Bayesian Estimation of the Stress-Strength Reliability Based on Generalized Order Statistics for Pareto Distribution

. Te aim of this paper is to obtain a Bayesian estimator of stress-strength reliability based on generalized order statistics for Pareto distribution. Te dependence of the Pareto distribution support on the parameter complicates the calculations. Hence, in literature, one of the parameters is assumed to be known. In this paper, for the frst time, two parameters of Pareto distribution are considered unknown. In computing the Bayesian confdence interval for reliability based on generalized order statistics, the posterior distribution has a complex form that cannot be sampled by conventional methods. To solve this problem, we propose an acceptance-rejection algorithm to generate a sample of the posterior distribution. We also propose a particular case of this model and obtain the classical and Bayesian estimators for this particular case. In this case, to obtain the Bayesian estimator of stress-strength reliability, we propose a variable change method. Ten, these confdence intervals are compared by simulation. Finally, a practical example of this study is provided.


Introduction
Tere are at least two factors in a system, one of which puts stress on the other and the other factor resists.In this case, the stress-strength parameter is raised.In a system where stress is applied to its component and members, it resists that stress.According to this model, the more stress is created on the system, the more likely the system will fail.In other words, the system continues to operate as long as the system's strength is greater than the stress applied to it.Te stress-strength parameter is defned as a probability R � P(X > Y), in which Y is the random variable of stress and X is the random variable of strength based on which the survival of a system can be controlled.Te term stress-strength model was frst coined by [1].Ten, many studies were performed on the stress-strength parameter based on diferent distributions and diferent conditions governing random variables.Some of the most recently used distributions include two-parameter bathtubshaped lifetime distribution [2], POLO distribution [3], fnite mixture distributions [4], standard two-sided power distribution [5], Kumaraswamy distribution [6], and Gompertz distribution [7].
Te Pareto distribution is one of the most important statistical distributions with heavy and skewed tails that is used as a model for many socioeconomic phenomena.Te Pareto distribution is also used to study the lifetime of organisms and the issue of reliability, as well as many statistical issues related to fnance, insurance, and hydrology.In recent years, the study of reliability based on the Pareto distribution has become an exciting topic.Reference [8] estimated the reliability of the Pareto distribution in the presence of outliers using maximum likelihood (ML), method of moments, least squares, and modifed maximum likelihood.Reference [9] studied the confdence interval estimation and approximate hypothesis testing for the reliability of the Pareto distribution based on progressively type-II-censored data with the generalized variable method.Reference [10] assumed the scale parameter of the Pareto distribution to be known and obtained the Bayesian reliability estimate using conjugate and Jefrey's priors based on type-II-censored data.Reference [11], with the generalized variable method, investigated the reliability of the generalized Pareto distribution.Reference [12] obtained the ML and Bayesian estimates and also the highest posterior density interval for multicomponent stress-strength reliability by considering diferent shape parameters and common scale parameters based on upper record values.
Te generalized order statistics (GOS) model can be considered as a unifed model for studying ordered random variables [13].Te GOS includes a wide range of statistics with a sequential nature, such as ordinary order statistics, progressively type-II, censored order statistics, type-II-censored order statistics, and frst n record values as subgroups.Te theorems expressed and proved for the GOS are also established in its subgroups.Te importance of using these models in terms of reliability cannot be ignored.Recently, the study of Pareto distribution based on the GOS has attracted the attention of many authors.Reference [14] obtained the ratio distribution of the GOS from the Pareto distribution.Te recurrence relations of moments for the Pareto distribution's GOS were presented by [15].Reference [16] estimated the parameters of the generalized Pareto distribution based on GOS using ML, bootstrap, and Bayesian under the LSE and LINEX loss functions.Reference [17] studied the properties, recurrence relations of moments, and ML estimate of the parameters for the generalized Pareto distribution based on the GOS.
Reference [18] studied the analysis of stress-strength reliability model based on the Pareto distribution using records, and [10] studied this model based on censored data.However, an analysis of the stress-strength reliability model for the Pareto distribution based on GOS is not available in the statistical literature.On the other hand, because the support of the Pareto distribution depends on the parameter, due to the difculty of having two unknown parameters in articles, one parameter is assumed to be fxed and analyses are performed.In this paper, for the frst time, we present the estimation of the stress-strength reliability of the Pareto distribution using classical and Bayesian inference based on the GOS, where both parameters are considered unknown.In the Bayesian method, the posterior distribution is not a closed form; so, to produce a sample of it, we propose the acceptance-rejection method.We also introduce a special case of this model.In estimating the reliability of the special case by the Bayesian method, we need to solve an integral that cannot be solved by analytical methods and we propose a method to solve it using variable change and Monte Carlo.
Te structure of the article includes seven sections.In Section 2, the generalized, bootstrap percentile, and bootstrap-t confdence intervals of stress-strength reliability (R) for the Pareto distribution are calculated, which we denote these confdence intervals by GCI, BPCI, and BTCI, respectively.In Section 3, R estimation based on GOS is obtained using the ML method.In Section 4, Bayesian inference is provided for this model by using the squared error-loss function.Section 5 obtains ML and Bayesian estimation for the specifc model of the Pareto distribution based on GOS.Te Monte Carlo simulation for comparing estimators and confdence intervals obtained by ML and Bayesian methods are presented in Section 6.Finally, in Section 7, these methods are applied to real data to demonstrate the application of the proposed methods.

The GCI, BPCI, and BTCI of R for Pareto Distribution
Te random variable X has a Pareto distribution with the shape parameter δ and scale parameter c when its cumulative distribution function (CDF) and probability density functions (PDFs) are as follows: We denote it by X ∼ Pareto(δ, c).To obtain R for Pareto distribution, let X ∼ Pareto(δ 1 , c 1 ) and Y ∼ Pareto(δ 2 , c 2 ) be independent.Tus, 2

Journal of Probability and Statistics
Te above relation can be restated as follows: and We construct GCI, BPCI, and BTCI for R of the Pareto distribution.

GCI.
Te GCI and generalized pivotal quantity (GPQ) were defned by [19].We propose a GPQ in the following theorem.
Proof.Te proof is similar to [9,20]. ( Here, W 1 , W 2 , Q 1 and Q 2 are independent.Our proposed GPQ for R is as follows: where We use Monte Carlo simulation to fnd GCI.Algorithm 1 is presented for this purpose.□ 2.2.BPCI and BTCI.One of the critical issues in statistical inference is the confdence interval for a parameter, which expresses the status of the parameter at a certain level of confdence.Usually, assuming the population distribution is normal, the z-standard and t-student confdence intervals for the mean population and the mean diference between two populations, the chi-square and Fisher confdence intervals for the variance, and the variance ratio of two populations are used.Nevertheless, the assumption of a normal society is not always established.Statistical studies have shown that when data are selected from a population with a skewed distribution or the sample size is small, the abovementioned confdence intervals do not have the required coverage accuracy.In search of ways to solve these problems, we can point to the bootstrap confdence intervals, which have a high coverage accuracy, and their efciency is further determined by the size of small samples.BPCI and BTCI are bootstrap confdence intervals [21].We obtain these two confdence intervals for R of the Pareto distribution with Algorithm 2.

Confdence Intervals
Where Proof.Te proof is similar to Teorem 1. W 1 ′ , W 2 ′ and Q ′ are independent.We suggest the following GPQ for R: where Similar to Algorithm 1, the GCI for R can be obtained for this case.BPCI and BTCI are obtained by Algorithm 3.
), where the ε th quantile of R 1 g , . . ., R N g is the same as R (ε) g .
(6) Obtain the 100 where  R ml is the ML estimation of R,

ML Estimation of R Based on GOS
Let G and g be continuous CDF and PDF, respectively.If the joint PDF of X (1,n,q,h) , . . ., X (n,n,q,h) are as follows: then X (1,n,q,h) , . . ., X (n,n,q,h) are GOS, where is the quantile function of G and and (q 1 , . . ., q n− 1 ) ∈ R n− 1 .Let X (1,n,q,h) , . . ., X (n,n,q,h) be GOS from Pareto(δ, c) and x be the observation vector.Te likelihood function is obtained as follows: Te log-likelihood function is where x i is the recorded value of the GOS sample and So, the ML estimator of c is  c � X (1,n,q,h) and taking the derivative of l relative to δ and putting it equal to zero where the ML estimator of δ is obtained by Now, we obtain the ML estimate of R. Let X (i,n,q,h) ∼ Pareto(δ 1 , c 1 ), i� 1, 2, . . ., n and Y (j,n ′ ,q ′ ,h ′ ) ∼ Pareto(δ 2 , c 2 ) be GOS such that X (i,n,q,h) and Y (j,n ′ ,q ′ ,h ′ ) are independent.According to invariance property of the ML estimator, the R estimate is given by where In the above equations, (1) Given  δ 0 ,  c 01 ,  c 02 , n, and (2) , and ′ /2(n+n ′ )  δ 0 are the estimations of the bootstrap sample for δ, c 1 , and c 2 ; (4) Perform steps 4, 5, and 6 in Algorithm 3. ALGORITHM 3: BPCI and BTCI for the special case δ 1 � δ 2 � δ.

Bayesian Estimation of R Based on GOS
We consider the prior distributions of parameters δ and c independently and propose their density as follows: Let x be the observation vector, then the joint posterior distribution of δ 1 and c 1 is We obtain the marginal posterior distribution of c 1 as follows: Also, the conditional posterior distribution of δ 1 is Similarly, let y be the observation vector.We obtain where ϵ 00 � min y (1) , 9 2  , We propose Algorithm 4 to obtain the Bayesian confdence interval.

Special Case δ
In this section, we obtain the ML and Bayesian estimators for R of Pareto distribution based on GOS for the special case δ 1 � δ 2 � δ.

Bayesian Estimation.
Consider the following prior distributions for parameters δ, c 1 , and c 2 : Te joint posterior distribution is where ϵ 0 � min x (1) , c 1  , ϵ 00 � min y (1) , c 2   and In this case, the R changes as follows: Based on the squared error loss function, the Bayesian
Journal of Probability and Statistics posterior distribution.By considering ϵ * � min ϵ 0 , ϵ 00  , we obtain where Integral 49 cannot be solved by analytical methods.To solve this integral, we propose a variable change method.Let where U ∼ U(0, 1) is from uniform distribution.We generate M samples form U(0, 1).Tus, under the strong law of large numbers, Similarly, we repeat the above steps for E[(c 1 /c 2 ) δ | x, y] as follows: where

Simulation
Te Monte Carlo simulation is used to compare GCI, BPCI, and BTCI of R for Pareto distribution and the specifc case of Pareto distribution.For this purpose, samples of Pareto distribution with diferent sample sizes (n, n ′ ) and diferent values of R � 0.148 ) and also for the special case, samples with diferent values of are generated.Te length (L) and coverage probability (CP) of these confdence intervals for Pareto distribution and its specifc case are summarized in Tables 1 and 2, respectively.Based on these two tables, the CPs of GCI are approximately equal to 0.95, the CPs of BPCI are less than 0.95, and the CPs of BTCI are greater than 0.95.For BPCI and BTCI, in most cases, with the increase of (n, n ′ ), the CPs approach 0.95.We can conclude GCI is better than BPCI and BTCI.Also, with increasing sample size (n, n ′ ), the L of all confdence intervals has decreased.We also compare the ML and Bayesian confdence intervals of R based on GOS for Pareto distribution and its specifc case.Consider (n, n ′ ) ∈ (10, 10), (10,15), (15,10), { (20, 20)} and the number of repetitions is 10,000.To generate a GOS sample, we perform the algorithm proposed in [22].Te random samples of Pareto distribution with parameters 8 Journal of Probability and Statistics Te Bayesian confdence interval of R is obtained by Algorithm 4. According to steps 1 and 2 of this algorithm, we need to generate samples from φ As can be seen from 31, 32, 36, and 37, these density functions do not have a simple form.To generate samples of these density functions, we propose the Algorithm 6. Te values of the hyperparameters are considered Finally,  R, L, and CP of R by using the ML and Bayesian methods for Pareto distribution are given in Table 3.
Te abovementioned steps are repeated with diferent values of parameters (δ, γ) and ξ 2 � 3, and the results are summarized in Table 4.For the special case of Pareto distribution, we produce a sample with parameters δ � 1, c 1 � 2, c 2 � 1(R � 0.75).We consider the hyperparameters ζ � 1, ϑ � 0, 9 1 � 5, 9 2 � 5, ξ 1 � 1 and ξ 2 � 1 for the Bayesian method and report the results in Table 5.As mentioned earlier, GOS includes many special cases.We consider three cases ordinary order statistics (OOS) with l � 1, q i � 0, η i � n − i + 1, frst n record values (FR) with l � 1, q i � − 1, η i � 1, and progressively type-II (PTII) with l � h + 1, q i � 0, η i � n + h − i + 1.From Tables 3-5, it can be concluded that for OOS, FR, and PTII, the CP values of the Bayesian method are almost equal to 0.95 but the ML method is far from 0.95, which in most cases approaches 0.95 with the increase of the sample sizes.Te L of confdence intervals decreases with increasing (n, n ′ ) in both methods for OOS, FR, and PTII.

Conclusion
Tis paper investigated classical and Bayesian stress-strength reliability estimators based on GOS for Pareto distribution for the frst time.It was proposed to calculate generalized confdence intervals and bootstrap Algorithms 1 and 2. Ten, the ML estimate of R was obtained based on GOS.To calculate the Bayesian confdence interval, Algorithm 4 was presented due to the complexity of the posterior distribution.In addition, classical and Bayesian inference was performed for a specifc case of this model (δ 1 � δ 2 � δ).In this case, for Bayesian estimation, we encountered a complex integral that could not be solved analytically and we proposed a change of variable method to solve this integral.In the simulation part, we considered three specifc GOS modes including OOS, FR, and PTII and concluded that the CP values of the Bayesian method are approximately equal to 0.95.As the sample size increases, the CP values of the ML method approach 0.95 and the L values decrease in all confdence intervals.
Case of Pareto Distribution.We consider X ∼ Pareto(δ, c 1 ) and Y ∼ Pareto(δ, c 2 ).So, we have Journal of Probability and Statistics