Interval Estimation of Stress-Strength Reliability for a General Exponential Form Distribution with Different Unknown Parameters

This paper deals with interval estimation of the stress-strength reliability, when the stress and strength variables follow a general exponential form distribution. The distribution parameters of both the stress and the strength are assumed to be unknown. Interval estimation for reliability is discussed, using different approaches. The results obtained are applicable to many well known distributions. For illustration of the general results obtained a simulation study is performed with application on Weibull distribution. Numerical comparison of the interval estimators is carried out based on average length, probability coverage, and tail errors.


Introduction
The stress-strength reliability can be expressed as R = P (X 1 <X 2 ), where the random variables X 1 and X 2 represent a random stress and a random strength, respectively.The first papers with P (X 1 <X 2 ) in their title were introduced by (Birnbaum, 1956), and (Birnbaum and McCarty, 1958).Because of the importance of what we talk about there are a lot of research, which took this topic in different ways and different distributions.One of these ways the point estimation of R, (see, (Saracoglu et al., 2009), and (Panahi and Asadi, 2010)).In many situations knowing an interval estimator is better than just knowing a point estimator because the interval estimator covers the unknown parameter R with a specified probability, confidence coefficient.Many authors have studied the interval estimation of R.Among them, (Kundu and Gupta, 2006), (Krishnamoorthy and Lin, 2010), (Asgharzadeh et al., 2011), and (Asgharzadeh et al., 2013).(Kotz et al., 2003) have comprehensively covered the problem of point and interval estimation of R. (Singh et al., 2015), and (Asgharzadeh et al., 2017) have also discussed the problem of point and interval estimation of R. Recently, (Mokhlis et al., 2017) introduced point and interval estimation of R = P (X 1 <X 2 ) by different methods when X 1 and X 2 follow a general exponential form or a general inverse exponential form with survival functions given by either or respectively, where g 1 (x;c) is free from the unknown parameter θ, continuous, monotone increasing, and differentiable function, with g 1 (x;c)→0 as x→0 and g 1 (x;c)→∞ as x→∞, while g 2 (x;c) is free from the unknown parameter η, continuous, monotone decreasing, and differentiable function, with g 2 (x;c)→∞ as x→0 and g 2 (x;c)→0 as x→∞, and c is a known parameter.
In this paper, we discuss the interval estimation of R =P (X 1 <X 2 ), where X 1 and X 2 follow a general exponential form distribution.Different methods of estimation are discussed.We obtain an approximate confidence interval of R via the maximum likelihood estimator of R. Using the generalized variable approach, a generalized confidence interval of R is presented.Two bootstrap confidence intervals (percentile and t-boot) of R are also obtained.Bayesian credible interval of R is also considered, using Markov chain Monte Carlo method (MCMC) in two scenarios.In the first scenario we apply gamma priors, while in the second one we apply gamma and uniform priors.The different interval estimators are illustrated using Weibull distribution, and a comparison is performed among the estimators obtained, on the basis of average length, average coverage probability, and tail errors.
This paper is organized as follows: In Section 2, the stress-strength reliability, R, and its maximum likelihood estimator are introduced.The approximate confidence interval of R is considered, in Section 3. In Section 4, the generalized confidence interval of R is presented, while in Section 5, the percentile and t-bootstrap intervals of R are obtained.In Section 6, the Bayesian credible intervals of R are introduced.In Section 7, an illustrative example of the obtained interval estimators of R is given by using the Weibull distribution.Then a numerical comparison of the intervals obtained is applying the Weibull distribution, in Section 8.

Maximum Likelihood Estimator of R
Let X 1 and X 2 be non-negative independent and continuous random variables, having general exponential form distributions, with the survival functions (SF) and the probability density functions (pdf) given by and where φ i (b i , c i ) is a differentiable function of the unknown parameters b i ∈ B i , and c i ∈ i , and B i , i are the parametric spaces of b i , and c i , respectively.The function g (x;c i ) is continuous, monotone increasing, differentiable function such that, g (x;c i ) → 0 as x→0 and g (x;c i ) → ∞ as x→∞, and g ′ (x;c i ) is the first derivative of g (x;c i ) w.r.t x.Notice that, if Then the stress-strength reliability is given by where In case of c 1 =c 2 = c, the stress-strength reliability can be expressed as R = φ 1 φ 1 +φ 2 as in (Mokhils et al., 2017).
, are two independent random samples from populations with distributions given by (1), then the likelihood function is given as where x i j is the j th observation in the sample X i ;j = 1, . .., n i , i = 1, 2, and Partially differentiating ln L ) with respect to θ, and equating to 0, we get and From (6) and knowing that ∂φ i ∂b i 0, the MLEs, φi of φ i , are given by φi = n i ∑ n i j=1 g(x i j ; ĉi ) The MLEs ĉi of c i can be obtained, by substituting φi into (7) and solving numerically.Once we obtain ĉi , the MLEs, φi of φ i are completely determined by substituting ĉi in (8).Using the invariance property, the MLEs, bi of b i can be deduced from φi The corresponding MLE, R of R can be obtained by replacing the parameters in (3) with their MLEs.

Approximate Confidence Interval of R (ACI)
As we know from the asymptotic maximum likelihood properties, the approximate (1−α)100% confidence interval for R is ) , where z (1−α/2) is the (1−α/2)th quantile of the standard normal distribution and σ2 R is the estimator of variance of R, σ2 R= A t B −1 A θ= θ, where θ is the MLE of θ, B −1 is the inverse of the Fisher information matrix B of θ, A t is the transpose of matrix A, (see, (Rao, 1965)), where We can see that the explicit expression of σ 2 R depends on the forms of φ i , g

Generalized Confidence Interval of R (GCI)
We obtain the generalized confidence interval for R by applying the generalized variable approach.The generalized pivotal quantity and G c i , where G φ i , G b i , and G c i denote the GPQs for φ i , b i , and c i ; i = 1, 2, respectively.The GPQ is a function of observed statistics and random variables whose distribution is free of unknown parameters.The (1−α)100% generalized confidence interval of R can be obtained as , where G R(α/2) and G R(1−α/2) are the (α/2)th and (1−α/2)th quantiles of R, respectively.

Bootstrap Confidence Intervals of R (boot)
The bootstrap confidence interval of R can be estimated using either percentile bootstrap or t-bootstrap.The bootstrap samples will be generated firstly, using the following bootstrap sampling algorithm, (see, (Efron, 1994)).

Repeat
Step 2, N times to obtain a set of bootstrap samples of R, say { R * * j ;j = 1, . . ., N } .Order R * * j , in an increasing order.

Bayesian Credible Interval of R (BCI)
We suggest two different scenarios to estimate the Bayesian credible interval of R by applying MCMC.For the first scenario, we assume gamma priors for φ 1 , φ 2 , c 1 , and c 2 , while in the second scenario, we consider independent gamma priors for φ 1 , φ 2 and uniform priors for c 1 , c 2 as the available priors information is weak.

Gamma Priors (G-BCI)
Let the prior density of φ i ; i = 1, 2 be gamma given by and also assume that φ 1 and φ 2 are independent.Moreover, assume that c i ; i = 1, 2 have gamma priors with probability density functions and c 1 and c 2 are independent.From ( 4), ( 9), and ( 10), the joint posterior density function of φ 1 , φ 2 , c 1 , and c 2 can be obtained as Since, the joint posterior density function cannot be obtained analytically, we apply MCMC method to estimate the Bayesian credible interval of R.There are generally two algorithms, the Gibbs sampling and the Metropolis-Hastings algorithm.If the conditional distribution for each parameter is a known distribution, the Gibbs sampling can be used.If the conditional distribution doesn't look like any known distribution, in this case the Metropolis-Hastings algorithm can be useful.To perform the MCMC method, we first find the marginal posterior distributions of φ i and c i .The marginal posterior distribution of φ i is The marginal posterior distribution of c i is where However, we shall use the union of the Metropolis-Hastings with Gibbs sampling, (see, (Asgharzadeh et al., 2013)).The procedure is shown by the following algorithm.

Mixed Priors (M-BCI)
Let the prior density function of φ i be as (9); i= 1, 2, and assume that, c i has uniform prior distribution with probability density function From the likelihood function in (4) and the prior density functions of φ 1 , φ 2 , c 1 , and c 2 , the joint density function can be obtained as The joint posterior density functions of φ 1 , φ 2 , c 1 , and c 2 based on ( 13) is given as The marginal posterior distribution of φ i is given by the same as in (11), while the marginal posterior distribution of c i becomes where As we notice that, the marginal posterior distributions of c 1 and c 2 do not have a known form.Using a technique similar to that in Algorithm 2 except for the posterior distribution of c i ;i = 1, 2. The (1−α)100% Bayesian credible interval of R can be obtained as ) , where R m(α/2) and R m(1−α/2) are the (α/2)th and (1−α/2)th quantiles of R, respectively.

Illustrative Example
We see that, the Weibull distribution follows the general exponential form in (1), with φ i = 1 b c i i and g (x;c i ) = x c i .If X i ; i = 1, 2, are two independent random samples from Weibull distributions with the survival function given as F X i (x;b i , c i ) =exp α)100% t-bootstrap confidence interval of R is given by ( R− t(1−α/2) S * * , R− t(α/2) S * * ) , where S * * be the sample standard deviation of { R * * j ;j = 1, . . ., N } and t(α) be the (α)th quantile of { R * * j − R S * * ;j = 1, . . ., N } .
Using the MLEs, ( ĉ1 , ĉ2 , φ1 , φ2 , R) , the approximate (1−α)100% confidence interval for R can be obtained.Using the Newton-Raphson iterative method, the MLE ĉi , of c i from (7) is given by 1 and the MLE, φi , of φ i from (8) can be expressed as φi =1 bĉ i i = n i ∑n i j=1 x ĉi ij, and also the MLE, bi , of b i can be deduced as bi = ĉ i ;i = 1, 2. The MLE, R of R is given as R= ĉ1

Figure 2 :
Figure 2: Average coverage probability for Weibull distribution.

Figure 2 .
Figure 2. Average coverage probability for Weibull distribution

Figure 4 :
Figure 4: Right tail error for Weibull distribution.

Figure 3 .
Figure 3. Left tail error for Weibull distribution