Interval Estimation of a P(X1 < X2) Model for Variables having General Inverse Exponential Form Distributions with Unknown Parameters

In this article the interval estimation of a P(X1 < X2) model is discussed when X and X are non-negative independent random variables, having general inverse exponential form distributions with different unknown parameters. Different interval estimators are derived, by applying different approaches. A simulation study is performed to compare the estimators obtained. The comparison is carried out on basis of average length, average coverage, and tail errors. The results are illustrated, using inverse Weibull distribution as an example of the general inverse exponential form distribution.


Introduction
There has been continuous interest in the problem of estimating the stress-strength reliability, R P X X , where X and X are independent random variables. The parameter R is referred to as the reliability parameter. This problem arises in the classical stress-strength reliability, where the random strength X of a component exceeds the random stress X to which the component is subjected. If X X , then the component fails. Kotz et al. [1] have surveyed most all the theoretical and the practical results on the theory and applications of the stress-strength reliability problem up to year 2003. However, after year 2003 several authors have considered the stress-strength reliability problem by different approaches for example, Al-Mutairi et al. [2], Amiri et al. [3], Rezaei et al. [4], among others.
Recently Mokhlis et al. [5] have obtained the point and the interval estimation of R P X X by different methods, under the assumption that X and X are non-negative independent and continuous random variables, having the general inverse exponential form with the cumulative distribution functions (CDFs) and probability density functions (pdfs) given respectively by where c is a common known parameter, η ∈ ζ ; i 1, 2, are unknown parameters,ζ is the parametric space,g x; c is a continuous, monotone decreasing, differentiable function , such that, g x; c → ∞ as x → 0 and g x; c → 0 as x → ∞ . They proved that the reliability function, R, is given by R . / 0 .
, if and only if, X and X have CDFs as in (1).
Also, Mokhlis et al. [6] obtained interval estimators of R, when c is unknown, and η b , c is a function of the unknown parameters b and c; i 1,2 . The reliability, R, takes the form The present paper, presents estimation of R, when X and X follow the general inverse exponential form distributions with CDFs and pdfs given respectively by F 4 x; η , c = exp −η b , c g x; c , and f 4 x; η , c = −η b , c g < x; c exp −η b , c g x; c ; i = 1,2 .
(3) whereη = η b , c is a differentiable function in two unknown parameters b and c , where c ∈ ∁ , b ∈ B , and ∁ and B are the parametric spaces of c and b , respectively. The function g x; c is a continuous, monotone decreasing, differentiable function, such that, g x; c → ∞ as x → 0 and g x; c → 0asx → ∞, g < x; c is the first derivative of g x; c w.r.t x. Using (3), the reliability is given by R = ? −η 2 b 2 , c 2 @ A g ′ z; c 2 expD−η 1 b 1 , c 1 g z; c 1 − η 2 b 2 , c 2 g z; c 2 E dz (4) The above integral can be evaluated numerically. If c = c = c, then the reliability can be expressed as in (2) as Mokhlis et al. [6].
Different interval estimators are constructed by applying different approaches. (i) An approximate confidence interval for R is constructed; using the maximum likelihood estimator (MLE) of R. (ii) A generalized confidence interval is obtained, using the generalized variable (GV) approach. (iii) Two bootstrap confidence intervals (percentile and t) are also presented. (iv) Two Bayesian credible intervals of R are obtained, using Markov chain Monte Carlo (MCMC) method, with different priors. The different interval estimators obtained are illustrated using inverse Weibull distributions as examples of the underlying distributions. A comparison is performed, by means of simulation among the estimators obtained on the basis of average length, average coverage, and tail errors. This paper is organized as follows: the approximate confidence interval for R is obtained, in Section 2. In Section 3, using generalized variable approach, the generalized confidence interval of R is derived. The bootstrap intervals are obtained, using percentile and t-bootstrap methods, in Section 4. In Section 5, using MCMC method, two Bayesian credible intervals of R are presented by applying two different sets of priors. In Section 6, taking the inverse Weibull distribution as an example of the underlying distributions, the results obtained are illustrated and a numerical comparison of the interval estimators is performed.

Approximate Confidence Interval (ACI) of R
Let X = FX , X , … , X H; i = 1, 2, be two independent random samples from populations with general inverse exponential form distributions given by (3). The likelihood function is where x ij is the jth observation in the sample X ; j = 1, …, n i . For simplicity write η = η b , c ; i = 1,2, the log-likelihood function is L= L x ,x , c c n ln ln g (x ;c ) g(x ;c ).
To derive the MLEs,η I , c I , and b J of η ,c , and b ; i = 1, 2, respectively, solve the following system of equations for η I , c K , and b J The MLE, c I of c ; i = 1, 2, can be obtained by substituting (9) in (8), and then solving (8) numerically. Finally, the MLE, b J of b ; i = 1, 2, is obtained by using the relation η I = η b J , c I . Consequently, the MLE, R L of R can be obtained by replacing the parameters in (4) with their MLEs. This means R L = ? −η I @ A g < z; c I exp −η I g z; c I − η I g z; c I dz. (10) Clearly, b J doesn't need to be in R L . The MLE, R L is asymptotically normal with mean R and variance σ N L = S P T S , where T is the inverse of the Fisher Clearly, the explicit expression of U N L depends on η , g < Fx V ; c H , and gFx V ; c H ; j = 1, … , n , i = 1, 2 . The 1 − α 100% ACI for R is FR L ±z [ ⁄ σ K N L H , where σ K N L = S P T S| 2 2 L , I , K is the estimator of U N L .

Generalized Confidence Interval (GCI) of R
Let X = FX , X , … , X H; i = 1, 2, be two independent random samples from populations with distributions given in (3), having unknown parameters c and b ; i = 1, 2 . It is known that, a GPQ of any parameter is a function of observed statistics and random variables whose distribution is free of unknown parameters. For constructing the GCI, applying a useful feature of the GV approach, which states that if G 2 and G are GPQs of b and c , then η G 2 , G is a GPQ of η ; i = 1, 2. This feature enables us obtaining a GPQ for R given as G N = R G / , G . , G / , G . by replacing the parameters in (4) with their GPQs, then using G N in constructing confidence interval for R. The ⁄ th and 1 − α 2 ⁄ th quantiles of R.

Bootstrap Confidence Interval (boot) of R
There are several ways to construct bootstrap confidence intervals. Clearly, the percentile and t-bootstrap confidence intervals are commonly used for the reliability, (see, Efron [7]). Algorithm 1 is applied for constructing the bootstrap confidence intervals.

Bayesian Credible Interval (BCI) of R
a. Gamma priors (G-BCI) Let X ; i = 1, 2 be two independent random samples from general inverse exponential form distributions in (3) with unknown parameters b and c . Consider, η = η b , c , as a single parameter, assume that the prior distributions of η ; i = 1, 2 are independent with pdfs Moreover assume that, c ; i = 1, 2 have independent gamma priors with pdfs π η π η π π π η η = π η π η π π η η η η η η It is observed that (13) cannot be obtained in a closed form. The BCI can be computed by a combination of Metropolis-Hastings and Gibbs sampling. Moreover the marginal posterior distribution of η is which is Gamman n + α , pβ Assume that X , i = 1, 2 are two independent random samples from populations with (3) having unknown parameters b and c ; i = 1, 2, and also assume that, η ; i = 1, 2 has independent gamma prior as in (11) The marginal posterior distribution of c ; i = 1, 2, is not a known form. To obtain the 1 − α 100%M-BCI for R of mixed priors, use Algorithm 2 with a difference in step 4 by generating c V from (16) instead of (15

Numerical Illustration
This section, presents a numerical illustration of the results obtained. The confidence intervals; ACI, GCI, P-boot, Tboot, G-BCI, and M-BCI for R with some general inverse exponential form distributions are compared. 1000 samples of sample sizes (n 1 , n 2 ) = (10, 10) and (30, 30) from the underlying distributions of X and X , with unknown parameters are generated. Taking α = 0.05, average length, average coverage probability, left and right tail errors of the 1 − α 100% confidence intervals are calculated. The parameter values that produce the values of R to be approximately 0.6, 0.7, 0.8, 0.9, 0.95, 0.97, and 0.99 are selected.
The inverse Weibull distribution is chosen as an example of the general inverse exponential form. The inverse Weibull distribution is flexible and includes a variety of distributions. For the inverse Weibull distributions, the CDFs with η b , c = 2 ‹ , g x; c = ‹ , and g < x; c = ‹ OE/ ; i = 1,2, and the reliability is obtained by substituting these values in (4) as As noticed from (9) Order G N k ; j = 1, … , N, ascending to obtain G N k ≤ ⋯ ≤ G N k g .
M-BCI: let α , β = 2, 4 , and α , β = 1, 2 . The comparisons on the basis of average length, average coverage, and left and right tail errors are introduced in Tables (1-4), respectively. From Tables 1-4, the following results are observed: 1. As expected, the average length of all confidence intervals decreases when n and R increase. 2. When R = 0.8068-0.9910, the average length of boot and the left tail error of T-boot are smallest, and the left tail error of all confidence intervals decrease when n and R increase. 3. The average coverage of GCI is approximately around the 1 − α 100%. 4. The average coverage of ACI, boot, and G-BCI decrease when R increases. But the average coverage of ACI and boot increase when n increases. 5. For all the confidence intervals, the right tail error is decreasing when R is increasing. Moreover the right tail error of all confidence intervals except BCI are decreasing when n is increasing. 6. The right tail error of BCI is the smallest.

Conclusion
In this article, several approaches for estimating the confidence intervals of Stress-Strength Reliability P(X 1 < X 2 ) are provided when random variables having general inverse exponential form distributions with different unknown parameters. A simulation study is performed to compare its performance using the inverse Weibull distribution as an example of the general inverse exponential form distribution. The comparison is carried out on basis of average length, average coverage, and tail errors. The results obtained were very close to each other.