Stress-Strength Reliability for Exponentiated Inverted Weibull Distribution with Application on Breaking of Jute Fiber and Carbon Fibers

Abu El Azm, Wael S.; Almetwally, Ehab M.; Alghamdi, Abdulaziz S.; Aljohani, Hassan M.; Muse, Abdisalam Hassan; Abo-Kasem, O. E.

doi:https://doi.org/10.1155/2021/4227346

Computational Intelligence and Neuroscience

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Volume 2021 | Article ID 4227346 | https://doi.org/10.1155/2021/4227346

Stress-Strength Reliability for Exponentiated Inverted Weibull Distribution with Application on Breaking of Jute Fiber and Carbon Fibers

Wael S. Abu El Azm,¹Ehab M. Almetwally,²Abdulaziz S. Alghamdi,³Hassan M. Aljohani,⁴Abdisalam Hassan Muse,⁵and O. E. Abo-Kasem¹

Academic Editor: Ahmed Mostafa Khalil

Received03 Aug 2021

Accepted28 Aug 2021

Published22 Sept 2021

Abstract

For the first time and by using an entire sample, we discussed the estimation of the unknown parameters , and and the system of stress-strength reliability for exponentiated inverted Weibull (EIW) distributions with an equivalent scale parameter supported eight methods. We will use maximum likelihood method, maximum product of spacing estimation (MPSE), minimum spacing absolute-log distance estimation (MSALDE), least square estimation (LSE), weighted least square estimation (WLSE), method of Cramér-von Mises estimation (CME), and Anderson-Darling estimation (ADE) when X and Y are two independent a scaled exponentiated inverted Weibull (EIW) distribution. Percentile bootstrap and bias-corrected percentile bootstrap confidence intervals are introduced. To pick the better method of estimation, we used the Monte Carlo simulation study for comparing the efficiency of the various estimators suggested using mean square error and interval length criterion. From cases of samples, we discovered that the results of the maximum product of spacing method are more competitive than those of the other methods. A two real‐life data sets are represented demonstrating how the applicability of the methodologies proposed in real phenomena.

1. Introduction

Since Birnbaum’s [1] pioneering research, statistical inference of a system stress-strength parameter has received increased attention and is widely used in a variety of engineering applications. If X and Y are two independent random variables that represent the strength and the stress, then is a measure of system performance that naturally arises in mechanical dependability. In this case, the system fails if and only if the applied stress is greater than its strength at any time. Several studies in the statistical literature have investigated the problem of estimating the stress-strength parameters of a system. Kundu and Gupta [2, 3] introduced the estimation of stress-strength reliability for generalized exponential and Weibull random variables, respectively. um likelihood method, max Raqab et al. [4] discussed the estimation of , where and are distributed as two independent three-parameter generalized exponential random variables. Rezaei et al. [5] considered the estimation of , where and are two independent generalized Pareto distributions.

Recently, many authors investigated the estimation of for different life testing schemes based on different distributions by using the maximum likelihood and Bayesian estimation methods; see, for example, Mokhlis [6], Kundu and Raqab [7], Asgharzadeh et al. [8], Asgharzadeh et al. [9], Asgharzadeh et al. [10], Valiollahi et al. [11], Rao et al. [12], Rao et al. [13], Mirjalili et al. [14], Jia et al. [15], Nadeb et al. [16], Alshenawy et al. [17], El-Sherpieny et al. [18], Nassr et al. [19], and Muhammad et al. [20]. As frequentist methods, the maximum likelihood method and the Bayesian estimation method were used in these studies. However, little thought was given to estimate using other methods, although, in some cases, they can provide better estimates than the maximum likelihood approach. Almetwally and Almongy [21] examined classical and Bayesian estimation methods for the stress-strength model of the power Lomax distribution.

Aside from the maximal likelihood estimation (MLE) approach, Almarashi et al. [22] offered nine other frequentist estimate methods to estimate the stress-strength reliability of the Weibull distribution, namely, least square, weighted least square, percentile, maximum product of spacing, minimum spacing absolute distance, minimum spacing absolute-log distance, method of Cramér-von Mises, and Anderson-Darling and right-tail Anderson-Darling. They compared the efficiency of the different proposed estimators by using Monte Carlo simulation study. In terms of relative biases and relative mean squared errors, the performance and finite sample properties of the various estimators are compared.

To the authors’ knowledge, the MLE and the maximum product of spacing method which were used to estimate the parameters , and of life of an EIW under a finite sample (MPSE), minimum spacing absolute-log distance estimation (MSALDE) method, least square estimation (LSE) method, weighted least square estimation (WLSE) method of Cramér-von Mises estimation (CME), and Anderson-Darling estimation (ADE) method have not yet been investigated. In this paper, our main purpose is to use eight approaches to derive estimates of the unknown parameters and stress-strength reliability , which we think if applied by statisticians/reliability engineers would be very interesting in a scaled exponentiated inverted Weibull distribution. Furthermore, the simulation study and real data analysis demonstrate that there are classical methods, rather than MLE methods, which can provide desirable estimates, justifying their use in applied areas. It should also be noted that this is the first time that eight estimation methods have been considered to estimate the unknown parameters , and and stress-strength reliability of the EIW distribution.

Flaih et al. [23] proposed the exponentiated inverted Weibull distribution as a generalization of the standard parent distribution known as the exponentiated-parent distribution and the standard inverted Weibull distribution. More research on the EIW distribution is needed, both theoretically (estimation methods) and practically (analysis further data). According to this study, the EIW can be used as an alternative to the inverted Weibull distribution and may perform better than the inverted Weibull distribution. For , it represents the standard inverted Weibull distribution, and for , it represents the exponentiated standard inverted exponential distribution. As a result, the exponentiated inverted Weibull distribution is a generalization of both the exponentiated inverted exponential and inverted Weibull distributions. The physical interpretation of the exponentiated inverted Weibull distribution is also available.

The EIW with scale parameter and shape parameter , denoted by EIW , has the following probability density function (PDF):and the corresponding cumulative distribution function (CDF) is given by

Let and be independent exponentiated inverted Weibull random variables and follow EIW and EIW , respectively; then can be written as follows (see Hassan et al. [24]):

Figure 1 shows different values for when and change.

The main objective of this study is to estimate unknown parameters , and and stress-strength reliability when and are independent of a scaled EIW distribution using the eight estimation methods listed above. Furthermore, for the stress-strength parameter, we use percentile bootstrap and bias-corrected percentile bootstrap confidence intervals. To compare the efficiency of the various estimates, we conduct an extensive Monte Carlo numerical simulation study, as well as an analysis of two real‐life data sets, the applicability of the methodologies proposed in real phenomena. The rest of this paper is organized as follows: In Section 2, we proposed the different estimation methods. Percentile bootstrap and bias-corrected percentile bootstrap confidence intervals are discussed in Section 3. In Section 4, a Monte Carlo numerical simulation research is carried out. In Section 5, two real-life data sets are examined. Finally, Section 6 concludes the paper.

2. Different Estimation Methods

In this section, the eight recurrent estimation methods considered in this paper to obtain the unknown parameters and different estimates of the stress-strength parameter will be discussed. These estimation methods would be of particular interest when comparing them with other maximum likelihood estimation procedures. For more examples of classical estimation method, see the works of Almetwally [25], El-Morshedy et al. [26], Almetwally et al. [27], and Sabry et al. [28].

2.1. Maximum Likelihood Estimation

Let and be random samples from EIW and EIW , respectively, and the likelihood function of the observed sample can be expressed as

We obtain by taking the natural logarithm likelihood function as

The MLEs of , , and denoted by , , and , respectively, can be obtained by solving the subsequent equations:

and can be obtained as a function of the unknown parameter from (6) and (7), respectively, as follows:

Substituting the estimators and obtained from (9) in (5), the profile log-likelihood function of parameter can then be obtained as follows:

To obtain , as a result of differentiating (10) with respect to and equating the result by zero,

After obtaining from (11) by using any iteration procedure, we can obtain and from (9). Now, the MLE of a system can be obtained as

2.2. Maximum Product of Estimation

Cheng and Amin [29] introduced the method of maximum product of spacing as an alternative to the maximum likelihood method to estimate the parameters of the lognormal distribution. Let denote the order statistics of a random sample n from EIW and let denote the order statistics of a random sample k from EIW ; the uniform spacings of the two samples can therefore be defined as follows:

Let denote the order statistics of a random sample from EIW.

The MPSEs of the unknown parameters are produced by maximization of the following function, as in the work of Cheng and Amin [30].

From (2) and (14), the MPSEs of the unknown parameters , and denoted by , , and can be obtained by maximizing, with respect to , and , the following function:

These estimates can be obtained equivalently by solving the following equations simultaneously:where and . Using the obtained estimates, we can obtain the MPSE of a system as

2.3. Minimum Spacing Distance Estimation

Torabi [31] was the first to propose the minimum spacing distance estimating method. The minimum spacing distance estimators (MSADEs) are obtained by minimizing the following function, using the same notations as in the previous subsections:where , , and is an appropriate distance. The most common selections of in (18) are called absolute distance and absolute-log distance . The MSADEs of the unknown parameters denoted by , , and can be determined by minimizing the the next function in terms of , and .

Simultaneously, the three following equations are solved:

Similarly, the MSALDEs of the unknown parameters , and denoted by , , and can be obtained by minimizing the function that follows:

The three following equations are solved:

Now, the MSADE and MSALDE of a system can be obtained, respectively:

2.4. Least Square and Weighted Least Square Estimation

Swain et al. [32] proposed the least squares and weighted least squares estimation methods for estimating the Beta distribution parameters. Let be the order statistics of a random sample of size from EIW and let be the order statistics of a random sample of size from EIW . The least square estimations (LEs) of the unknown parameters , and denoted by , , and can be obtained by minimizing the following function with respect to , and as follows:

Instead of minimizing (18), the estimates , , and can be obtained by simultaneously solving the three following equations:

Upon obtaining the estimates , , and , the LSE of can be obtained as follows:

Similarly, the unknown parameters’ WLSEs , and denoted by , , and can be obtained by minimizing the following function:where and . These estimates can also be obtained by simultaneously solving the three following equations:

The WLSE of can be obtained as

2.5. Cramér-von Mises Estimation

Cramér [33] and von Mises [34] introduced the Cramér-von Mises method of estimation to estimate the unknown parameters , and denoted by , , and which are obtained by minimizing the following goodness-of-fit statistic:with respect to , and , where and . These estimates can also be obtained by solving the three following equations simultaneously:

The CME of can be obtained as follows:

2.6. Anderson-Darling Estimation

Another type of minimum distance estimator is the Anderson-Darling estimation, which is obtained by minimizing Anderson-Darling statistics. Right-tail Anderson-Darling estimation (ADEs) statistics were introduced by Luceño [35] as a modification to the Anderson-Darling estimation (ADEs) statistics (RADEs). The unknown parameters of ADEs , and denoted by , , and are obtained by minimizing the following function:with respect to , and . These estimates can also be obtained by solving the three following equations simultaneously:

The ADE of can be obtained, respectively, by

3. Bootstrap Confidence Intervals

There are two confidence intervals for parameters , and in this section, and parametric bootstrap methods will be proposed. Percentile bootstrap (Boot-P) and bias-corrected percentile bootstrap (Boot-BCP) confidence intervals are shown as two distinct parametric confidence intervals. The steps below will show how to estimate the confidence intervals of .

3.1. Boot-P Confidence

(1)Generate samples and to obtain the bootstrap estimates of , and from the original data, where , and are the estimates obtained from the different estimation.(2)Use and to generate a bootstrap sample and and to generate a bootstrap sample .(3)Based on and , obtain the bootstrap estimate of a system , say .(4)Repeat steps 1–3 times to have .(5)Arrange the bootstrap estimates in step 4 in ascending order as .(6)A two-side Boot-P confidence interval of is given by .

3.2. Boot-BCP Confidence Interval

(1)The same steps as (1–4) in Boot-P(2)A two-side Boot-BCP confidence interval for the unknown parameters is given by where where is the CDF of the standard normal distribution, , and can be obtained as follows:

4. Simulation Study

In the simulation section, a Monte Carlo simulation is done to estimate the unknown parameters of EIW distribution to get stress-strength reliability for MLE, MPSE, MSADE, MSALDE, LSE, WLSE, CME, and ADE methods using R-program are described as follows: Step 1: Generate 10000 random samples, in strength variable , the sample size is , and from the EIW distribution, and in stress variable the sample size is , and from the EIW distribution. Step 2: Use the quantile , where and are distributed as EIW for different parameters and different cases of actual parameters values are selected; see Tables 1–3. Case 1: and 3, , and . Case 2: and 2, , and . Case 3: and 1.5, , and . Step 3: The MLE, MPSE, MSADE, MSALDE, LSE, WLSE, CME, and ADE of the model parameters are obtained by solving the nonlinear equations for the stress-strength model. Step 4: The mean and mean square errors (MSE) of the parameters are obtained. Step 5: The length of CI by using bootstrapping of the stress-strength reliability is obtained in Tables 4–6. Step 6: The numerical results of parameters estimation of EIW distribution are listed in Tables 1–3.

The simulation outcomes of point estimation are recorded in Tables 1–3. The following concluding remakes are noticed based on these tables:(i)In some cases, as the sample size of strength increases and for a fixed sample size of stress, the MSEs associated with the parameter estimates decrease for all methods of estimation.(ii)In some cases, as the sample size of stress increases and for a fixed sample size of strength, the MSEs associated with the parameter estimates decrease for all methods of estimation.(iii)In Case 1, as increases and for a fixed sample size of strength and stress, the MSEs of parameters of EIW are increasing for all methods of estimation.(iv)In Case 2, as increases and for a fixed sample size of strength and stress, the MSEs for most of the parameters of EIW are increasing for all methods of estimation.

The simulation outcomes of interval estimation of stress-strength reliability are recorded in Tables 4–6. The following concluding remakes are noticed based on these tables:(i)In some cases, as the sample size of strength increases and for a fixed sample size of stress, the length of CI of stress-strength reliability estimates decreases for all methods of estimation.(ii)In some cases, as the sample size of stress increases and for a fixed sample size of strength, the length of CI of stress-strength reliability estimates decreases for all methods of estimation.(iii)In some cases, as a level of interval increases and for a fixed sample size of stress and strength, the length of CI of stress-strength reliability estimates increases for all methods of estimation.

5. Application of Real Data

In this section, we consider two applications of the stress-strength reliability model by using breaking strengths of jute fiber and carbon fibers data to describe all the details for illustrative purposes. We used Kolmogorov-Smirnov statistics (KSS) with P value to check the fit of the model and standard errors (SE) of estimators.

5.1. Breaking Strengths of Jute Fiber Data

A pair of real data sets are studied for demonstration purposes. The breaking strengths of jute fiber at two different gauge lengths are depicted in these data. Xia et al. [36] used these two data sets in their study, where X represents the breaking strength of jute fiber with a diameter of 10 mm and represents the breaking strength of a 20 mm diameter jute fiber.

The breaking strengths of jute fiber with a gauge length of 10 mm are “X = 693.73, 704.66, 323.83, 778.17, 123.06, 637.66, 383.43, 151.48, 108.94, 50.16, 671.49, 183.16, 257.44, 727.23, 291.27, 101.15, 376.42, 163.40, 141.38, 700.74, 262.90, 353.24, 422.11, 43.93, 590.48, 212.13, 303.90, 506.60, 530.55, and 177.25.”

The breaking strengths of jute fibers with a gauge length of 20 mm are “Y = 71.46, 419.02, 284.64, 585.57, 456.60, 113.85, 187.85, 688.16, 662.66, 45.58, 578.62, 756.70, 594.29, 166.49, 99.72, 707.36, 765.14, 187.13, 145.96, 350.70, 547.44, 116.99, 375.81, 581.60, 119.86, 48.01, 200.16, 36.75, 244.53, and 83.55.”

From Table 7, we can see that although the EIW distribution fits the data because the diﬀerence between the values of KSS is very small and the P value is more than 0.05, for more illustration, Figures 2 and 3 show the fitted CDF with empirical CDF, fitted PDF with histogram, and P-P plot for strength and stress, respectively, computed at the estimated parameters of EIW distribution.

The estimates of the parameters model of stress-strength reliability for EIW distribution are obtained in Table 8. MSADE has the smallest SE and the largest reliability.

5.2. Carbon Fibers Data

In this subsection, we look at two data sets and discuss all of the specifics for the sake of illustration. The two data sets were first published by Bader and Priest [37]; and they reflected the GPA strength of single carbon fibers with lengths of 10 mm (Data Set I) and 10 mm (Data Set II), respectively, with sample sizes of n = 63 and m = 69. These data were analyzed previously by Hassan et al. [38]. The following are the data sets: Data Set I (length of 10 mm): X (n = 63): “1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020.” Data Set II (length of 20 mm): Y (m = 69): “1.312, 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 2.14, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 2.511, 2.514, 2.535, 2.554, 2.566, 2.57, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773, 2.800, 2.809, 2.818, 2.821, 2.848, 2.88, 2.954, 3.012, 3.067, 3.084, 3.090, 3.096, 3.128, 3.233, 3.433, 3.585, 3.585.”

From Table 9, we can see that although the EIW distribution fits the carbon fibers data because the diﬀerence between the values of KSS is very small and the P value is more than 0.05, for more illustration, Figures 4 and 5 show the fitted CDF with empirical CDF, fitted PDF with histogram, and P-P plot for strength and stress, respectively, computed at the estimated parameters of EIW distribution.

The estimates of the parameters model of stress-strength reliability for EIW distribution are obtained in Table 10. MSADE has the smallest SE and the largest reliability.

6. Conclusion

In this paper, we assumed that and are two independent EIW distributions with the same scale parameter, and by using eight methods of estimation, we could propose the estimations of the unknown parameters , and and the system of stress-strength parameter . The eight methods of estimations are MLEs, MPSEs, MSADEs, MSALDEs, LSEs, WLSEs, CMEs, and ADEs. The percentile bootstrap and bias-corrected percentile bootstrap confidence intervals which are two parametric bootstrap confidence intervals of were introduced. Breaking strengths of jute fiber and carbon fibers data were used as two real data sets to demonstrate the performance of the unknown parameters , and and the system of stress-strength reliability in practical applications, the goodness of fit of the methods estimators for each real data set was examined using the KSS, and the results were sufficient and satisfactory.

We investigated the proposed point and interval estimates using simulation studies, and they performed admirably for a variety of sample sizes, as evidenced by their MSE and confidence intervals. In both techniques, the MSE decreases as the sample size increases; however, the method of maximum product of spacing outperforms other estimation methods. By comparing the estimators using an extensive Monte Carlo numerical simulation study and analyzing a real-world data set, in all sample cases, the MPSEs method outperformed the MLEs. Overall, simulation results show that the maximum product of spacing methods outperforms the other methods in terms of minimum MSE and confidence interval length in the majority of cases. In terms of minimum confidence interval lengths, Boot‐PCP outperforms Boot‐P confidence intervals.

Data Availability

All the data are included in the manuscript.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study was funded by Taif University Researchers Supporting Project (no. TURSP-2020/279), Taif University, Taif, Saudi Arabia.

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Copyright © 2021 Wael S. Abu El Azm et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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