Reliability estimation of a consecutive k-out-of-n system for non-identical strength components with applications to wind speed data

ABSTRACT In the stress-strength reliability literature, although identical components assumption may not be realistic due to the structure of a system, independent and identically distributed components have been commonly used. In this study, we aim to contribute to the literature by studying of non-identical distributed components in a consecutive -out of- system when both stress and strength components follow the proportional hazard rate models. Estimation methods for the stress-strength reliability of this system are investigated from frequentist and Bayesian perspectives. Maximum likelihood and uniformly minimum variance unbiased estimations are obtained in the frequentist approach. Based on the suitability of the structure, approximate Bayes estimates methods: Lindley’s approximation, Markov Chain Monte Carlo through Metropolis-Hastings or Gibbs sampling algorithms and exact Bayes estimates are derived. Asymptotic confidence and highest posterior density credible intervals are also constructed for all cases. We provide comprehensive simulation experiments for investigating the performances of the considered estimates. Wind speed data from NASA’s satellite data source project are used in the application of the considered model and methods. We present the comparison of wind energy potentials of two districts on the Aegean coast of Turkey using our model structure after determining their wind speed distributions.


Introduction
Statistical inference of stress-strength reliability has been widely considered in the literature.In the simplest form, system reliability is defined as the probability of the strength component ðYÞ overcomes the stress ðXÞ imposed on it, namely, R ¼ PðY > XÞ.Birnbaum (Birnbaum, 1956) is the first to introduce this idea, and since then statistical inference of R has been continuing to be studied under different assumptions.For some recent studies about this topic, one can see Dong et al (Dong et al., 2013), Akgül and Senoğlu (Akgül & Şenoğlu, 2017), Ghanbari et al (Ghanbari et al., 2022), Asadi and Panahi (Asadi & Panahi, 2022), Tripathi et al (Tripathi et al., 2022) and references in there.
Extension of this model with two or more components was introduced by by Bhattacharyya and Johnson (Bhattacharyya & Johnson, 1974), and it was called the multicomponent stress-strength stress-strength reliability with non-identical strength components based on bathtub-shaped distribution under the adaptive Type-II hybrid progressive censoring samples.
The reliability of consecutive k-out of-n:G system in a stress-strength set-up was obtained by Eryilmaz and Demir (Eryilmaz & Demir, 2007) when the random strengths Y 1 ; . . .; Y n were independent and identical, and independent of the random stress X. Eryilmaz (Eryilmaz, 2008) obtained the stress-strength reliability of consecutive k-out of-n:G system when the strength components were independent and non-identical for a special case of k and n.
Let Y 1 ; . . .; Y n are independent random strengths with the first n 1 ones having a common cumulative distribution function (cdf) F ð1Þ and the remaining n 2 ¼ n À n 1 ones having another common cdf F ð2Þ , and independent of the random stress X having cdf F X .The stress-strength reliability of consecutive k-out-of-n:G system under these assumptions when 2k � n ¼ n 1 þ n 2 and n 1 is known was derived by Eryilmaz (Eryilmaz, 2008) as follows: where and In the aforementioned studies, the statistical inferences of multicomponent stress-strength reliability have paid attention when the strength components have non-identical distributions.However, the same problem for the stress-strength reliability of consecutive k-out of-n : G system has not been considered except for the next two studies.In our knowledge, the maximum likelihood (ML) and uniformly minimum variance unbiased (UMVU) estimates of the stress-strength reliability of a consecutive k-out of-n : G system was first studied by Eryilmaz (Eryilmaz, 2008) when all the components distribution was exponential, and the strength components were non-identical.Then, ML and UMVU estimates of the stress-strength reliability of a consecutive k-out of-n : G system with a change point in stress for the exponential distribution was studied by Akıcı (Akici, 2010).Since the distribution families have flexibility in modelling the lifetimes of components, these are extensively used in reliability analysis, see Marshall and Olkin (Marshall & Olkin, 2007) for detailed information.PHR family is one of the most commonly used families.That is why these motivates us to consider the more flexible lifetime distributions in this non-identical stress-strength reliability set-up.
Let X be a random variable from the PHR family of continuous distributions with cdf and corresponding probability density function (pdf) where F 0 ðx; λÞ ¼ 1 À F 0 ðx; λÞ is the survival function of the baseline random variable, α and λ are the positive interested parameters.It is denoted by X,PHRðα; λÞ.Some well-known distributions such as Burr Type XII, Gompertz, Kumaraswamy, Lomax, Pareto, Rayleigh, Weibull and so on belong to the PHR family.In survival analysis, this type of models are commonly known as Cox's proportional hazards model which was introduced by Lehmann (Lehmann, 1953).PHR family is also commonly studied in reliability analysis because of its flexibility and including well-known distributions.In some recent studies, Wang et al (B.X. Wang et al., 2019).considered PHR and also proportional reversed hazard rate families for statistical inference of simple stress-strength reliability.Hassanein and Seyam (Hassanein & Seyam, 2022) studied bivariate Rayleigh proportional hazard rate model for COVID-19 data set.That is why we consider PHR family in this study, so we can obtain and present more general results.
To the best of our knowledge, the classical and Bayesian estimates of the stress-strength reliability of a consecutive k-out of-n : G system with non-identical strength components has not been studied for the PHR family.Since the PHR family includes exponential distribution, some result of this study is a generalization of Eryilmaz (Eryilmaz, 2008) study from the statistical inference perspective.Since it includes series and parallel systems, and more complex systems than multicomponent system, our results can give some ideas and clues to who encounter this kind of systems from the perspective of the system structure.Another importance is the implementation of the obtained methods to NASA's POWER (Prediction Of Worldwide Energy Resource) data (for more detail see https://power.larc.nasa.gov/).For this aim, wind energy potentials of two locations are compared by using the considered reliability model results based on NASA's source data.From this point, this study will be the first study for using this kind of data in the reliability literature.
The rest of the paper is organized as follows.In Section 2, the ML and Bayes estimates for the stress-strength reliability of the considered system are derived when the second parameters of underlying distributions are common and unknown.Lindley's approximation and MCMC method with Metropolis-Hastings algorithm are used in the Bayesian part.The asymptotic confidence and highest posterior density (HPD) credible intervals are also constructed.In Section 3, all the estimates are studied when the second parameters of the underlying distributions are different and known.The UMVU estimate is also obtained.In Bayesian approach, exact Bayes and also two approximate Bayes estimates (Lindley's approximation and MCMC method with Gibbs algorithm) are derived.In addition, the asymptotic confidence and HPD credible intervals are developed.In Section 4, all the proposed estimates are compared via Monte Carlo simulations.In Section 5, an illustration of how the proposed model and methods may be used in real life is presented with the analysis of wind speed data from NASA's satellite data source project.Finally, comments and concluding remarks of the study are presented in Section 6.

Estimation of R n;k when the second parameters are common and unknown
In this section, the estimation methods of the stress-strength reliability of a consecutive k-out-of -n:G system with non-identical strengths are examined when the second parameters of underlying distributions λ 1 ; λ 2 and λ 3 are common and unknown.

MLE ofR n;k
Let the system consists of n strength variables such that first n 1 ones Y ð1Þ j 1 ,PHRðα 1 ; λÞ; j 1 ¼ 1; ::; n 1 , the remaining n 2 ¼ n À n 1 ones Y ð2Þ j 2 ,PHRðα 2 ; λÞ; j 2 ¼ n 1 þ 1; ::; n; and stress variable X, PHRðβ; λÞ. Then, gða; b; c;  ð Þ are observed when m systems are put on a life-testing experiment to obtain the estimates of R n;k .Then, the likelihood function of the observed sample is and the log-likelihood function can be written as where The ML estimates of α 1 , α 2 and β are given by The MLE of λ, say λ, is the solution of the following non-linear equation where @f 0 ðx; λÞ=@λ;f 0 λ ðx; λÞ and @F 0 ðx; λÞ=@λ;F 0λ ðx; λÞ.Since λ has not a closed-form, we need to apply numerical methods such as Newton-Raphson or quasi-Newton method for it.We use the limited-memory modification of the BFGS, namely, L-BFGS-B, quasi-Newton method (see (Byrd et al., 1995)) to obtain the λ in our simulations by optim function in the statistical software R.Then, the ML estimates of α 1 ; α 2 and β are evaluated using λ from Equation (7).Thus, the MLE of R n;k , say RMLE n;k , is computed using the invariance property of MLE from Equations ( 1) and (6).

Asymptotic distribution and confidence interval forR n;k
The observed information matrix of θ ¼ ðα 1 ; α 2 ; β; λÞ is given by Then, the elements of the observed information matrix are obtained as The expectation of these elements cannot be always obtained analytically since they depend on the baseline pdf and cdf.Based on the considered baseline distribution, the Fisher information matrix of θ, i.e.IðθÞ ¼ EðJðθÞÞ can be computed numerically or observed information matrix can be used as a consistent estimator of IðθÞ.Hence, the observed information matrix JðθÞ is used in the asymptotic normality of the MLE when the Fisher information matrix is not derived analytically.
The MLE of R n;k is asymptotically normal with mean R n;k and asymptotic variance where I À 1 ij is the ði; jÞth element of IðθÞ (see Rao (Rao, 1965)).Then, we have Notice that IðθÞ is replaced by JðθÞ when IðθÞ is not obtained.Therefore, an asymptotic 100ð1 À γÞ% confidence interval of R n;k is given by R n;k 2 ð RMLE n;k � z γ=2 σR n;k Þ where z γ=2 is the upper γ=2th quantile of the standard normal distribution and σR n;k is the value at MLE of the parameters.

Bayes Estimation of R n;k
In this subsection, approximate Bayes estimates of R n;k are obtained when all the parameters α 1 , α 2 , β and λ have statistically independent gamma distributions with parameters ða i ; b i Þ, i ¼ 1; 2; 3; 4, respectively, as prior distributions.If X be a gamma random variable with parameters ða; bÞ, then its pdf is where x > 0; a; b > 0.Then, the joint posterior density function of α 1 , α 2 ; β and λ is where Iðx; y 1 ; y 2 Þ is a normalizing constant and given by Hence, Bayes estimate of R n;k is given as follows under the squared error (SE) loss function.Two approximation methods Lindley's approximation and MCMC method are applied to obtain Bayes estimate of R n;k due to the multiple integrals in Equation ( 9) are not computed analytically and difficulties in numerical computations of these integrals.

Lindley's approximation
An approximate method for the computation of the ratio of two integrals was introduced by Lindley (Lindley, 1980).Since then this method has been used for obtaining an approximation of the posterior expectation of the function uðθÞ for a given x is where QðθÞ ¼ lðθÞ þ ρðθÞ, lðθÞ is the logarithm of the likelihood function and ρðθÞ is the logarithm of the prior density of θ.Using Lindley's method, EðuðθÞ x j Þ is approximately obtained as follows: where l=@θ i @θ j ; L ijk ¼ @ 3 l=@θ i @θ j @θ k , ρ j ¼ @ρ=@θ j , and σ ij ¼ ði; jÞth element in the inverse of the matrix À L ij � � all evaluated at the MLE of the parameters.We can refer to Lindley (Lindley, 1980) for more details about this method.
In our case, we have four unknown parameters as θ ¼ ðα 1 ; α 2 ; β; λÞ; and so Lindley's approximation leads to evaluated at θ ¼ ðα 1 ; α2 ; β; λÞ where u ¼ uðθÞ Þ À b 4 ; and σ ij ; i ¼ 1; 2; 3; 4 terms are obtained using L ij ; i; j ¼ 1; 2; 3; 4 where where Furthermore, the derivatives of u ¼ R n;k in Equation (1) with respect to λ are zero, and others are not given here for the sake of brevity.Hence, the constants of RLin n;k;B in Equation (10) are obtained as

Estimation of R n;k when the second parameters are different and known
In this section, the estimation methods of the stress-strength reliability of a consecutive k-out-of -n:G system with non-identical strengths are examined when the second parameters of underlying distributions λ 1 ; λ 2 and λ 3 are different and known.

MLE ofR n;k
Let the system consists of n strength variables such that first n 1 ones Y ð1Þ j 1 ,PHRðα 1 ; λ 1 Þ; j 1 ¼ 1; . . .; n 1 the remaining n 2 ones Y ð2Þ j 2 ,PHRðα 2 ; λ 2 Þ; j 2 ¼ n 1 þ 1; ::; n, stress variable X,PHRðβ; λ 3 Þ and ðλ 1 ; λ 2 ; λ 3 Þ are known constants.Then, the stress-strength reliability R n;k is computed by using the same Equations in (1) and ( 6).The likelihood function of the observed sample for this case is given by and the log-likelihood function is where t ð1Þ ;t ð1Þ The ML estimates of α 1 ; α 2 and β are easily obtained as α1 ¼ n 1 m=t ð1Þ ; α2 ¼ n 2 m=t ð2Þ and b β ¼ m=t.Hence, the MLE of R n;k is calculated by using the invariance property of MLE and Equations ( 1) and ( 6).
An asymptotic confidence interval of R n;k is obtained using the Fisher information matrix.In this case, θ ¼ ðα 1 ; α 2 ; βÞ and the elements of the Fisher information matrix are Hence, MLE of R n;k is asymptotically normal with mean R n;k and asymptotic variance (see Rao (Rao, 1965)) Thus, the asymptotic 100ð1 À γÞ% confidence interval of R n;k is given by R n;k 2 ð RMLE n;k � z γ=2 σR n;k Þ where z γ=2 is the upper γ=2th quantile of the standard normal distribution and σR n;k is the value at MLE of parameters.

Bayes estimation ofR n;k
In this subsection, the exact and approximate Bayes estimates of R n;k are obtained when the parameters α 1 , α 2 and β follow statistically independent gamma distributions with parameters ða i ; b i Þ, i ¼ 1; 2; 3, respectively.The joint posterior density function of α 1 ; α 2 and β is The marginal posterior densities of α 1 ; α 2 and β have gamma distributions with parameters . More details about the derivation of Q 1 ; Q 2 and Q 3 integrals are given in Appendix.
Therefore, b R n;k;B is computed analytically using Q 1 ; Q 2 and Q 3 integrals in Equation ( 14).

Lindley's approximation
In this case, we have three unknown parameters θ ¼ ðα 1 ; α 2 ; βÞ, then the Lindley's approximation of R n;k is derived as where uðα 1 ; α

MCMC method
In this case, Bayes estimate of R n;k is computed using Gibbs sampling algorithm since the marginal posterior densities of α 1 ; α 2 and β are known distributions.The following algorithm is used for the generation of random samples of these marginal posterior densities.
Step 2: Generate α Step 7: Repeat Steps 2 through 6, N times and obtain the posterior sample R ðiÞ n;k ; i ¼ 1; . . .; N.Then, the Bayes estimate of R n;k under the SE loss function is computed by where M is the burn-in period.The HPD 100ð1 À γÞ% credible interval of R n;k is obtained by the method of Chen and Shao (Chen & Shao, 1999).

Example
Kumaraswamy and Burr Type XII distributions from the PHR family are considered as examples in the case of λ unknown, and Weibull distribution is considered for the λ known case.In this context, the following derivations are obtained for each case.
When λ is unknown, Kumaraswamy distributions with parameters ðα 1 ; λÞ and ðα 2 ; λÞ are used to generate strength variables, and stress variables are generated from Kumaraswamy distribution with parameters ðβ; λÞ.Then, the survival functions are F 0 ðy ð1Þ and λ is the solution of the following nonlinear equation Some elements of observed information matrix are derived as Some elements for Bayes estimation of R n;k using Lindley's approximation are obtained as The marginal posterior density of λ is derived as for implementation of MCMC method.
When it is assumed that strength and stress variables are generated from Burr Type XII distribution with parameters ðα 1 ; λÞ, ðα 2 ; λÞ and ðβ; λÞ, then the baseline survival functions are , respectively.We have and Some elements of observed information matrix are derived as Some elements for Bayes estimation of R n;k using Lindley's approximation are obtained as The marginal posterior density of λ for this case is obtained as for implementation of MCMC method.

Simulation study
In this section, a Monte Carlo simulation study is performed for the comparison of the derived estimates of R n;k for both cases: i. the second parameters of underlying distributions are common and unknown; ii. the second parameters of underlying distributions are different and known.
In the point estimation procedure, mean square error (MSE), estimated risks (ER) and biases are used for the comparisons in ML and Bayes estimates, respectively.When θ is estimated by θ, the bias of θ estimate is given by and the ER of Bayes estimate θ under the SE loss function or MSE of θ are given by The performances of 95% asymptotic confidence and credible intervals are considered by their average lengths (ALs) and coverage probabilities (CPs).All simulations study results are obtained by using the statistical software R (R Core Team, 2021) based on 2500 replications.Point and interval estimates of R n;k are given under different scenarios in tables and figures.In the estimates tables, the first three rows represent the estimate of R n;k , its bias and MSE (or ER) values, respectively.
In the simulations, the log-likelihood function is maximized by solving the non-linear equation system with respect to the unknown parameters.It is solved by using L-BFGS-B quasi-Newton method with optim function in R. In each replication of this optimization process, the initial value of any unknown parameter (for instance θ) is picked randomly from the interval ð0:5θ; 1:5θÞ.Then, the solution of the non-linear equation system is obtained based on these initial values, and the convergence of the system is observed by the convergence value of optim function result.It is possible to encounter non-convergent results in a few replications.In this case, we update the initial values a couple of times for obtaining the convergent result and also increase the maximum number of iterations in the algorithm.After this, we rarely see non-convergent optimization results (in our experience this is generally less than 5 in total 2500 replications), and we discard this replication from our result.

When the second parameters are unknown
In this subsection, Monte Carlo simulation study is performed for the comparison of ML and Bayes estimates of R n;k when the second parameters of underlying distributions are common ðλÞ and unknown.Numerical results based on point and interval estimation are presented for Kumaraswamy and Burr Type XII distributions from the PHR family with different sample sizes m ¼ 10; 20; 30; 40.The true values of the parameters are taken as ðα 1 ; α 2 ; β; λÞ ¼ ð0:75; 1:5; 8; 2:5Þ and ð1:5; 3; 15; 5Þ, respectively.
Point estimates of R n;k and corresponding interval estimates are presented in Tables 1-4, for Kumaraswamy and Burr Type XII distributions.The consecutive k-out of-n : G system is considered for the different combinations of ðk; n 1 ; n 2 Þ under the condition of 2k � n.Bayes estimates are obtained by using both Lindley's approximation and MCMC method based on the following informative and non-informative priors: Prior 1: The informative priors parameters are chosen such that their mean is equal to the true parameter values.Since the non-informative prior, i.e.ða i ; b i Þ ¼ ð0; 0Þ; i ¼ 1; 2; 3; 4 provides prior distributions are not proper, we adopt the suggestion of Congdon (Congdon, 2001), and we choose the non-informative priors as ða i ; b i Þ ¼ ð0:0001; 0:0001Þ; i ¼ 1; 2; 3; 4: In the MCMC case, two MCMC chains and 10,000 iterations for each one are generated.In each chain, we discard the first 5000 results as burn-in period which reduces the effect of the starting distribution.Then, we obtain MCMC Bayes estimate using every 5 th sampled values in chains in thinning procedure.Moreover, the convergence of the MCMC chains has been monitored by using the Gelman and Rubin's convergence diagnostic in Gelman and Rubin (Gelman & Rubin, 1992), We obtain its average value are found to be below 1.1 in all cases.It is an acceptable value for the convergence MCMC chains.
From Tables 1 and 3, it is observed that MSE, ERs and biases of all estimates generally decrease when the sample size increases which shows the consistency of estimators.Also, the proposed Bayes estimators based on informative priors have more effective performance than corresponding ML estimates.In Table 1 Bayes estimate using Lindley's approximation outperforms the MCMC results in terms of ERs except for m ¼ 10 under informative priors while these estimators show similar performances under non-informative priors.In Table 3, Bayes estimate using MCMC method outperforms Lindley's approximation in terms of ERs except for m ¼ 40 under informative priors, while these estimators show similar results under non-informative priors.In general, ERs of Bayes estimates based on informative priors are closing each other as the sample size increases.Bayes estimates based on non-informative priors have similar performance in terms of ER and MSEs when compared to ML estimate results.Also, these estimates and their error values are getting closer to each other as the sample size increases.
From Tables 2 and 4, the ALs of all intervals decrease with the increase in sample sizes, as expected, and the CPs of all intervals are satisfactory.The HPD credible intervals based on informative priors have the smallest AL while HPD credible intervals based on non-informative priors are similar to the asymptotic confidence intervals.Therefore, the HPD credible intervals may be preferred if the prior information is available or not.
We also compare the performances of Bayes estimates using Lindley's approximation (based on informative and non-informative priors) and MLE by plots.Since we observe similar performance results for these point estimates, these figures are given in the Supplemental Materials Figures 1  and 2.

When the second parameters are known
In this subsection, Monte Carlo simulation study is performed for the comparison of ML, UMVU and Bayes estimates of R n;k when the second parameters parameters of underlying distributions λ 1 ; λ 2 ; λ 3 ð Þ are known.Point and interval estimation results are presented for one-parameter Weibull distribution with different sample sizes.
From Table 5, MSE, ERs and biases decrease with the increase in sample sizes, as expected.It is observed that ML and UMVU estimates show similar results, and they are getting close to each other as the sample size increases.Bayes estimators based on informative priors show better performance than these classical estimates while Bayes estimates based on non-informative priors give similar results to them.Except only one case, Bayes estimates using Lindley's approximation based on informative priors show the best performance in terms of error values.Moreover, it is also observed that approximate Bayes estimates and their corresponding ER values are generally close to the exact Bayes estimate values.From Table 6, the ALs of all intervals decrease with the increase in sample sizes, as expected, and the coverage probabilities of all intervals are quite satisfactory.HPD credible intervals of R n;k based on informative prior have the smallest AL while the asymptotic confidence intervals of R n;k generally have the largest AL.
Moreover, we also prepared more tables like Tables 5 and 6 for the different parameters.Since the same results are observed, these tables are given in the Supplemental Materials Table 1-3.In our simulation studies, we encounter some difficulties in the evaluation of the exact Bayes estimate of R n;k for the large values of m; n and k.It is seen that the integral in the exact Bayes estimate of R n;k can create some problems for some values of constants in there.We observed that performance of two approximate Bayes estimates: Lindley's approximation and MCMC algorithm with Gibbs sampling have almost same with the exact Bayes estimate in the tables.Therefore, we listed ML, UMVU and two approximate Bayes estimates results in Table 7 for large values of m; n and k.Moreover, the asymptotic confidence and HPD intervals of R n;k for these estimates are given in Table 8.In these tables, the true values of the parameters are taken as ðα 1 ; α 2 ; βÞ ¼ ð0:75; 2; 20Þ, λ 1 ; λ 2 ; λ 3 ð Þ ¼ ð6; 9; 3Þ and Prior 4: ða 1 ; b 1 Þ ¼ ð0:75; 1Þ; ða 2 ; b 2 Þ ¼ ð2; 1Þ; ða 3 ; b 3 Þ ¼ ð20; 1Þ is used as an informative prior in Bayesian case.
From Table 7, it is observed that estimates of R n;k provide similar performances as in the Table 5.Also, intervals results in Table 8 show similar performances as in the Table 6.In addition, more tables for the different parameters are also given in Supplemental Materials Tables 4-6.
Step 2: For given m; samples from Weibull distribution are generated for the strength and the stress variables.
Step 4: Steps 2-3 are repeated N ¼ 2500 times, the MSE or ER for estimates of R 14;k7 are calculated as by using 1, ML, UMVU and Bayes estimates (based on informative and non-informative prior) have greater errors when R n;k is around 0:5 and have smaller errors when R n;k is close to the extreme values.Bayes estimate based on informative prior has the smallest error while Bayes    estimate based on non-informative prior, ML and UMVU estimates have similar results.Hence, it can be concluded that the results of Tables 5 and 7 are compatible with the figure.

Real data analysis
Wind energy is an important renewable energy source.It is used as a significant renewable energy source in electricity production in some countries.However, usage and modelling of renewable energy sources have been paying attention to the governments as well as researchers when we consider the problems in the consumption of fossil fuels such as global warming and climate change.
Getting reliable information about the wind capacity of a candidate region is vital for planning the installation of wind power plants.In wind energy studies, two-parameter Weibull and Rayleigh distributions are most commonly used to model the wind speed distributions, see Ozay and Celiktas (Ozay & Celiktas, 2016), Akgül and Senoğlu (Akgül & Şenoğlu, 2019) and their references.Some important research questions for the installation of the wind turbines can be answered by using the wind speed distribution of the area before the installation of the wind turbines, see Eryilmaz and Devrim (Eryilmaz & Devrim, 2019).That is why a comparison of the wind energy potentials of the candidate locations are potentially useful to make decisions.
In this section, we use the NASA's POWER (Prediction Of Worldwide Energy Resource) data source.The POWER project aims to improve current renewable energy data set and to create new data sets from new satellite systems.We consider a kind of comparison of the wind speed data of two districts on the Aegean coast of Turkey for an illustration of our stress-strength model.To our    knowledge, this is the first study considering wind speed data from NASA's POWER source for the stress-strength reliability analysis.We use NASA's POWER satellite data for Fethiye and Datça stations which are on the Aegean coast of Turkey.Since this region has high wind energy potential, it will be valuable for doing prestudy about their wind capacities using only satellite data without any investment.These stations geographical information are the same as in Akgül and Senoğlu (Akgül & Şenoğlu, 2019) study and locations are given in Figure 2. Data from NASA's POWER source can be taken from directly by using its data access viewer (https://power.larc.nasa.gov/data-access-viewer/)or using the nasapower (see Sparks (Sparks, 2018(Sparks, , 2021))) package in R.
In this study, we use the wind speed observations (m/s) at the 10 m height on an hourly basis in February 2019.It is assumed that Fethiye stations data from 7 am to 8 pm for each day of February 2019 are considered the first type of strength data Y 1 ; and data from 8 pm to 7 am are considered the second type of strength data Y 2 .The daily average wind speed data of Datça station is considered as stress data X.Then, we have n 1 ¼ 13, n 2 ¼ 11, m ¼ 28 for our model.
In this structure, we can construct the following scenario for comparison of the wind energy potential of Fethiye and Datça districts.Since the hourly data is used for each day, we have the consecutive k-out-of-24 : G system based on these stress and strength data sets, i.e. system reliability represents whether the Fethiye district could be a suitable location for the optimal installation of the wind power plant when compared to Datça or not.Hence, wind energy investors can make decisions which district is more suitable for the installation of wind power plants.fortheir investments.If the stress-strength reliability of the consecutive k-out-of-24 : G system for k ¼ 12; 13; 14; 15 is greater than 0:50, they will consider that the wind energy potential of Fethiye can be more desirable with regard to Datça for a detailed feasibility study.Moreover, if the system reliability is less than 0:50, they will think of the reverse result of the aforementioned scenario.
Descriptive statistics for the interested wind speed data of Y 1 ; Y 2 and X are given in Table 9.We check whether stress data set X and strength data sets Y 1 and Y 2 come from the interested underlying distribution of PHR family or not.Kolmogorov-Smirnov (K-S), Anderson-Darling (A-D) and Cramer-von Mises (C-VM) tests are applied for the goodness-of-fit by using the stats package in R. The test statistics and corresponding p À values (given in brackets) are computed based on the MLEs of the unknown parameters and presented in Table 10.Moreover, the root mean square (RMSE) and coefficient of determination (R 2 ) measures are used to determine which Table 5.Estimates of R n;k for Weibull distribution when ðα 1 ; α 2 ; βÞ ¼ ð2; 0:75; 12Þ and.λ 1 ; λ 2 ; λ 3 ð Þ ¼ ð4; 2; 8Þ.
Bayes (Prior 3) Bayes (Non-inf.prior)  where b F i is the estimated cdf for the ith ordered observation, b , n is the sample size.It is known that lower values of RMSE indicate better fit while the higher of R 2 demonstrate better modelling.These measures are commonly used in the wind speed data analysis see Quarda et al (Quarda et al., 2016).Furthermore, fitting performance of some distributions from the PHR family are visualized with the histograms of data and fitted density plots in Figure 3. From Table 10, it is observed that two-parameter Weibull distribution provides a good fit than other considered distributions for all data sets.
We also consider some diagnostic plots for showing the convergence of the simulated Markov chains using diagMCMC function in CalvinBayes package (Pruim, 2021) in R software.We present the trace plot which is a plot of the iteration number i against the value of the R ðiÞ 24;k at each iteration, autocorrelation plot of MCMC chains, Gelman and Rubin's diagnostic plot which shows the pattern of Gelman and Rubin's shrink factor as the number of iterations increases, and the density plot of the posterior distribution of R 24;k .Weibull and Burr Type XII distributions cases for k ¼ 15 with using Prior 7 are given in Figures 4 and 5.In these figures, two MCMC chains with 10,000 iterations for each one are generated, and the first 5000 are discarded as burn-in, then every 5 th sample values are used for the taking average.ESS and MCSE are the effective sample size and Monte Carlo standard error, respectively.From Figures 4 and 5, it is observed that trace plots fluctuate around its center with a similar variation, the autocorrelation plots show these chains achieve stationarity, the shrink factor values are smaller than the general acceptable top value 1.1 and the density plots are symmetric and unimodal.Hence, we conclude that these two MCMC chains are converged for each figure.Moreover, the overall convergence of MCMC chains is monitored and satisfactory in all cases of the real data, and these are omitted for the sake of brevity.
It is concluded that Bayes estimates based on non-informative prior and the MLE of R 24;k are very close to each other as in the simulation results.Moreover, both approximate Bayes estimates of R n;k are very similar.Bayes estimates of R 24;k based on Prior 7 are greater than that of other estimates, and the corresponding HPD credible interval has the largest length with respect to other intervals.It is observed that since all the estimates of R n;k for k ¼ 12; 13; 14; 15 values are less than 0:50, Datça district should be paid attention to for more investigations of wind energy power plant investment based on the considered scenario.

Conclusions
In this paper, we studied the stress-strength reliability estimation of a consecutive k-out of-n system with non-identical strength components.We derived all the point and interval estimations of this reliability assuming both stress and strength components follow the proportional hazard rate model.Frequentist estimation methods and Bayesian approach with two approximate methods Lindley's approximation and MCMC method were utilized when the second parameters of underlying distributions are unknown and known.
We performed a Monte Carlo simulation study to compare the performance of proposed estimators.From the results of the simulation study, we concluded that the performances of Bayes estimators based on informative priors were better than other estimators while ML and Bayes estimators based on non-informative priors show similar performances.The implementation of the proposed model was made by using the wind speed data sets from NASA's POWER project source for comparison of two locations' wind energy potentials.This study has novelty with regard to the considered system structure which includes some sub-models as well as real data application of the obtained estimates.
Finally, the considered problem in this study can be extended in different ways such as by changing the underlying distributions or data types, assuming two or more change points in strength and/or stress variables.Moreover, it can be considered to create an R package for easily apply of our and some related results.These will be among our future research problems.

Table 1 .
Estimates of R n;k for Kumaraswamy distribution when α

Table 3 .
Estimates of R n;k for Burr Type XII distribution when.ðα

Table 9 .
Descriptive statistics of wind speed data (m/s).The histograms and fitted densities for the wind speed data.

Table 10 .
ML estimates of the parameters, goodness-of-fit test, RMSE and R 2 values for Y 1 ;Y 2 and X data sets.

Table 11 .
Estimates of R 24;k for different distributions.