Inference on the stress strength reliability with exponentiated generalized Marshall Olkin-G distribution

Neama Salah Youssef Temraz

doi:10.1371/journal.pone.0280183

Abstract

In this paper, an inference on stress-strength reliability model is introduced in case of the exponentiated generalized Marshall Olkin G family of distributions. The maximum likelihood estimator of the stress-strength reliability function is deduced. An asymptotic confidence and bootstrap confidence intervals for the stress-strength reliability function are derived. A Bayesian inference is introduced for the stress-strength reliability. A simulation is introduced to obtain the maximum likelihood and Bayesian estimates for the stress strength reliability. Real data applications are provided to show the results for the stress-strength model and compare the exponentiated generalized Marshall Olkin-G distribution with other distributions.

Citation: Salah Youssef Temraz N (2023) Inference on the stress strength reliability with exponentiated generalized Marshall Olkin-G distribution. PLoS ONE 18(8): e0280183. https://doi.org/10.1371/journal.pone.0280183

Editor: Oluwafemi Samson Balogun, University of Eastern Finland, FINLAND

Received: October 10, 2022; Accepted: April 20, 2023; Published: August 9, 2023

Copyright: © 2023 Neama Salah Youssef Temraz. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper.

Funding: This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project (PSAU-2022/01/19479).

Competing interests: The authors have declared that no competing interests exist.

1. Introduction

In life testing, it is very important to study the stress-strength reliability which refers to the quantity R = P (Y < X) and it is assumed that two independent random variables X and Y where X represents the stress of a component and Y represents the strength of the component such that the component fails in case that the stress exceeds the strength.

Al-Mutairi et. al [1] presented an inference on stress-strength reliability from Lindley distribution. Rao et. al [2] introduced an estimation of stress–strength reliability from inverse Rayleigh distribution. Singh et al. [3] presented an estimation of the stress strength reliability subject to the inverted exponential distribution. Al-Mutairi [4] discussed an inference on stress-strength reliability from weighted Lindley distributions. Sharma [5] presented a stress-strength reliability model subject to the inverse Lindley distribution with application to head and neck cancer data.

Mokhlis et al. [6] proposed the stress-strength reliability with general form distributions. Çetinkaya and Genç [7] introduced the stress–strength reliability estimation under the standard two-sided power distribution. Almarashi [8] introduced an estimation of the stress-strength reliability for Weibull distribution with application. Muhammad et al. [9] proposed an estimation of the reliability of a stress–strength system from Poisson half logistic distribution. Nojosa and Rathie [10] introduced Stress–strength reliability estimation involving generalized gamma and Weibull distributions.

Alamri et al. [11] studied the stress-strength reliability where the strength (X) follows Rayleigh-half-normal distribution and stress follows Rayleigh-half-normal distribution, exponential distribution, Rayleigh distribution, and half-normal distribution, respectively. Hassan et al. [12] introduced stress–strength reliability for the generalized inverted exponential distribution using median ranked set sampling. Abu El Azm et al. [13] presented a study for stress-strength reliability subject to exponentiated inverted Weibull distribution with application on breaking of jute fiber and carbon fibers. Jha et al. [14] discussed the multicomponent stress-strength reliability estimation based on unit generalized Rayleigh distribution.

Jha et al. [15] proposed the multicomponent stress-strength reliability estimation based on unit generalized exponential distribution. Maurya et al. [16] introduced a reliability estimation in a multicomponent stress-strength model based on inverse Weibull distribution. Jovanovic et al. [17] proposed an inference on reliability of stress-strength model with Peng-Yan extended Weibull distributions. Sabry et al. [18] presented a Monte Carlo simulation of the stress-strength model and reliability estimation for extension of the exponential distribution. Zarei and Shahrestani [19] proposed the Bayes and empirical Bayes estimator of reliability function in multicomponent stress-strength system based on generalized Rayleigh distribution.

The exponentiated Marshall Olkin family distribution is used in practice since it makes the kurtosis of data more flexible compared to the baseline model and is used to construct heavy-tailed distributions for modeling real data. Special models can be constructed from the exponentiated Marshall Olkin family distribution with all types of the hazard rate functions. The exponentiated Marshall Olkin family distribution is used to provide consistently better fits than other generated models under the same baseline distribution. The Marshall Olkin family distribution is introduced and discussed in many papers in literature such as [20–26].

This paper deals with the inference of the stress-strength reliability in case of the stress and strength variables follow the exponentiated generalized Marshall Olkin G family of distributions. The maximum likelihood estimation of the stress-strength reliability is discussed. An asymptotic confidence and bootstrap confidence intervals for the stress-strength reliability function are discussed. Bayesian inference and the credible interval for the stress-strength reliability function are introduced. A simulation is carried out to obtain the results for the stress strength reliability in case of the exponentiated generalized Marshall Olkin Weibull distribution. Real data applications are introduced to study the goodness of fit of the real datasets.

2. Exponentiated generalized Marshall Olkin-G distribution

Handique et al. [20] introduced the exponentiated generalized Marshall Olkin G family of distributions (EGMO-G) with cumulative distribution function and probability density function defined as follows and where G(x; λ_k) and g(x; λ_k) are the cumulative distribution function and the probability density function of a random variable X, respectively, with parameter vector λ_k, k = 1,2,…,l.

For example, the cumulative distribution function and the probability density function of a random variable X that follows the exponentiated generalized Marshall Olkin Weibull distribution (EGMO-W) are given by and where a, b, ϑ and γ are the shape parameters and β is the scale parameter.

And the cumulative distribution function and the probability density function of a random variable X that follows the exponentiated generalized Marshall Olkin exponential distribution (EGMO-E) are given by and where a, b, ϑ and γ are the shape parameters and λ is the scale parameter.

3. Stress-strength reliability

Let X and Y are two independent random variables follow the EGMO-G distribution, then the stress-strength reliability function will be given by

Solving this integral, an expression for the stress-strength reliability is obtained as (1)

4. Maximum likelihood estimation

Assuming that the two independent random samples (X₁, X₂,…,X_n) and (Y₁, Y₂,…,Y_m) are selected from the EGMO-G distribution with the parameters (b₁, a, ϑ, λ_k) and (b₂, a, ϑ, λ_k), respectively. The likelihood function is given as follows (2)

The log-likelihood function is obtained as follows

The partial derivatives of the log-likelihood function with respect to (b₁, b₂, ϑ, a, λ_k) are obtained as follows (3) (4) (5) (6) (7)

Equating the partial derivatives to zero and then solving the equations numerically yields the maximum likelihood estimators for the parameters (b₁, b₂, ϑ, a, λ_k).

The maximum likelihood estimators for the parameters b₁ and b₂ are obtained as (8) (9)

The maximum likelihood estimates of the stress-strength reliability is obtained by substituting in Eq (1).

(10)

The partial derivatives with respect to the unknown parameters cannot be solved directly, so a simulation depends on methods like the Newton-Raphson method will be used to obtain the estimates of the unknown parameters and hence calculate the estimate of the stress-strength reliability.

5. Asymptotic Confidence Interval (A.C.I)

The observed Fisher information matrix of the stress strength reliability parameters is given by

The elements of this matrix are obtained by differentiating the Eqs from (3) to (7) with respect to (b₁, b₂, ϑ, a, λ_k), respectively, the results are obtained as

The asymptotic variances of the parameters (b₁, b₂, ϑ, a, λ_k) are given by

The asymptotic variance of an estimate is given by

and the asymptotic 100(1−δ)% confidence interval of R is given by

The asymptotic confidence interval is used for large samples and do not perform well for small samples. So, the bootstrap confidence interval method will be proposed.

6. Bootstrap Confidence Interval (B.C.I.)

The bootstrap method for constructing confidence intervals is illustrated by the following algorithm.

Step 1: Generate independent random samples x₁, x₂,…,x_n from EGMO-G(b₁, ϑ, a, λ_k) and y₁, y₂,…,y_m from EGMO-G(b₂, ϑ, a, λ_k)

Step 2: Generate new samples and taken by sampling with replacement from the samples in step (1) and compute the MLE estimate of stress strength reliability by using Eq (9).

Step 3: Repeat step 2, S times to obtain the estimates .

Step 4: Rearrange the estimates obtained in step (3) such that . The 100(1−δ)% bootstrap confidence interval is given by

7. Bayesian estimation

Bayesian estimation method will be discussed into cases. The first case when the parameters are unknown and the second case when the parameters are known.

Case I: When the parameters are unknown

The Bayesian estimator for the stress-strength reliability will be obtained assuming that the parameters b₁, b₂, ϑ, a and λ_k are independent random variables with prior follow gamma distribution as follows

The joint posterior density function of b₁, b₂, ϑ, a and λ_k given the data (x, y) is given by where

The marginal posterior distributions of b₁, b₂, ϑ, a and λ_k can be deduced as

It is obvious that seen that posterior samples for b₁, b₂, ϑ and a can be generated using gamma distribution. However, λ_k cannot be directly simulated from its posterior distribution as it is not in known form and in this case the Metropolis-Hastings algorithm can be applied to simulate random samples from the posterior density of λ_k.

The Bayesian estimator of the reliability function R under the squared error loss function using the posterior mean is given by

This integral has no analytical solution and the Markov Chain Monte Carlo (MCMC) simulation method can be applied to obtain the Bayesian estimation of the stress-strength reliability. In the method of the Markov chain Monte Carlo, samples are generated from the posterior density function and in turn to compute the Bayes estimates of the reliability function.

7.1 MCMC method

Markov chain Monte Carlo (MCMC) method comprises a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis–Hastings algorithm. The steps of applying MCMC method and the Metropolis–Hastings algorithm are presented as follows.

Step1: Choose initial values

Step 2: Set t =1

Step 3: Generate from Gamma(n+η₁, H₁)

Step 4: Generate from Gamma(m+η₂, H₂)

Step 5: Generate ϑ^(t) from Gamma(a(n+m)+η₃, ξ₃)

Step 6: Generate a^(t) from Gamma(n+m+η₄,H₃)

Step 7: Generate from using MH algorithm as the following

Generate proposals from .
Evaluate the acceptance probabilities
Generate u_k from Uniform(0, 1).
If u_k<z_k, accept the proposal and set , else set

Step 8: Compute R^(t) from Eq (9)

Step 9: Set t = t+1

Step 10: Repeat steps from 3 to 9, T times.

Step 11: Stop for sufficiently large value of T, the Bayes estimate of the stress-strength reliability function under the squared error loss will be given as

Step 12: To construct the credible interval for R, order R^(t) as R⁽¹⁾<R⁽²⁾<⋯<R^(T). Then a 100(1−ε)% credible interval of R becomes

Case II: When the parameters are known.

In this case the Bayes estimator of the stress-strength reliability under the squared error loss will be given by

Let where 0<w₁<1, w₂>0. Hence, the Bayes estimator of the stress-strength reliability will be given by

Where

The Bayes estimator of the stress strength reliability is deduced as

In some situations, it is difficult to find the stress strength reliability from the previous relations and in this case, the MCMC method can be used to find an estimate for the stress strength reliability based on simulated random samples and the steps are given as follows.

Step1: Choose initial values

Step 2: Set t =1

Step 3: Generate b₁ from Gamma(n+η₁, H₁)

Step 4: Generate b₂ from Gamma(m+η₂, H₂)

Step 5: Compute R^t from Eq (9)

Step 6: Set t = t+1

Step 7: Repeat steps from 3 to 6, T times,

Step 8: Stop for sufficiently large value of T, the Bayes estimate of the stress-strength reliability function under the squared error loss will be given as

7.2 Credible interval

The credible interval for the stress strength reliability can be deduced as follows. From the relations between the gamma distribution and chi-square distribution, it can be shown that and

The posterior distribution of R can be written as

And therefore a 100(1−ε)% credible interval for R will be given by

8. Simulation study

Monte Carlo simulation is performed with replications of 1000 samples. The simulated samples are generated using different values of the parameters from the exponentiated generalized Marshall Olkin Weibull distribution (EGMO-W). The simulated samples are taken of different sizes: (5, 5), (10, 10), (20, 20), (30, 30), (50, 50) and (100, 100). The results for the maximum likelihood estimates of the strength-stress reliability with the mean squared error values, the 95% bootstrap confidence intervals with (B = 1000) and the 95% asymptotic confidence intervals are obtained. The MCMC method is performed with T = 1000, the Bayes estimates of the strength-stress reliability and the 95% credible intervals are deduced in two cases: The first case of prior-I (ξ_i = 2, η_i = 3, i = 1,…5) and the second case of prior-II (ξ_i = 1, η_i = 10, i = 1,…5). All the simulations and computations were performed using the software program R. The results are presented in Tables 1–4. Steps of applying the Monte Carlo simulation is illustrated as follows.

Step 1: Set initial values of the parameters (b₁, b₂, ϑ, a, γ, β).

Step 2: Choose the samples sizes (n, m).

Step 3: Generate random values of the random variables X_i and Y_j at the initial values of the parameters (b₁, b₂, ϑ, a, γ, β) by applying the following inversion formula method

For EGMO-W distribution the inversion formula will be given by

Step 4: Solve the differential Eqs (3)–(5) and using Eqs (8) and (9) to obtain the estimates of the parameters by using the Newton-Raphson method and the aid of software program R.

Step 5: Obtain the estimate of the stress-strength reliability by substituting in Eq (10).

Step 6: Repeat steps from 3 to 5, 1000 times. In each time of simulation, the same values of the initial parameters and the same samples sizes are considered but the values of generating random samples are varying each time. So, we have 1000 values of estimates of the stress-strength reliability . The mean squared error (MSE) can be obtained from the following relation

Download:

Table 1. Simulation results for MLE and Bayesian estimation with MSE when a = b₁ = b₂ = ϑ = γ = β = 0.5.

https://doi.org/10.1371/journal.pone.0280183.t001

Download:

Table 2. Results for confidence, bootstrap confidence and credible intervals when a = b₁ = b₂ = ϑ = γ = β = 0.5.

https://doi.org/10.1371/journal.pone.0280183.t002

Download:

Table 3. Simulation results for MLE and Bayesian estimation with MSE when a = 0.5, b₁ = 1.5, b₂ = 2.5, ϑ = 0.5, γ = 3.5, β = 1.5.

https://doi.org/10.1371/journal.pone.0280183.t003

Download:

Table 4. Results for confidence, bootstrap confidence and credible intervals when a = 0.5, b₁ = 1.5, b₂ = 2.5, ϑ = 0.5, γ = 3.5, β = 1.5.

https://doi.org/10.1371/journal.pone.0280183.t004

In Table 1, simulation results are obtained for the stress strength reliability estimates by applying the maximum likelihood method and Bayesian estimation method in case of prior I and prior II when the values of the parameters are given by a = b₁ = b₂ = ϑ = γ = β = 0.5. In Table 2, results for confidence, bootstrap confidence and credible intervals in case of prior I and prior II are deduced when the values of the parameters are given by a = b₁ = b₂ = ϑ = γ = β = 0.5.

In Table 3, simulation results are obtained for the stress strength reliability estimates by applying the maximum likelihood method and Bayesian estimation method in case of prior I and prior II when the values of the parameters are given by a = 0.5, b₁ = 1.5, b₂ = 2.5, ϑ = 0.5, γ = 3.5, β = 1.5. In Table 4, results for confidence, bootstrap confidence and credible intervals in case of prior I and prior II are deduced when the values of the parameters are given by a = 0.5, b₁ = 1.5, b₂ = 2.5, ϑ = 0.5, γ = 3.5, β = 1.5. All results obtained in Tables 1–4 are performed in case of the parameters are unknown (case I) and in case of the parameters are known.

From the results obtained in Tables 1–4, it can be concluded the following:

The values of the mean squared errors decrease as the sample sizes increase.
The lengths of the asymptotic confidence intervals, bootstrap confidence intervals and credible intervals decrease as the sample sizes increase.
Stress strength reliability estimates obtained in case of Bayesian estimation method is greater than stress strength reliability estimates obtained in case of maximum likelihood method.

9. Real data Application 1

A real data for the confirmed, recovered cases and deaths for covid 19 in Saudi Arabia from 1 April 2020 to 15 May 2020 are given in Table 5. These data obtained from the website (https://datahub.io/core/ covid-19#resource-covid-19_zip/). It is considered that the stress and strength follow exponentiated generalized Marshall Olkin exponential distribution (EGMO-E) and death rate represents the stress and the recovery rate represents the strength, where

Calculations of recovery and death rates based on the confirmed, recovered cases and deaths of covid 19 data are presented in Table 5. Summary of the measures of the death and recovery rate is illustrated in Table 6.

Download:

Table 5. Recovery and death rates based on the confirmed, recovered cases and deaths of covid 19.

https://doi.org/10.1371/journal.pone.0280183.t005

Download:

Table 6. Summary of the measures of the death and recovery rate.

https://doi.org/10.1371/journal.pone.0280183.t006

The goodness of fit for the exponentiated generalized Marshall Olkin exponential distribution (EGMO-E) is compared with the generalized Marshall Olkin exponential distribution (GMO-E), the exponentiated generalized exponential distribution (EG-E), the Marshall Olkin exponential distribution (MO-E) and the exponential distribution (E). The results of the goodness of fit for the death and recovery rate of covid 19 data are shown in Table 7.

Download:

Table 7. MLE estimates, Log L, AIC, BIC and KS for the death and recovery rate.

https://doi.org/10.1371/journal.pone.0280183.t007

The best model is chosen as the one having lowest AIC (Akaike Information Criterion). The results obtained in Table 7 indicated that the exponentiated generalized Marshall Olkin exponential distribution (EGMO-E) is a better distribution to model the death and recovery rate of covid 19 data than the other distributions.

The maximum likelihood estimate of the reliability of the stress-strength model according to the death and recovery rate is obtained as 0.5175 and the 95% asymptotic confidence interval of R is [0.4143, 0.6206]. The Bayesian estimate of the reliability of the stress-strength model according to prior I is obtained as 0.8085 and the 95% credible confidence interval of R (with T = 1000) is [0.5810, 0.9603]. The Bayesian estimate of the reliability of the stress-strength model according to prior II is obtained as 0.7297 and the 95% credible confidence interval of R (with T = 1000) is [0.5802, 0.8560]. The death rate in Saudi Arabia from 1 April 2020 to 15 May 2020, the estimated PDF of the death rate, the estimated PDF with histogram of the death rate, the estimated CDF of the death rate and the log likelihood function of the death rate are presented in Figs 1–5, respectively. The recovery rate in Saudi Arabia from 1 April 2020 to 15 May 2020, the estimated PDF of the recovery rate, the estimated PDF with histogram of the recovery rate, the estimated CDF of the recovery rate and the log likelihood function of the recovery rate are presented in Figs 6–10, respectively.

Download:

Fig 1. The death rate in Saudi Arabia from 1 April 2020 to 15 May 2020.

https://doi.org/10.1371/journal.pone.0280183.g001

Download:

Fig 2. The estimated PDF of the death rate.

https://doi.org/10.1371/journal.pone.0280183.g002

Download:

Fig 3. The estimated PDF with histogram of the death rate.

https://doi.org/10.1371/journal.pone.0280183.g003

Download:

Fig 4. The estimated CDF of the death rate.

https://doi.org/10.1371/journal.pone.0280183.g004

Download:

Fig 5. The log likelihood function of the death rate.

https://doi.org/10.1371/journal.pone.0280183.g005

Download:

Fig 6. The recovery rate in Saudi Arabia from 1 April 2020 to 15 May 2020.

https://doi.org/10.1371/journal.pone.0280183.g006

Download:

Fig 7. The estimated PDF of the recovery rate.

https://doi.org/10.1371/journal.pone.0280183.g007

Download:

Fig 8. The estimated PDF with histogram of the recovery rate.

https://doi.org/10.1371/journal.pone.0280183.g008

Download:

Fig 9. The estimated CDF of the recovery rate.

https://doi.org/10.1371/journal.pone.0280183.g009

Download:

Fig 10. The log likelihood function of the recovery rate.

https://doi.org/10.1371/journal.pone.0280183.g010

10. Real data Application 2

A real data application is introduced to demonstrate how the proposed estimation techniques can be applied in practice. The following datasets were provided by Badar and Priest [21]. Here are the single fibers of 20 mm (dataset 1) and 10 mm (dataset 2) in gauge lengths.

Dataset 1 (n = 69): 0.312, 0.314, 0.479, 0.552, 0.700, 0.803, 0.861, 0.865, 0.944, 0.958, 0.966, 0.997, 1.006, 1.021, 1.027, 1.055, 1.063, 1.098, 1.140, 1.179, 1.224, 1.240, 1.253, 1.270, 1.272, 1.274, 1.301, 1.301, 1.359, 1.382, 1.382, 1.426, 1.434, 1.435, 1.478, 1.490, 1.511, 1.514, 1.535, 1.554, 1.566, 1.570, 1.586, 1.629, 1.633, 1.642, 1.648, 1.684, 1.697, 1.726, 1.770, 1.773, 1.800, 1.809, 1.818, 1.821, 1.848, 1.880, 1.954, 2.012, 2.067, 2.084, 2.090, 2.096, 2.128, 2.233, 2.433, 2.585, 2.585.

Dataset 2 (m = 63): 0.101, 0.332, 0.403, 0.428, 0.457, 0.550, 0.561, 0.596, 0.597, 0.645, 0.654, 0.674, 0.718, 0.722, 0.725, 0.732, 0.775, 0.814, 0.816, 0.818, 0.824, 0.859, 0.875, 0.938, 0.940, 1.056, 1.117, 1.128, 1.137, 1.137, 1.177, 1.196, 1.230, 1.325, 1.339, 1.345, 1.420, 1.423, 1.435, 1.443, 1.464, 1.472, 1.494, 1.532, 1.546, 1.577, 1.608, 1.635, 1.693, 1.701, 1.737, 1.754, 1.762, 1.828, 2.052, 2.071, 2.086, 2.171, 2.224, 2.227, 2.425, 2.595, 3.220.

The maximum likelihood estimates of the parameters for different distributions are calculated with their standard errors for the two datasets. Also, the log-likelihood function (Log L), Akaike information criteria, Bayesian information criteria and Kolmogorov–Smirnov (KS) test statistics are calculated for the two datasets. The goodness of fit for the exponentiated generalized Marshall Olkin Weibull distribution (EGMO-W) is compared with the exponentiated generalized Weibull distribution (EG-W), the generalized Marshall Olkin Weibull distribution (GMO-W), the exponentiated Marshall Olkin Weibull distribution (EMO-W), the generalized Weibull distribution (G-W), the Marshall Olkin Weibull distribution (MO-W) and Weibull distribution (W). The maximum likelihood estimators (MLE) for the parameters, Log L, AIC, BIC and KS for the dataset 1 and dataset 2 are shown in Tables 8 and 9, respectively.

Download:

Table 8. MLE estimates, Log L, AIC, BIC and KS for the data set 1.

https://doi.org/10.1371/journal.pone.0280183.t008

Download:

Table 9. MLE estimates, Log L, AIC, BIC and KS for the data set 2.

https://doi.org/10.1371/journal.pone.0280183.t009

The best model is chosen as the one having lowest AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion). The results obtained in Tables 8 and 9, indicated that the exponentiated generalized Marshall Olkin Weibull distribution (EGMO-W) is a better distribution to model the datasets than the other distributions.

The maximum likelihood estimate of the reliability of the stress-strength model according to data set 1 and data set 2 is obtained as 0.458 and the 95% asymptotic confidence interval of R is [0.391, 0.524]. The non-informative priors (ξ_i = η_i = 0, i = 1,2,3,4) Bayesian estimate for the reliability of the stress-strength model according to dataset 1 and dataset 2 is obtained as 0.561 and the 90% credible confidence interval (with T = 10000) of R is [0.391, 0.534]. The informative priors (ξ_i = η_i = 0.5, i = 1,2,3,4) Bayesian estimate for the reliability of the stress-strength model according to data set 1 and data set 2 is obtained as 0.560 and the 90% credible confidence interval (with T = 10000) of R is [0.392, 0.533]. the estimated cumulative distribution function and empirical cumulative distribution function, estimated probability density function with histogram, Q-Q plot, and P-P plot for the EGMO-W distribution for dataset 1 and dataset 2 are shown graphically in Figs 11 and 12, respectively.

Download:

Fig 11. Estimated CDF, estimated PDF with histogram, Q-Q and P-P plots for EGMO-W distribution for dataset 1.

https://doi.org/10.1371/journal.pone.0280183.g011

Download:

Fig 12. Estimated CDF, estimated PDF with histogram, Q-Q and P-P plots for EGMO-W distribution for dataset 2.

https://doi.org/10.1371/journal.pone.0280183.g012

11. Conclusion

The study of the stress-strength reliability model subject to the exponentiated generalized Marshall Olkin G family of distributions is introduced. The maximum likelihood estimator, the asymptotic confidence and bootstrap confidence intervals for the stress-strength reliability function are obtained. Bayesian estimators and the credible interval for the stress-strength reliability function are derived. A simulation study is introduced in which EGMO-W distribution is applied. All results obtained in the simulation study is consistent. Applications based on real data are introduced to show the results for the stress-strength model and compare the EGMO-E and EGMO-W distributions with other different distributions. These comparisons show that EGMO-E and EGMO-W distributions can be considered as better models to fit the datasets.

References

1. Al-Mutairi D. K., Ghitany M. E. and Kundu D. (2013). Inferences on Stress-Strength Reliability from Lindley Distributions. Communications in Statistics-Theory and Methods. 42(8). https://doi.org/10.1080/03610926.2011.563011.
- View Article
- Google Scholar
2. Rao G.S., Kantam R.L., Rosaiah K. and Reddy J.P. (2013). Estimation of Stress–Strength Reliability from Inverse Rayleigh Distribution. Journal of Industrial and Production Engineering. 30(4), 256–263.
- View Article
- Google Scholar
3. Singh S.K., Singh U., Yadav A.S. and Vishwkarma P.K. (2015). On the Estimation of Stress Strength Reliability Parameter of Inverted Exponential Distribution. International Journal of Scientific World. 3(1), 98–112.
- View Article
- Google Scholar
4. Al-Mutairi D.K., Ghitany M.E. and Kundu D. (2015). Inferences on Stress-Strength Reliability from Weighted Lindley Distributions. Communications in Statistics-Theory and Methods. 44(19), 4096–4113.
- View Article
- Google Scholar
5. Sharma V.K., Singh S.K., Singh U. and Agiwal V. (2015). The Inverse Lindley Distribution: A Stress-Strength Reliability Model with Application to Head and Neck Cancer Data. Journal of Industrial and Production Engineering. 32(3), 162–173.
- View Article
- Google Scholar
6. Mokhlis N.A., Ibrahim E.J and Gharieb D.M. (2017). Stress−Strength Reliability with General form Distributions. Communications in Statistics-Theory and Methods. 46(3), 1230–1246.
- View Article
- Google Scholar
7. Çetinkaya Ç. and Genç A.İ. (2019). Stress–Strength Reliability Estimation under the Standard Two-Sided Power Distribution. Applied Mathematical Modelling. 65, 72–88. https://doi.org/10.1016/j.apm.2018.08.008.
- View Article
- Google Scholar
8. Almarashi A.M., Algarni A. and Nassar M. (2020). On Estimation Procedures of Stress-Strength Reliability for Weibull Distribution with Application. PLOS ONE. 15. https://doi.org/10.1371/journal.pone.0237997.
- View Article
- Google Scholar
9. Muhammad I., Wang X., Li C., Yan M. and Chang M. (2020). Estimation of the Reliability of a Stress–Strength System from Poisson Half Logistic Distribution. Entropy. 22. pmid:33287072
- View Article
- PubMed/NCBI
- Google Scholar
10. Nojosa R., Rathie P. N., Stress–Strength Reliability Models Involving Generalized Gamma and Weibull Distributions. International Journal of Quality & Reliability Management. 37, 538–551. https://doi.org/10.1108/IJQRM-06-2019-0190.
- View Article
- Google Scholar
11. Alamri O.A., Abd El-Raouf M.M., Ismail E.A., Almaspoor Z., Alsaedi B.O., Khosa S.K. and Yusuf M. (2021). Estimate Stress-Strength Reliability Model Using Rayleigh and Half-Normal Distribution. Computational Intelligence and Neuroscience. Article ID 7653581. pmid:34285693
- View Article
- PubMed/NCBI
- Google Scholar
12. Hassan A.S., Al-Omari A. and Nagy H.F. (2021), Stress–Strength Reliability for the Generalized Inverted Exponential Distribution Using MRSS, Iranian Journal of Science and Technology, Transactions A: Science. 45, 641–659, https://doi.org/10.1007/s40995-020-01033-9.
- View Article
- Google Scholar
13. Abu El Azm W.S., Almetwally E.M., Alghamdi A.S., Aljohani H.M., Muse A.H. and Abo-Kasem O.E. (2021). Stress-Strength Reliability for Exponentiated Inverted Weibull Distribution with Application on Breaking of Jute Fiber and Carbon Fibers. Computational Intelligence and Neuroscience, 1–21. pmid:34603431
- View Article
- PubMed/NCBI
- Google Scholar
14. Jha M.K., Tripathi Y.M. and Dey S. (2020). Multicomponent Stress-Strength Reliability Estimation Based on Unit Generalized Rayleigh Distribution. International Journal of Quality & Reliability Management. 38, 2048–2079. https://doi.org/10.1108/IJQRM-07-2020-0245.
- View Article
- Google Scholar
15. Jha M.K., Dey S., Alotaibi R., Alomani G. and Tripathi Y.M. (2022). Multicomponent Stress-Strength Reliability Estimation Based on Unit Generalized Exponential Distribution, Ain Shams Engineering Journal. 13, 1–26.
- View Article
- Google Scholar
16. Maurya R.K., Tripathi Y.M. and Kayal T. (2022). Reliability Estimation in a Multicomponent Stress-Strength Model Based on Inverse Weibull Distribution. Sankhya: The Indian Journal of Statistics. 84, 364–401. https://doi.org/10.1007/s13571-021-00263-0.
- View Article
- Google Scholar
17. Jovanovic M., Milosevic B., Obradovic M. and Vidovic Z. (2021). Inference on Reliability of Stress-Strength Model with Peng-Yan Extended Weibull Distributions. Filomat. 35, 1927–1948. https://doi.org/10.2298/FIL2106927J.
- View Article
- Google Scholar
18. Sabry M.A., Almetwally E.M. and Almongy H.M. (2022). Monte Carlo Simulation of Stress-Strength Model and Reliability Estimation for Extension of the Exponential Distribution. Thailand Statistician. 20,124–143.
- View Article
- Google Scholar
19. Zarei R. and Shahrestani S.Y. (2021). Bayes and Empirical Bayes Estimator of Reliability Function in Multicomponent Stress-Strength System based on Generalized Rayleigh Distribution. Journal of Statistical Sciences, 14, 429–448, https://doi.org/10.29252/jss.14.2.429.
- View Article
- Google Scholar
20. Handique L., Chakraborty S. and de Andrade T.N. (2019). The Exponentiated Generalized Marshall–Olkin Family of Distribution: Its Properties and Applications. Annals of Data Science. 6, 391–411. https://doi.org/10.1007/s40745-018-0166-z.
- View Article
- Google Scholar
21. Badar M.G. and Priest A.M. (1982), Progress in Science and Engineering Composites, In Hayashi v Kawata K. and Umekawa S., Eds., ICCM-IV, Tokyo, Japan, 1129–1136.
22. Dias B., Cordeiro C.R. and Alizadeh G.M. (2016). Exponentiated Marshall-Olkin Family of Distributions. Journal of Statistical Distributions and Applications. 3, Article number 15. https://doi.org/10.1186/s40488-016-0051-2.
- View Article
- Google Scholar
23. Korkmaz M.Ç., Yousof H.M., Hamedani G.G. and Ali M.M. (2018). The Marshall-Olkin Generalized G Poisson Family of Distributions, Pakistan Journal of Statistics. 34(3), 251–267.
- View Article
- Google Scholar
24. Afify A.Z., Kumar D. and Elbatal I. (2020). Marshall–Olkin Power Generalized Weibull Distribution with Applications in Engineering and Medicine. Journal of Statistical Theory and Applications. 19(2), 223–237.
- View Article
- Google Scholar
25. Yousof H.M, Afify A.Z., Nadarajah S., Hamedani G. and Aryal G.R. (2018). The Marshall-Olkin Generalized-G Family of Distributions with Applications. Statistica. 78(3), 273–295. https://doi.org/10.6092/issn.1973-2201/7662.
- View Article
- Google Scholar
26. Chesneau C., Karakaya K., Bakouch H.S. and Kus C. (2022). An Alternative to the Marshall-Olkin Family of Distributions: Bootstrap, Regression and Applications. Communications on Applied Mathematics and Computation. 4, 1229–1257.
- View Article
- Google Scholar

[ref1] 1. Al-Mutairi D. K., Ghitany M. E. and Kundu D. (2013). Inferences on Stress-Strength Reliability from Lindley Distributions. Communications in Statistics-Theory and Methods. 42(8). https://doi.org/10.1080/03610926.2011.563011.
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Rao G.S., Kantam R.L., Rosaiah K. and Reddy J.P. (2013). Estimation of Stress–Strength Reliability from Inverse Rayleigh Distribution. Journal of Industrial and Production Engineering. 30(4), 256–263.
View Article
Google Scholar

[5] View Article

[6] Google Scholar

[ref3] 3. Singh S.K., Singh U., Yadav A.S. and Vishwkarma P.K. (2015). On the Estimation of Stress Strength Reliability Parameter of Inverted Exponential Distribution. International Journal of Scientific World. 3(1), 98–112.
View Article
Google Scholar

[8] View Article

[9] Google Scholar

[ref4] 4. Al-Mutairi D.K., Ghitany M.E. and Kundu D. (2015). Inferences on Stress-Strength Reliability from Weighted Lindley Distributions. Communications in Statistics-Theory and Methods. 44(19), 4096–4113.
View Article
Google Scholar

[11] View Article

[12] Google Scholar

[ref5] 5. Sharma V.K., Singh S.K., Singh U. and Agiwal V. (2015). The Inverse Lindley Distribution: A Stress-Strength Reliability Model with Application to Head and Neck Cancer Data. Journal of Industrial and Production Engineering. 32(3), 162–173.
View Article
Google Scholar

[14] View Article

[15] Google Scholar

[ref6] 6. Mokhlis N.A., Ibrahim E.J and Gharieb D.M. (2017). Stress−Strength Reliability with General form Distributions. Communications in Statistics-Theory and Methods. 46(3), 1230–1246.
View Article
Google Scholar

[17] View Article

[18] Google Scholar

[ref7] 7. Çetinkaya Ç. and Genç A.İ. (2019). Stress–Strength Reliability Estimation under the Standard Two-Sided Power Distribution. Applied Mathematical Modelling. 65, 72–88. https://doi.org/10.1016/j.apm.2018.08.008.
View Article
Google Scholar

[20] View Article

[21] Google Scholar

[ref8] 8. Almarashi A.M., Algarni A. and Nassar M. (2020). On Estimation Procedures of Stress-Strength Reliability for Weibull Distribution with Application. PLOS ONE. 15. https://doi.org/10.1371/journal.pone.0237997.
View Article
Google Scholar

[23] View Article

[24] Google Scholar

[ref9] 9. Muhammad I., Wang X., Li C., Yan M. and Chang M. (2020). Estimation of the Reliability of a Stress–Strength System from Poisson Half Logistic Distribution. Entropy. 22. pmid:33287072
View Article
PubMed/NCBI
Google Scholar

[26] View Article

[27] PubMed/NCBI

[28] Google Scholar

[ref10] 10. Nojosa R., Rathie P. N., Stress–Strength Reliability Models Involving Generalized Gamma and Weibull Distributions. International Journal of Quality & Reliability Management. 37, 538–551. https://doi.org/10.1108/IJQRM-06-2019-0190.
View Article
Google Scholar

[30] View Article

[31] Google Scholar

[ref11] 11. Alamri O.A., Abd El-Raouf M.M., Ismail E.A., Almaspoor Z., Alsaedi B.O., Khosa S.K. and Yusuf M. (2021). Estimate Stress-Strength Reliability Model Using Rayleigh and Half-Normal Distribution. Computational Intelligence and Neuroscience. Article ID 7653581. pmid:34285693
View Article
PubMed/NCBI
Google Scholar

[33] View Article

[34] PubMed/NCBI

[35] Google Scholar

[ref12] 12. Hassan A.S., Al-Omari A. and Nagy H.F. (2021), Stress–Strength Reliability for the Generalized Inverted Exponential Distribution Using MRSS, Iranian Journal of Science and Technology, Transactions A: Science. 45, 641–659, https://doi.org/10.1007/s40995-020-01033-9.
View Article
Google Scholar

[37] View Article

[38] Google Scholar

[ref13] 13. Abu El Azm W.S., Almetwally E.M., Alghamdi A.S., Aljohani H.M., Muse A.H. and Abo-Kasem O.E. (2021). Stress-Strength Reliability for Exponentiated Inverted Weibull Distribution with Application on Breaking of Jute Fiber and Carbon Fibers. Computational Intelligence and Neuroscience, 1–21. pmid:34603431
View Article
PubMed/NCBI
Google Scholar

[40] View Article

[41] PubMed/NCBI

[42] Google Scholar

[ref14] 14. Jha M.K., Tripathi Y.M. and Dey S. (2020). Multicomponent Stress-Strength Reliability Estimation Based on Unit Generalized Rayleigh Distribution. International Journal of Quality & Reliability Management. 38, 2048–2079. https://doi.org/10.1108/IJQRM-07-2020-0245.
View Article
Google Scholar

[44] View Article

[45] Google Scholar

[ref15] 15. Jha M.K., Dey S., Alotaibi R., Alomani G. and Tripathi Y.M. (2022). Multicomponent Stress-Strength Reliability Estimation Based on Unit Generalized Exponential Distribution, Ain Shams Engineering Journal. 13, 1–26.
View Article
Google Scholar

[47] View Article

[48] Google Scholar

[ref16] 16. Maurya R.K., Tripathi Y.M. and Kayal T. (2022). Reliability Estimation in a Multicomponent Stress-Strength Model Based on Inverse Weibull Distribution. Sankhya: The Indian Journal of Statistics. 84, 364–401. https://doi.org/10.1007/s13571-021-00263-0.
View Article
Google Scholar

[50] View Article

[51] Google Scholar

[ref17] 17. Jovanovic M., Milosevic B., Obradovic M. and Vidovic Z. (2021). Inference on Reliability of Stress-Strength Model with Peng-Yan Extended Weibull Distributions. Filomat. 35, 1927–1948. https://doi.org/10.2298/FIL2106927J.
View Article
Google Scholar

[53] View Article

[54] Google Scholar

[ref18] 18. Sabry M.A., Almetwally E.M. and Almongy H.M. (2022). Monte Carlo Simulation of Stress-Strength Model and Reliability Estimation for Extension of the Exponential Distribution. Thailand Statistician. 20,124–143.
View Article
Google Scholar

[56] View Article

[57] Google Scholar

[ref19] 19. Zarei R. and Shahrestani S.Y. (2021). Bayes and Empirical Bayes Estimator of Reliability Function in Multicomponent Stress-Strength System based on Generalized Rayleigh Distribution. Journal of Statistical Sciences, 14, 429–448, https://doi.org/10.29252/jss.14.2.429.
View Article
Google Scholar

[59] View Article

[60] Google Scholar

[ref20] 20. Handique L., Chakraborty S. and de Andrade T.N. (2019). The Exponentiated Generalized Marshall–Olkin Family of Distribution: Its Properties and Applications. Annals of Data Science. 6, 391–411. https://doi.org/10.1007/s40745-018-0166-z.
View Article
Google Scholar

[62] View Article

[63] Google Scholar

[ref21] 21. Badar M.G. and Priest A.M. (1982), Progress in Science and Engineering Composites, In Hayashi v Kawata K. and Umekawa S., Eds., ICCM-IV, Tokyo, Japan, 1129–1136.

[ref22] 22. Dias B., Cordeiro C.R. and Alizadeh G.M. (2016). Exponentiated Marshall-Olkin Family of Distributions. Journal of Statistical Distributions and Applications. 3, Article number 15. https://doi.org/10.1186/s40488-016-0051-2.
View Article
Google Scholar

[66] View Article

[67] Google Scholar

[ref23] 23. Korkmaz M.Ç., Yousof H.M., Hamedani G.G. and Ali M.M. (2018). The Marshall-Olkin Generalized G Poisson Family of Distributions, Pakistan Journal of Statistics. 34(3), 251–267.
View Article
Google Scholar

[69] View Article

[70] Google Scholar

[ref24] 24. Afify A.Z., Kumar D. and Elbatal I. (2020). Marshall–Olkin Power Generalized Weibull Distribution with Applications in Engineering and Medicine. Journal of Statistical Theory and Applications. 19(2), 223–237.
View Article
Google Scholar

[72] View Article

[73] Google Scholar

[ref25] 25. Yousof H.M, Afify A.Z., Nadarajah S., Hamedani G. and Aryal G.R. (2018). The Marshall-Olkin Generalized-G Family of Distributions with Applications. Statistica. 78(3), 273–295. https://doi.org/10.6092/issn.1973-2201/7662.
View Article
Google Scholar

[75] View Article

[76] Google Scholar

[ref26] 26. Chesneau C., Karakaya K., Bakouch H.S. and Kus C. (2022). An Alternative to the Marshall-Olkin Family of Distributions: Bootstrap, Regression and Applications. Communications on Applied Mathematics and Computation. 4, 1229–1257.
View Article
Google Scholar

[78] View Article

[79] Google Scholar

Figures

Abstract

1. Introduction

2. Exponentiated generalized Marshall Olkin-G distribution

3. Stress-strength reliability

4. Maximum likelihood estimation

5. Asymptotic Confidence Interval (A.C.I)

6. Bootstrap Confidence Interval (B.C.I.)

7. Bayesian estimation

Case I: When the parameters are unknown

7.1 MCMC method

Case II: When the parameters are known.

7.2 Credible interval

8. Simulation study

9. Real data Application 1

10. Real data Application 2

11. Conclusion

References