variables. Then R = P (X > Y ) = = = = = = 0 0 P (X > Y Y = y). f Y (y) dy S X (y). f Y (y) dy θ2 c+1 { (θ1 + c) (θ 1 + c)(θ 2 + c) Γ 2 (c) 0 +θ c 1 Γ(c, θ 1 y). (y c 1 + y c )e θ 2y dy 0 (y 2c 1 + y 2c )e (θ 1+θ 2 )y dy } θ2 c+1 { (θ 1 + c) (θ 1 + c)(θ 2 + c) Γ 2 (c) θ1 c+1 Γ(c, t). (θ 1 t c 1 + t c )e (θ 2/θ 1 )t dt 0 + θc 1 Γ(2c) (θ 1 + θ 2 ) (θ 2c θ 2 + 2c) } θ1 c θ2 c+1 Γ(2c) { (θ 1 + c)(θ 1 + θ 2 ) 2F (θ 1 + c)(θ 2 + c)(θ 1 + θ 2 ) 2c+1 Γ 2 1 (2c, 1; c + 1; (c) c + 2c(θ 1 + c) 2F 1 (2c + 1, 1; c + 2; c + 1 θ1 c θ2 c+1 Γ(2c) { θ 1 + θ 2 θ 2 2F (θ 2 + c)(θ 1 + θ 2 ) 2c+1 Γ 2 1 (2c, 1; c + 1; ) (c) c θ 1 + θ 2 θ 2 θ 1 + θ 2 ) θ 2 ) + θ 1 + θ 2 + 2c } θ 1 + θ 2 + 2c θ 2 c F 1 (2c + 1, 1; c + 2; ) + θ 2 + c θ 1 + θ 2 θ 1 + c + 1}

8 8 Remarks. (i) When θ 1 = θ 2, R = 0.5. This, of course, is expected since, in this case, X and Y are i.i.d. and there is an equal chance that X is bigger than Y. 3.2 Maximum Likelihood Estimator of R Suppose x 1, x 2,..., x n1 is a random sample of size n 1 from the WL(c, θ 1 ) and y 1, y 2,..., y n2 is an independent random sample of size n 2 from the WL(c, θ 2 ). The log-likelihood function l l(c, θ 1, θ 2 ) = l(θ) based on the two independent random samples is given by l = n 1 n 2 ln[f X (x i )] + ln[f Y (y j )] i=1 j=1 = n 1 [(c + 1) ln(θ 1 ) ln(θ 1 + c) ln(γ(c))] + ln(1 + x i ) + (c 1) ln(x i ) n 1 θ 1 x + n 2 [(c + 1) ln(θ 2 ) ln(θ 2 + c) ln(γ(c))] + n 1 i=1 n 2 ln(1 + y j ) + (c 1) n 1 i=1 n 2 j=1 j=1 ln(y j ) n 2 θ 2 ȳ, where x and ȳ are the the sample means of x 1,..., x n1 and y 1,..., y n2, respectively. The maximum likelihood estimator (MLE) θ of θ is the solutions of the non-linear equations: l c = n 1 [ ln(θ 1 ) 1 θ 1 + c n 1 n 2 + ln(x i ) + ln(y i ) = 0, i=1 j=1 l c(θ 1 + c + 1) = n 1 n 1 x = 0, θ 1 θ 1 (θ 1 + c) l c(θ 2 + c + 1) = n 2 n 2 ȳ = 0, θ 2 θ 2 (θ 2 + c) ] [ + n 2 ln(θ 2 ) 1 ] (n 1 + n 2 )ψ(c) θ 2 + c where ψ(c) = d ln(γ(c)) is the digamma function. It is clear that the MLEs of the unknown dc parameters cannot be obtained in explicit forms. The MLEs are as follows: θ 1 θ 1 (ĉ) = ĉ( x 1) + ĉ 2 ( x 1) 2 + 4ĉ(ĉ + 1) x 2 x

9 9 θ 2 θ 2 (ĉ) = ĉ(ȳ 1) + ĉ 2 (ȳ 1) 2 + 4ĉ(ĉ + 1)ȳ, 2ȳ where ĉ is the solution of the non-linear equation: 1 n 1 [ ln( θ 1 (ĉ)) θ 1 (ĉ) + ĉ ]+n 1 2[ ln( θ 2 (ĉ)) θ 2 (ĉ) + ĉ ] (n 1 +n 2 )ψ(ĉ)+ ln(x i )+ ln(y i ) = 0. Some iterative procedure needs to be used to solve the above non-linear equation. n 1 i=1 n 2 j=1 Now we provide the elements of the expected Fisher information matrix, which will be needed to construct the asymptotic confidence intervals of the unknown parameters. We will use the following notations. p 1 = lim n 1,n 2 n 1 n 1 + n 2, p 2 = lim n 1,n 2 n 2 n 1 + n 2 and H(θ) = 2 l c 2 2 l θ 1 c 2 l θ 2 c 2 l c θ 1 2 l θ l θ 2 θ 1 2 l c θ 2 2 l θ 1 θ 2 The elements of the symmetric expected Fisher information matrix of (c, θ 1, θ 2 ) is given by In this case 2 l θ 2 2. [ ] I(θ) = ((I ij )) = lim E 1 H(θ). n 1,n 2 n 1 + n 2 I 11 = ψ (c) I 22 = p 1 [ c + 1 θ 2 1 I 33 = p 2 [ c + 1 θ 2 2 p 1 (θ 1 + c) 2 p 2 (θ 2 + c) 2, I 12 = p 1 [ 1 θ 1 + I 13 = p 2 [ 1 θ 2 + I 23 = 0, 1 (θ 1 + c) 2 ], 1 ], (θ 2 + c) 2 1 ], (θ 1 + c) 2 1 (θ 2 + c) 2 ],

10 10 where ψ (c) = d ψ(c) is the trigamma function. Since WL family satisfies all the regularity dc of the consistency and asymptotic normality of the MLEs, see for example Lehmann and Casella (1998), pp , we have the following result. Theorem 3: The asymptotic distribution of the MLE θ of θ satisfies n1 + n 2 ( θ θ) d N 3 (0, I 1 (θ)), where d denotes convergence in distribution and I 1 (θ) is the inverse of the matrix I. The asymptotic variance-covariance matrix of the MLE θ is given by V ar(ĉ) Cov(ĉ, ˆθ 1 ) Cov(ĉ, ˆθ 2 ) 1 I 1 (θ) = Cov(ĉ, n 1 + n 2 ˆθ 1 ) V ar( ˆθ 1 ) Cov( ˆθ 1, ˆθ 2 ). Cov(ĉ, ˆθ 2 ) Cov( ˆθ 1, ˆθ 2 ) V ar( ˆθ 2 ) We need the following integral representation for further development, see for example Andrews (1998), p. 364: 2F 1 (a, b; c; z) = Γ(c) Γ(b)Γ(c b) t b 1 (1 t) c b 1 (1 zt) a dt, zt < 1, c > b > 0, 2F 1 (2c, 1; c + 1; z) = c (1 t) c 1 (1 zt) 2c dt 0 J 1 (c, z) = { } 1 c c 2 F 1 (2c, 1; c + 1; z) ( ) 1 1 t = (1 t) c 1 (1 zt) 2c ln dt 0 (1 zt) 2 1 2F 1 (2c + 1, 1; c + 2; z) = (c + 1) (1 t) c (1 zt) 2c 1 dt 0 J 2 (c, z) = { } 1 c c F 1 (2c + 1, 1; c + 2; z) ( ) 1 1 t = (1 t) c (1 zt) 2c 1 ln dt. 0 (1 zt) 2 In order to establish the asymptotic normality of R, we further define where T is the transpose operation and d(θ) = ( R c, R θ 1, R θ 2 ) T = ( d1, d 2, d 3 ) T,

11 11 d 1 = θ1 c θ2 c+1 Γ(2c) { (θ1 + θ (θ 2 + c)(θ 1 + θ 2 ) 2c+1 Γ 2 2 ) J 1 (c, (c) + 2 θ 2 θ 1 + θ 2 ) + 2c J 2 (c, ) θ 1 + θ 2 θ 2 c F 1 (2c + 1, 1; c + 2; ) + θ 1 θ } 2 θ 1 + θ 2 (θ 1 + c) 2 +R { ( ) θ1 θ 2 ln (θ 1 + θ 2 ) 2 θ 2 + 2ψ(2c) 2ψ(c) 1 θ 2 + c } d 2 = d 2 (θ 1, θ 2 ) = Γ(c ){ 4c 3 + (θ 1 + 1)(θ 1 + θ 2 ) 2 + c 2 (2 + 8θ 1 + 4θ 2 ) + c[5θ1 2 + θ 2 (θ 2 + 2) + θ 1 (4 + 6θ 2 )] } 2 1 2c θ c, π Γ(c)(θ1 + c) 2 (θ 2 + c)(θ 1 + θ 2 ) 2c+2 1 θ2 c 1 d 3 = d 2 (θ 2, θ 1 ). Therefore, using Theorem 3 and delta method, we obtain the asymptotic distribution of R, the MLE of R, as n1 + n 2 ( ˆR R) d N ( 0, d T (θ) I 1 (θ) d(θ) ). (7) From (7) the asymptotic variance of R is obtained as V ar( R) = 1 n 1 + n 2 d T (θ) I 1 (θ) d(θ) = d 2 1 V ar(ĉ) + d 2 2 V ar( θ 1 ) + d 2 3 V ar( θ 2 ) + 2 d 1 d 2 Cov(ĉ, θ 1 ) +2 d 1 d 3 Cov(ĉ, θ 2 ) + 2 d 2 d 3 Cov( θ 1, θ 2 ). Hence, using (7), an asymptotic 100(1 α)% confidence interval for R can be obtained as R z α 2 V ar( ˆR), where z α 2 is the upper α/2 quantile of the standard normal distribution. Since, the expression of asymptotic variance of R is quite involved, we consider different bootstrap confidence intervals of R.

12 12 4 Bootstrap Confidence Intervals In this section we provide different parametric and non-parametric bootstrap confidence intervals of R. It is assumed that we have independent random samples x 1,, x n1 and y 1,, y n2 obtained from WL(c, θ 1 ) and WL(c, θ 2 ), respectively. First we propose to use the following method to generate non-parametric bootstrap samples, as suggested by Efron and Tibshirani (1998), from the given random samples. Algorithm: (Non-parametric bootstrap sampling) Step 1: Generate independent bootstrap samples x 1,, x n 1 and y 1,, y n 2 taken with replacement from the given samples x 1,, x n1 and y 1,, y n2, respectively. Based on the bootstrap samples, compute the MLE (ĉ, θ 1, θ 2) of (c, θ 1, θ 2 ) as well as R = R(ĉ, θ 1, θ 2) of R. Step 2: Repeat Step 1, B times to obtain a set of bootstrap samples of R, say { R j, j = 1,, B}. Using the above bootstrap samples of R we obtain three different bootstrap confidence interval of R. The ordered R j for j = 1, B will be denoted as: R (1) < < R (B). (i) Percentile bootstrap (p-boot) confidence interval: Let R (τ) be the τ percentile of { R j, j = 1, 2,..., B}, i.e. R (τ) is such that 1 B B j=1 I( R j R (τ) ) = τ, 0 < τ < 1, where I( ) is the indicator function.

13 13 A 100(1 α)% p-boot confidence interval of R is given by ( R (α/2), R (1 α/2) ). (ii) Student s t bootstrap (t-boot) confidence interval: Let se ( R) be the sample standard deviation of { R j, j = 1, 2,..., B}, i.e. se ( R) = 1 B B ( R j R ) 2. j=1 Also, let t (τ) be the τ percentile of { R j R se ( R), j = 1, 2,..., B}, i.e. t (τ) is such that 1 B B j=1 I( R j R se ( R) t (τ) ) = τ 0 < τ < 1. A 100(1 α)% t-boot confidence interval of R is given by R ± ˆt (α/2) se ( R). (iii) Bias-corrected and accelerated bootstrap (BC a -boot) confidence interval: Let z (τ) and ẑ 0, respectively, be such that z (τ) = Φ 1 (τ) and ẑ 0 = Φ 1( 1 B B j=1 I( R j R) ), where Φ 1 (.) is the inverse CDF of the standard normal distribution. The value ẑ 0 is called bias-correction. Also, let â = ni=1 ( R (.) R (i) ) 3 6 [ ni=1 ( R (.) R (i) ) 2 ] 3/2 where R (i) is the MLE of R based of (n 1) observations after excluding the ith observation and R ( ) = 1 n ni=1 R (i). The value â is called acceleration factor.

14 14 A 100(1 α)% BC a -boot confidence interval of R is given by ( R (ν 1), R (ν 2) ), where ν 1 = Φ ( ẑ 0 + ẑ 0 + z (α/2) ), ν2 = Φ ( ẑ 0 + z (1 α/2) ) ẑ 1 â(ẑ 0 + z (α/2) 0 +. ) 1 ˆα(ẑ 0 + z (1 α/2) ) Now we provide the following method to generate parametric bootstrap samples from the given random samples, and they can be used to construct different parametric bootstrap confidence intervals. Algorithm: (Parametric bootstrap sampling) Step 1: Compute the MLEs of c, θ 1 and θ 2 from the given random samples, say ĉ, θ 1 and θ 2, respectively. Generate independent bootstrap samples x 1,, x n 1 and y1,, yn 2, from WL(ĉ, θ 1 ) and WL(ĉ, θ 2 ), respectively. Based on the bootstrap samples, compute the MLE (ĉ, θ 1, θ 2) of (c, θ 1, θ 2 ) as well as R = R(ĉ, θ 1, θ 2) of R. Step 2: Repeat Step 1, B times to obtain a set of bootstrap samples of R, say { R j, j = 1,, B}. Using the above bootstrap samples of R we can obtain three different parametric bootstrap confidence interval of R same as the non-parametric ones. It may be mentioned that all the bootstrap confidence intervals can be obtained even in the logit scale also, and we have presented those results in Section 6. In the next section we provide two estimators of R when the common shape parameter c is known.

15 15 5 Estimation of R when c is Known 5.1 Maximum Likelihood Estimator of R Based on the random samples {x 1, x 2,..., x n1 } from WL(c, θ 1 ) and {y 1, y 2,..., y n2 } from the WL(c, θ 2 ), the MLEs of θ 1 and θ 2 can be obtained as ˆθ 1 = c( x 1) + c 2 ( x 1) 2 + 4c(c + 1) x (8) 2 x ˆθ 2 = c(ȳ 1) + c 2 (ȳ 1) 2 + 4c(c + 1)ȳ. (9) 2ȳ The asymptotic distribution of the MLE ( θ 1, θ 2 ) of (θ 1, θ 2 ) satisfies n1 + n 2 { θ 1 θ 1, θ 2 θ 2 } d N 2 (0, A), where A = [ I 1 ] I33 1, and the asymptotic distribution of the MLE R of R satisfies n1 + n 2 ( ˆR R) d N ( ) 0, d 2 I d 3 I33 1. The elements I 22, I 33, d 2 and d 3 are same as defined in Section 3.2. The asymptotic variance of ˆR is given by V ar( ˆR) = 1 ( ) d 2 n 1 + n 2 I d 2 3I The asymptotic 100(1 α)% confidence interval for R is given by R z α 2 V ar( ˆR), where z α 2 is the upper α 2 quantile of the standard normal distribution.

16 Uniformly Minimum Variance Unbiased Estimator of R n If x 1,..., x n is a random sample from WL(c, θ), then x i is a complete sufficient statistic i=1 for θ when c is known. We need the following result for further development. Theorem 4: If X 1,..., X n are i.i.d. Lindley (c, θ) random variables, then the conditional n PDF of X 1 given Z = X i is where i=1 f X1 Z=z(x) = xc 1 (1 + x) Γ(c)A n (z) n 1 k=0 ( ) n 1 c n k 1 C k,n = k Γ(nc + n c k 1) Proof: It mainly follows from the fact and using Theorem 1. C k,n (z x) nc+n c k 2 ; 0 < x < z, and A n (z) = n k=0 f X1 Z=z(x) = f X 1 (x)f X X n (z x) f Z=z (z) ( ) n z nc+n k 1 k Γ(nc + n k). Now we can provide the UMVUE of R in the following Theorem. Theorem 5: Suppose form n 1 x i = u and i=1 n 2 y j = v, then the UMVUE of R has the following j=1 R UMV UE = here 1 A n1 (u)a n2 (v)[γ(c)] 2 I(u, v, a, b) = n 1 1 m=0 min{u,v} u 0 n 1 i=1 y n 2 1 n=0 C m,n1 C n,n2 I(u, v, n 1 c+n 1 m c 2, n 2 c+n 2 n c 2), (xy) c 1 (1 + x)(1 + y)(u x) a (v y) b dxdy. (10) Proof: Let us denote U = X i and V = n 2 j=1 Y j. For known c, since (U, V ) is a complete sufficient statistic for (θ 1, θ 2 ), therefore, the UMVUE of R can be obtained as R UMV UE = E(ϕ(X 1, Y 1 ) U = u, V = v),

17 17 where Therefore, R UMV UE = 1 if x > y ϕ(x, y) = 0 if x y min{u,v} u 0 y f X1 U=u(x)f Y1 V =v(y)dxdy. Now the result follows using Theorem 4. It may be noted that if c is an integer, then a and b in (10) can be taken to be integers, and in that case I(u, v, a, b, ) can be expressed in terms of double summation. For general a and b I(u, v, a, b) = u a+c min{u,v} 0 here B(x; a, b) is the incomplete Beta function, i.e. [ y c 1 (v y) b (1+y) B (1 y ) u ; a + 1, c + ub (1 y )] u ; a + 1, c + 1 dy, B(x; a, b) = x 0 t a 1 (1 t) b 1 dt. Clearly, the UMVUE has a quite complicated form, and it will not be pursued further. 6 Monte Carlo Simulations In this section we present some Monte Carlo simulation results mainly to compare the performances of different methods for different sample sizes and for different parameter values. We mainly investigate the performance of the point and interval estimation of the reliability R = P (X > Y ) based on maximum likelihood procedure when all the parameters are unknown. Specifically, we investigate the bias and mean square error (MSE) of the simulated MLEs. Also, we investigate the coverage probability and the length of the simulated 95% confidence intervals based on maximum likelihood method and different bootstrap procedures presented in Section 4.

18 18 For this purpose, we have generated 3, 000 samples with B = 1, 000 bootstrap samples from each of independent WL(c, θ 1 ) and WL(c, θ 2 ) distributions where (c, θ 1, θ 2 ): (0.75, 2.5, 1), (2, 1, 1) and (4, 2, 5). These parameter values correspond to R = 0.26, 0.5 and 0.91, respectively. We have taken different sample sizes namely (n 1, n 2 ): (20,20), (20,30), (20,50), (30,20), (30,30),(30,50), (50,20), (50,30), (50,50). In Tables 3-5, we report the average biases and the mean squared errors of the estimates of R based on MLEs, parametric bootstrap and non-parametric bootstrap methods. In Tables 6-8, we provide the coverage percentages and the average lengths of the confidence intervals of R based on MLE of R and using different bootstrap methods. Some of the points are quite clear from these experiments. Even for small sample sizes the performances of the MLEs are quite satisfactory in terms of biases and MSEs. It is observed that when n 1 = n 2 = n, and increases, the bias and MSEs decrease. It verifies the consistency property of the MLE of R. For fixed n 1 (n 2 ) as n 2 (n 1 ) increases, the MSEs decrease. Comparing the average confidence lengths and coverage percentages, it is observed that the performance of the confidence intervals of R, based on the asymptotic distribution of MLE is quite satisfactory. It maintains the nominal coverage percentages even for small sample sizes. Among the different bootstrap confidence intervals the biased corrected parametric bootstrap confidence intervals perform the best in terms of the coverage percentages. In most of the cases considered here, the confidence intervals based on MLE performs better than the biased corrected bootstrap confidence intervals in terms of shorter average confidence intervals. 7 Data Analysis In this section we consider two data sets and describe all the details for illustrative purposes. The two data sets were originally reported by Bader and Priest (1982), on failure stresses

19 19 (in GPa) of single carbon fibers of lengths 20 mm and 50 mm, respectively. We present the data sets below. Length 20 mm: X (n 1 = 69): 1.312, 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 2.14, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 2.511, 2.514, 2.535, 2.554, 2.566, 2.57, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773, 2.800, 2.809, 2.818, 2.821, 2.848, 2.88, 2.954, 3.012, 3.067, 3.084, 3.090, 3.096, 3.128, 3.233, 3.433, 3.585, Length 50 mm: Y (n 2 = 65): 1.339, 1.434, 1.549, 1.574, 1.589, 1.613, 1.746, 1.753, 1.764, 1.807, 1.812, 1.84, 1.852, 1.852, 1.862, 1.864, 1.931, 1.952, 1.974, 2.019, 2.051, 2.055, 2.058, 2.088, 2.125, 2.162, 2.171, 2.172, 2.18, 2.194, 2.211, 2.27, 2.272, 2.28, 2.299, 2.308, 2.335, 2.349, 2.356, 2.386, 2.39, 2.41, 2.43, 2.431, 2.458, 2.471, 2.497, 2.514, 2.558, 2.577, 2.593, 2.601, 2.604, 2.62, 2.633, 2.67, 2.682, 2.699, 2.705, 2.735, 2.785, 3.02, 3.042, 3.116, Before progressing further first we perform some preliminary data analysis. The scaled TTT plots of both X and Y (see Figure 1) indicate that the empirical hazard functions will be increasing functions in both the cases. Therefore, WL can be used to analyze the two data sets. First it is assumed that X WL(c 1, θ 1 ) and Y WL(c 2, θ 2 ). The MLEs of the unknown parameters are as follows: ĉ 1 = , θ 1 = , ĉ 2 = , θ 2 = , and the associated log-likelihood value is L 1 = Second suppose that X WL(c, θ 1 ) and Y WL(c, θ 2 ). Then, the MLEs of the unknown parameters are as follows: ĉ = , θ 1 = , θ 2 = , and the associated log-likelihood value is L 0 = We perform the following testing of hypothesis; H 0 : c 1 = c 2, vs. H 1 : c 1 c 2, and in this case 2(L 0 L 1 ) = Hence, the null hypothesis cannot be rejected.

20 20 Therefore, in this case the assumption of c 1 = c 2 is justified (a) (b) Figure 1: Scaled TTT plots: (a) Length 20 mm data (b) Length 50 mm data. To compute the MLEs of c, θ 1 and θ 2, we use the profile likelihood method. Figure 2 provides the profile log-likelihood of c. It indicates that it has a unique maximum. In Table 1 we provide the MLEs of the parameters c, θ 1, θ 2 as well as the Kolmogorov-Smirnov (K-S) and Anderson-Darling (A-D) goodness-of-fit tests. The table shows that both the tests accept the null hypothesis that each data set is drawn from WL distribution. Such conclusions are also supported by various diagnostic plots in Figures 3 and 4. Based on the MLEs ĉ, ˆθ 1, ˆθ 2, the point estimate of R is and the 95% confidence interval of R is (0.5164, ) with confidence length MLE, parametric and non-parametric bootstrap estimates of R are provided in Table 2.

21 21 Table 1: MLEs and K-S and A-D goodness-of-fit tests. Plane MLEs K-S statistic p-value A-D statistic p-value Length 20 mm ĉ = ˆθ 1 = Length 50 mm ĉ = ˆθ 2 = Profile log likelihood of c c Figure 2: The profile log-likelihood of c for the given data. Table 2: Statistical inference of R for the different estimation methods. Estimation Method ˆR 95% C.I. of R confidence length MLE (0.5286, ) Parametric Bootstrap p-boot (0.5265, ) t-boot (0.5265, ) BC a -boot (0.5218, ) Non-Parametric Bootstrap p-boot (0.5364, ) t-boot (0.5364, ) BC a -boot (0.5328, )

22 22 Empirical and fitted densities QQ plot Density Sample Quantiles Failure Stress (in GPa) Fitted Weighted Lindley Quantiles Empirical and fitted CDFs PP plot CDF Empirical CDF Failure Stress (in GPa) Fitted Weighted Lindley CDF Figure 3: Diagnostic plots for the fitted weighted Lindley model for length 20 mm data.

23 23 Empirical and fitted densities QQ plot Density Sample Quantiles Failure Stress (in GPa) Fitted Weighted Lindley Quantiles Empirical and fitted CDFs PP plot CDF Empirical CDF Failure Stress (in GPa) Fitted Weighted Lindley CDF Figure 4: Diagnostic plots for the fitted weighted Lindley model for length 50 mm data.

24 24 8 Concluding Remarks In this paper we consider the statistical inference of R = P (X > Y ), where X and Y are independent weighted Lindley random variables with a common shape parameter. This probability is a measure of discrimination between two groups and has been studied quite extensively under different conditions by various authors. We investigate Maximum likelihood, p-, t-, and BC a -bootstrapping estimation methods (point and interval) of R and their performances are examined by extensive simulations. It is observed that the maximum likelihood method provides very satisfactory results both for point and interval estimation of R. An example is provided to illustrate these results. It is hoped that our investigation will be useful for researchers dealing with the kind of data considered in this paper. One natural question arises what will happen if the shape parameter c is different in two populations? In this case, the expression of R will be different, although it can be written in terms of hyper-geometric functions. The estimation of R and the construction of confidence interval of R can be performed along the same line. References [1] Al-Mutairi, D.K., Ghitany, M.E. and Kundu, D. (2013). Inference on stress-strength reliability from Lindley distribution, Communications in Statistics - Theory and Methods, (to appear) [2] Andrews, L.C. (1998). Special Functions of Mathematics for Engineers. Second edition. Oxford University Press, Oxford. [3] Bader, M.G. and Priest, A.M. (1982). Statistical aspects of fiber and bundle strength in hybrid composites, In Progress in Science and Engineering Composites, (T. Hayashi, K. Kawata, and S. Umekawa, eds.), vol. ICCM-IV, pp [4] Birnbaum, Z.W. (1956). On a use of Mann-Whitney statistics, In Proceedings of the third Berkley Symposium in Mathematics, Statistics and Probability, vol. 1, [5] Church, J.D. and Harris, B. (1970). The estimation of reliability from stress-strength relationships. Technometrics 12:49 54.

25 25 [6] Efron, B., Tibshirani, R.J. (1998). An Introduction to Bootstrap. New York: Chapman & Hall. [7] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1954). Tables of Integral Transforms. McGraw-Hill, New York. [8] Ghitany, M.E., Atieh, B., Nadarajah, S. (2008). Lindley distribution and its application, Mathematics and Computers in Simulation 78: [9] Ghitany, M.E., Al-Qalaf, F., Al-Mutairi, and Husain, H.A. S. (2011). A two-parameter weighted Lindley distribution and its applications to survival data, Mathematics and Computers in Simulation 8: [10] Gupta, R.C., Ramakrishnan, S. and Zhou, X. (1999) Point and interval estimation of P(X < Y ): The normal case with common coefficient of variation. Annals of the Institute of Statistical Mathematics 51: [11] Gupta, R.C. and Brown, N. (2001) Reliability studies of the skew-normal distribution and its application to strength-stress models. Communications in Statistics, Theory and Methods 30: [12] Gupta, R.C. and Li, X. (2006) Statistical inference for the common mean of two lognormal distributions and some applications in reliability. Computational Statistics and Data Analysis 50: [13] Gupta, R.C. and Peng, C. (2009) Estimating reliability in proportional odds ratio models. Computational Statistics and Data Analysis 53: [14] Jiang L. and Wong, A.C.M. (2008). A note on inference for P (X < Y ) for right truncated exponentially distributed data, Statistical Papers 49: [15] Kim, C. and Chung, Y. (2006). Bayesian estimation of P (Y < X) from Burr-Type X model containing spurious observations, Statistical Papers 47: [16] Kotz, S., Lumelskii, Y. and Pensky, M. (2003). The Stress-Strength Model and its Generalizations: Theory and Applications. Singapore: World Scientific Press. [17] Krishnamoorthy, K., Mukherjee, S. and Guo, H. (2007). Inference on reliability in twoparameter exponential stress-strength model. Metrika 65: [18] Kundu, D. and Raqab, M.Z. (2009). Estimation of R = P (Y < X) for three-parameter Weibull distribution. Statistics and Probability Letters 79: [19] Kundu, D. and Gupta, R.D. (2007). A convenient way of generating gamma random variables. Computational Statistics and Data Analysis 51: [20] Kundu, D. and Gupta, R.D. (2006). Estimation of R = P [Y < X] for Weibull distributions. IEEE Transactions on Reliability 55: [21] Kundu, D. and Gupta, R.D. (2005). Estimation of P [Y < X] for generalized exponential distribution. Metrika 61:

26 26 [22] Lehmann, L.E. and Casella, G. (1998) Theory of Point Estimation, 2nd Edition, New York: Springer. [23] Lindley, D.V. (1958). Fudicial distributions and Bayes theorem. Journal of the Royal Statistical Society B 20: [24] Raqab, M.Z. and Kundu, D. (2005). Comparison of different estimators of P [Y < X] for a scaled Burr type X distribution. Communications in Statistics-Simulation and Computation 34: [25] Raqab. M.Z., Madi, M.T. and Kundu, D. (2008). Estimation of R = P (Y < X) for the 3-parameter generalized exponential distribution, Communications in Statistics - Theory and Methods 37: [26] Saraçoğlu, B., Kaya, M.F. (2007). Maximum likelihood estimation and confidence intervals of system reliability for Gompertz distribution in stress-strength models, Seluk Journal of Applied Mathematics 8: [27] Saraçoğlu, B., Kaya, M.F., Abd-Elfattah, A.M. (2009). Comparison of estimators for stress-strength reliability in Gompertz Case, Hacettepe Journal of Mathematics and Statistics 38: [28] Saraçoğlu, B., Kinaci, I. and Kundu, D. (2012). On estimation of R = P (Y < X) for exponential distribution under progressive type-ii censoring. Journal of Statistical Computation and Simulation 82: Table 3: Average bias (mean squared error) of different estimators of R = P (X > Y ) when c = 0.75, θ 1 = 2.5 and θ 2 = 1. n 1, n 2 MLE Parametric Boot Non-Parametric Boot 20, (0.0046) (0.0047) (0.0047) 20, (0.0037) (0.0038) (0.0037) 20, (0.0029) (0.0029) (0.0030) 30, (0.0037) (0.0039) (0.0037) 30, (0.0029) (0.0031) (0.0029) 30, (0.0023) (0.0023) (0.0024) 50, (0.0031) (0.0033) (0.0032) 50, (0.0023) (0.0023) (0.0023) 50, (0.0018) (0.0017) (0.0018)

27 27 Table 4: Average bias (mean squared error) of different estimators of R = P (X > Y ) when c = 2, θ 1 = 1 and θ 2 = 1. n 1, n 2 MLE Parametric Boot Non-Parametric Boot 20, (0.0078) (0.0079) (0.0080) 20, (0.0065) (0.0065) (0.0067) 20, (0.0055) (0.0051) (0.0056) 30, (0.0064) (0.0067) (0.0065) 30, (0.0050) (0.0052) (0.0052) 30, (0.0040) (0.0041) (0.0040) 50, (0.0052) (0.0056) (0.0053) 50, (0.0040) (0.0042) (0.0040) 50, (0.0030) (0.0030) (0.0030) Table 5: Average bias (mean squared error) of different estimators of R = P (X > Y ) when c = 4, θ 1 = 2 and θ 2 = 5. n 1, n 2 MLE Parametric Boot Non-Parametric Boot 20, (0.0016) (0.0015) (0.0016) 20, (0.0013) (0.0012) (0.0013) 20, (0.0010) (0.0011) (0.0010) 30, (0.0014) (0.0013) (0.0013) 30, (0.0011) (0.0010) (0.0011) 30, (0.0009) (0.0009) (0.0008) 50, (0.0010) (0.0011) (0.0010) 50, (0.0008) (0.0008) (0.0008) 50, (0.0006) (0.0007) (0.0006)

28 28 Table 6: Coverage probability (average confidence length) of different estimators of R = P (X > Y ) when c = 0.75, θ 1 = 2.5 and θ 2 = 1. n 1, n 2 MLE Parametric Bootstrap Non-parametric Bootstrap p-boot t-boot BC a -boot p-boot t-boot BC a -boot 20, (0.2473) (0.2546) (0.3050) (0.2635) (0.2496) (0.3026) (0.2566) 20, (0.2265) (0.2299) (0.2764) (0.2379) (0.2252) (0.2739) (0.2310) 20, (0.2048) (0.2075) (0.2474) (0.2142) (0.2003) (0.2432) (0.2057) 30, (0.2277) (0.2348) (0.2648) (0.2397) (0.2312) (0.2624) (0.2347) 30, (0.2044) (0.2086) (0.2392) (0.2129) (0.2048) (0.2358) (0.2085) 30, (0.1815) (0.1845) (0.2128) (0.1884) (0.1805) (0.2101) (0.1840) 50, (0.2090) (0.2163) (0.2288) (0.2174) (0.2113) (0.2234) (0.2119) 50, (0.1842) (0.1879) (0.2036) (0.1897) (0.1855) (0.2017) (0.1868) 50, (0.1599) (0.1621) (0.1792) (0.1640) (0.1597) (0.1770) (0.1614)

29 29 Table 7: Coverage probability (average confidence length) of different estimators of R = P (X > Y ) when c = 2, θ 1 = 1 and θ 2 = 1. n 1, n 2 MLE Parametric Bootstrap Non-parametric Bootstrap p-boot t-boot BC a -boot p-boot t-boot BC a -boot 20, (0.3172) (0.3361) (0.3391) (0.3380) (0.3355) (0.3398) (0.3365) 20, (0.2918) (0.3060) (0.3135) (0.3071) (0.3035) (0.3122) (0.3041) 20, (0.2690) (0.2791) (0.2904) (0.2795) (0.2756) (0.2880) (0.2759) 30, (0.2920) (0.3059) (0.3031) (0.3073) (0.3040) (0.3012) (0.3049) 30, (0.2635) (0.2740) (0.2760) (0.2749) (0.2724) (0.2743) (0.2728) 30, (0.2373) (0.2444) (0.2496) (0.2447) (0.2432) (0.2492) (0.2433) 50, (0.2693) (0.2786) (0.2713) (0.2797) (0.2747) (0.2661) (0.2755) 50, (0.2372) (0.2448) (0.2422) (0.2455) (0.2423) (0.2394) (0.2427) 50, (0.2069) (0.2121) (0.2132) (0.2124) (0.2107) (0.2117) (0.2109)

30 30 Table 8: Coverage probability (average confidence length) of different estimators of R = P (X > Y ) when c = 4, θ 1 = 2 and θ 2 = 5. n 1, n 2 MLE Parametric Bootstrap Non-parametric Bootstrap p-boot t-boot BC a -boot p-boot t-boot BC a -boot 20, (0.1641) (0.1473) (0.1305) (0.1731) (0.1426) (0.1254) (0.1646) 20, (0.1490) (0.1368) (0.1226) (0.1543) (0.1346) (0.1200) (0.1507) 20, (0.1325) (0.1262) (0.1155) (0.1363) (0.1235) (0.1124) (0.1330) 30, (0.1485) (0.1344) (0.1190) (0.1546) (0.1305) (0.1147) (0.1478) 30, (0.1331) (0.1235) (0.1100) (0.1377) (0.1208) (0.1071) (0.1337) 30, (0.1179) (0.1113) (0.1007) (0.1200) (0.1102) (0.0994) (0.1183) 50, (0.1327) (0.1227) (0.1094) (0.1370) (0.1191) (0.1054) (0.1319) 50, (0.1170) (0.1097) (0.0980) (0.1203) (0.1077) (0.0955) (0.1175) 50, (0.1020) (0.0981) (0.0884) (0.1047) (0.0960) (0.0862) (0.1023)