# Estimating System Reliability Using Neoteric and Median RSS Data for Generalized Exponential Distribution

Authors: Amal S. Hassan; Rasha S. Elshaarawy; Ronald Onyango; Heba F. Nagy
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this work, we show how to estimate stress strength (SS) reliability when the stress (Y) and strength (X) distributions are generalized exponentials with a common scale parameter. The SS reliability estimator is considered in view of neoteric ranked set sampling (NRSS) and median ranked set sampling (MRRS). We acquire an estimate of the reliability (R) when such samples of the stress and strength random variables are gathered using the same NRSS technique. Furthermore, the reliability estimator is derived when the stress distribution data are in the pattern of MRSS with just an odd/even set size and the strength distribution data are derived from NRSS and vice versa. The simulation results are used to evaluate and understand the adequacy of a variety of estimators for the suggested schemes. Based on our simulated results, we found that NRSS-based stress strength reliability estimates are more efficient than MRSS-based stress strength reliability estimates. The analysis of real-world data is used to implement the recommended estimators.


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## Body

## 1. Introduction

In the literature, the issue of inferring the SS model, represented byR = P (Y < X), has gotten a lot of attention. Engineering, statistics, and biostatistics are just a few of the disciplines where it is employed. When the stress surpasses the system’s strength, it is self-evident that the system breaks down; otherwise, it will operate normally. The SS reliability (SSR) is calculated in useful applications such as rocket-motor constructions and deterioration, fatigue failure of aircraft structures, and ageing concrete pressure vessels. The SS model was firstly proposed by Birnbaum [1], in which both X and Y follow specific distributions based on full samples, and several researchers have explored the estimate of the SSR model. The reader can refer to Sathe and Shah [2], Constantine et al. [3], Weerahandi and Johnson [4], Grimshaw [5], Saraçoyğlu and Kaya [6], and Krishnamoorthy and Lin [7].To estimate the population mean, McIntyre [8] introduced the ranked set sampling (RSS) strategy for selecting a more informative sample than a simple random sample (SRS). A small number of units can be quickly ranked, either by eye in relation to the research variable or on the basis of an arbitrary function. Once precise sample measurement for a particular unit is complicated, costly, or time-consuming, RSS can indeed be employed, such as environmental management, ecology, sociology, and agriculture. The following is the fundamental principle of selecting a sample in RSS:(i)
Divide the target population inton groups of size b∗ by allocating b∗2 randomly chosen units.(ii)
Despite knowing the values for the variable of interest, rank the units inside every set in order of importance. This could be influenced by personal professional evaluation or a variable that is connected to the variable of interest.(iii)
Take the least ranked unit from the first set, the second smallest ranked unit from the second set, and so on until the biggest ranked unit from the last set is picked for real quantification.(iv)
Repeat this techniquep times to get a sample of size B∗=b∗p for the actual measurement.In the literature, several researchers use RSS to estimate the parameters and SSR of particular distributions. Using RSS, Bhoj [9] estimated the parameters of the extreme value distribution. Fei et al. [10] examined the performance of RSS in the estimate process for more information about RSS. Under unbalanced RSS, Kvam and Samaniego [11] presented an estimate of the population mean and population distribution function. Lam et al. [12] utilized RSS to estimate the exponential distribution’s two parameters. Under various loss functions, Ali [13] provided the Bayesian estimators for the mean residual life function, Lorenz curve, and SSR. Hassan [14] used the RSS technique to explore maximum-likelihood and Bayesian estimators for the generalized exponential distribution (GED). The SSR estimates for Burr XII under some RSS designs were considered by Hassan et al. [15, 16], and Bantan et al. [17] handled the Zubair Lomax parameter estimators under RSS. Safariyan et al. [18] handled the point and interval SSR estimators using RSS. Al-Omari et al. [19] and Akgul et al. [20] derived, respectively, SSR estimators of exponentiated Pareto and generalized inverse Lindley distributions. Almarashi et al. [21] investigated the estimation of R = P[Y < X], in the case where Y and X are two independent Topp–Leone random variables. Bantan et al. [22] introduced and discussed parameter estimators of half logistic inverted Topp–Leone distribution from RSS scheme. Esemen et al. [23] proposed the SSR for the GED using maximum-likelihood (ML) and Bayesian methods via SRS and RSS designs.Several RSS modifications have been proposed in the literature; two important ranking systems are discussed in the following subsections.


### 1.1. NRSS Design

The NRSS system, which varies from the original RSS method, was presented by Zamanzade and Al-Omari [24]. If compared with SRS and RSS methods, this method has been demonstrated to give more accurate estimators for the population mean and variance. It is employed in circumstances where sorting sample observations are considerably easier than getting their precise values. The NRSS system is described in the following way:(i)
Assignn2 randomly selected units from the target population and rank the sample units according to the predetermined ordering criteria.(ii)
Selectn+1/2+i−1nth as the ranking unit for i = 1, …, n if n is odd. If n is even, choose the J+i−1nth ranked unit, where J=n/2 if i is an even and J=n+2/2 if i is an odd for i = 1, …, n.(iii)
Repeat this processp times to investigate a final sample of size N = np.Taconeli and Cabral [25] developed several two-stage sampling strategies based on NRSS and evaluated their performance. Koyuncu and Karagoz [26] used the NRSS architecture in statistical quality control to develop robust control charts. The inverse Weibull distribution parameters were estimated using the NRSS method by Sabry and Shaaban [27].


### 1.2. MRSS Design

The MRSS was proposed by Muttlak [28] as a modification of the RSS to reduce RSS lost efficiency due to ranking errors and to improve the effectiveness of the population mean estimator. The approach is described in the following manner:(i)
Distribute then∗2 randomly chosen units from the target population into n∗ sets, each of size n∗.(ii)
Despite knowing the values for the variable of interest, rank the units within each set in order of importance. It might have been defined as a subjective professional evaluation or a variable that is connected to the variable of interest.(iii)
If the set size is odd, usen∗+1/2th, the lowest rank unit from each sample for real measurements.(iv)
If the number of pieces in the set is even, choose the smallest ranked unit from the firstn∗/2 samples, the n∗/2th for the measurement, and the smallest ranked unit from the second n∗/2 samples, the n∗/2+1th for the measurement.(v)
To acquire a final sample of sizeN∗=n∗p, repeat this method p times.The asymptotic distribution of the parameter estimators of the simple linear regression model using MRSS was studied by Alodat et al. [29]. Al-Nasser and Gogah [30] presented a single sampling approach for ranking data under the assumption of a GED. Using one and two auxiliary variables, Koyuncu [31] developed novel regression estimators in MRSS and NRSS schemes. Hassan et al. [32] discussed the SSR when X and Y are generalized inverted exponential distributions.To date, no research has been published that deal with estimating SSR using the NRSS technique. Here, we address SSR-based estimation when the stressY and the strength X are both independent generalized exponential random variables using NRSS and MRSS schemes. As a result, we do the following:(i)
A comparison of stress strength reliability estimates under the NRSS and MRSS systems is taken into account.(ii)
When the stress and strength random variables are chosen from the same sample design (NRSS/MRSS), this problem is examined.(iii)
When the stress and strength random variables are chosen from various sample designs, this issue is taken into account.(iv)
A simulation study is conducted to validate the theoretical study and two applications to real data are provided.The following sections are arranged in the following order: Section2 discusses the significance of GED and its reliability parameter. The SSR estimator utilizing NRSS is shown in Section 3. When the observed data for X are obtained from MRSS and the observed data for Y are received from NRSS and the opposite, Section 4 produces the SSR estimator. Sections 5 and 6 give the experimental results and data analysis, respectively. Finally, concluding remarks are handled in Section 7.


## 1.1. NRSS Design

The NRSS system, which varies from the original RSS method, was presented by Zamanzade and Al-Omari [24]. If compared with SRS and RSS methods, this method has been demonstrated to give more accurate estimators for the population mean and variance. It is employed in circumstances where sorting sample observations are considerably easier than getting their precise values. The NRSS system is described in the following way:(i)
Assignn2 randomly selected units from the target population and rank the sample units according to the predetermined ordering criteria.(ii)
Selectn+1/2+i−1nth as the ranking unit for i = 1, …, n if n is odd. If n is even, choose the J+i−1nth ranked unit, where J=n/2 if i is an even and J=n+2/2 if i is an odd for i = 1, …, n.(iii)
Repeat this processp times to investigate a final sample of size N = np.Taconeli and Cabral [25] developed several two-stage sampling strategies based on NRSS and evaluated their performance. Koyuncu and Karagoz [26] used the NRSS architecture in statistical quality control to develop robust control charts. The inverse Weibull distribution parameters were estimated using the NRSS method by Sabry and Shaaban [27].


## 1.2. MRSS Design

The MRSS was proposed by Muttlak [28] as a modification of the RSS to reduce RSS lost efficiency due to ranking errors and to improve the effectiveness of the population mean estimator. The approach is described in the following manner:(i)
Distribute then∗2 randomly chosen units from the target population into n∗ sets, each of size n∗.(ii)
Despite knowing the values for the variable of interest, rank the units within each set in order of importance. It might have been defined as a subjective professional evaluation or a variable that is connected to the variable of interest.(iii)
If the set size is odd, usen∗+1/2th, the lowest rank unit from each sample for real measurements.(iv)
If the number of pieces in the set is even, choose the smallest ranked unit from the firstn∗/2 samples, the n∗/2th for the measurement, and the smallest ranked unit from the second n∗/2 samples, the n∗/2+1th for the measurement.(v)
To acquire a final sample of sizeN∗=n∗p, repeat this method p times.The asymptotic distribution of the parameter estimators of the simple linear regression model using MRSS was studied by Alodat et al. [29]. Al-Nasser and Gogah [30] presented a single sampling approach for ranking data under the assumption of a GED. Using one and two auxiliary variables, Koyuncu [31] developed novel regression estimators in MRSS and NRSS schemes. Hassan et al. [32] discussed the SSR when X and Y are generalized inverted exponential distributions.To date, no research has been published that deal with estimating SSR using the NRSS technique. Here, we address SSR-based estimation when the stressY and the strength X are both independent generalized exponential random variables using NRSS and MRSS schemes. As a result, we do the following:(i)
A comparison of stress strength reliability estimates under the NRSS and MRSS systems is taken into account.(ii)
When the stress and strength random variables are chosen from the same sample design (NRSS/MRSS), this problem is examined.(iii)
When the stress and strength random variables are chosen from various sample designs, this issue is taken into account.(iv)
A simulation study is conducted to validate the theoretical study and two applications to real data are provided.The following sections are arranged in the following order: Section2 discusses the significance of GED and its reliability parameter. The SSR estimator utilizing NRSS is shown in Section 3. When the observed data for X are obtained from MRSS and the observed data for Y are received from NRSS and the opposite, Section 4 produces the SSR estimator. Sections 5 and 6 give the experimental results and data analysis, respectively. Finally, concluding remarks are handled in Section 7.


## 2. Generalized Exponential Model

Since 1995, exponentiated distributions have been extensively researched in statistics, with a number of authors developing distinct classes of these distributions. Gupta and Kundu [33] proposed the GED, which is a very appealing extension of the exponential distribution. In the literature, the GED is also known as the exponentiated exponential distribution. It is often used in evaluating lifetime or survival statistics and is of considerable interest. The cumulative distribution function (CDF) and the probability density function of the GED are given, respectively, as follows:(1)Fx;β,φ=1−e−φxβ;x,β,φ>0,(2)fx;β,φ=φβe−φx1−e−φxβ−1;x,β,φ>0,where β and φ are shape and scale parameters, respectively. Many authors have looked into the properties of the GED; for example, see Gupta and Kundu [34], Raqab [35], Zheng and Park [36], Shirke et al. [37], Srivastava et al. [38], and Gupta and Kundu [39].The GED has a wide range of applications. The analysis of Los Angeles rainfall data has been studied by Madi and Raqab [40]. Subburaj et al. [41] suggested software reliability growth models for critical quality measures. Sarhan [42] published GED parameter estimates for a competing risk model based on censored and incomplete data. In addition, the censoring times of 36 small electrical equipments subjected to an automated life test were considered. Biondi et al. [43] investigated models for episode peak and length for eco-hydro-climatic applications. Cota-Felix et al. [44] used the GED to estimate the mean life of power system equipment with limited end-of-life failure data. Kannan et al. [45] investigated the performance of the cure rate model based on GED. They used data from a study on drug treatment programs to demonstrate the model’s performance. Rao [46] suggested a multicomponent SSR estimator of GED with an application to data from an airplane’s air conditioning system. Sadeghpour et al. [47] proposed an SSR estimator based on a lower record ranked set sampling scheme from the GED, and analysis to rocket-motor data was provided.LetX be an GED with parameters β,φ and Y be another GED with parameters γ,φ, where X and Y are independent. The SSR is calculated assuming that the scale parameter in both models is the same but with different shape parameters β and γ; that is, X ∼ GED β,φ and Y ∼ GED γ,φ. Therefore, the SSR is given as follows:(3)R=PY<X=∫0∞Fyxfxdx=∫0∞βφe−φx1−e−φxγ+β−1dx=βγ+β.Expression (3) is a function of parameters β and γ.


## 3. Reliability Estimator Based on NRSS

In this section, we obtain the SSR when bothX and Y are independent GEDs based on NRSS. Suppose that Xηit,i=1,2,…,n;t=1,2,…,px, is selected NRSS from GED β,φ with sample size N = npx. Let Yηja,j=1,2,…,m;a=1,2,…,py, be the chosen NRSS from GED γ,φ with sample size M = mpy. The observed NRSS design (e.g., n = 3) is illustrated in Table 1.Table 1
Design of NRSS scheme.
SampleX1X2X3X4X5X6X7X8X9↓↓↓ObservationX2X5X8The likelihood function (LF) of the observed samples from strengthX via NRSS is given as follows:(4)LL1=∏t=1px∏i=1nC1fxηit∏i=1n+1Fxηit−xηi−1tηi−ηi−1−1,where C1=τ!∏i=1n+1(ηi−ηi−1−1)!−1,η(0) = 0, η(n + 1) = τ + 1, x(η0)=−∞,τ = n2, and xηi+1=∞. Similarly, the LF of the observed samples from stress Y via NRSS is as follows:(5)LL2=∏a=1py∏j=1mC2fyηja∏j=1m+1Fyηja−Fyηj−1aηj−ηj−1−1,where C2=ε!∏j=1m+1(ηj−ηj−1−1)!−1η(0) = 0, η(m + 1) = ε + 1, y(η0)=−∞,ε = m2, and yηj+1=∞. The log LF based on NRSS is offered as follows:(6)InL1∝Nlnφ+lnβ+∑t=1px∑i=1nβ−1ln1−e−φxηi−1t−∑t=1px∑i=1nφxηit−∑a=1py∑j=1mφyηja+∑t=1px∑i=1nηi−ηi−1−1ln1−e−φxηitβ−1−e−φxηi−1tβ+∑a=1py∑j=1mγ−1ln1−e−φyη(j)a+Mlnφ+lnγ+∑a=1py∑j=1mηj−ηj−1−1ln1−e−φyηjaγ−1−e−φyηj−1aγ.The partial derivatives with respect toφ,γ, and β are computed as follows:(7)∂lnL1∂φ=N+Mφ+∑t=1px∑i=1n(β−1)xηiteφxη(it−1−∑t=1px∑i=1nxη(i)t+∑a=1py∑j=1m(γ−1)yηjaeφyηja−1−∑a=1py∑j=1myηja+∑t=1px∑i=1n+1βAi1−ziβ−1(zi)xη(i)t−1−zi−1β−1((zi−1)xηt1−ziβ−1−zi−1β+∑a=1py∑j=1m+1γBj1−wjγ−1wjyηja−1−wj−1γ−1wj−1yηj−1a1−wjγ−1−wj−1γ,where for simplified we write Ai=ηi−ηi−1−1,Bj=ηj−ηj−1−1zi=e−φxηit,zi−1=e−φxηi−1t,wj=e−φyηja, and wj−1=e−φyηj−1a.(8)∂lnL1∂β=Nβ+∑t=1px∑i=1nln1−zi+∑t=1px∑i=1n+1Ai1−ziβln1−zi−1−zi−1βln1−zi−11−ziβ−1−zi−1β,(9)∂lnL1∂γ=Mγ+∑a=1py∑j=1mln1−wj+∑a=1py∑j=1m+1Bj1−wjγln1−wj−1−wj−1γln1−wj−11−wjγ−1−wj−1γ.To produce ML estimators of the population parameters, equations (7) and (8), and (9) are quantitatively determined using an iterative approach. Consequently, the SSR estimator is obtained using (3).


## 4. Reliability Estimator Using NRSS versus MRSS

Herein, we estimate the SSR where bothX and Y are independent GEDs viz. MRSS and NRSS. Four different estimators are derived: the first and second SSR estimators are derived when the collected samples of stress random variable are generated from NRSS and strength random variable is generated from MRSS for odd set size (MRSSO) and vice versa. The third and fourth SSR estimators are calculated where the collected data of stress are taken from NRSS and strength data are drawn from MRSS for even set size (MRSSE) and conversely.


### 4.1. ML Estimators ofR=PXMRSSO<YNRSS

Suppose thatXigt,i=1,2,…,n∗;t=1,2,…,px;g=n∗+1/2th, is a MRSSO observed from GED β,φ distribution, with sample size N∗=n∗px. The observed MRSSO is illustrated in Table 2.Table 2
MRSS for odd set size scheme.
ObservationsSelected MRSSOX11…X1g…X1n∗X1g⋮…⋮…⋮⋮Xg−11…Xg−1g…Xg−1n∗Xg−1gXg1…Xgg…Xgn∗Xgg⋮…⋮…⋮⋮Xn∗−11…Xn∗−1g…Xn∗−1n∗Xn∗−1gXn∗1…Xn∗g…Xn∗n∗Xn∗gLetYηja,j=1,…,m,a=1,…,py, be a NRSS observed from GED γ,φ distribution, with sample size M = mpy. The LF of observed data would be as described as follows:(10)L2=∏t=1px∏i=1n∗C3Fxigtg−11−Fxigtg−1fxigt,∏a=1py∏j=1mC2f(yηja)∏j=1m+1Fyηja)−F(yη(j−1)aηj−η(j−1)−1,where C3=n∗!/g−1!2. The log LF of the GED β,φ and GED γ,φ distributions based on NRSS and MRSSO schemes is as follows:(11)lnL2∝N∗lnβ+∑t=1px∑i=1n∗βg−1ln1−e−φxigt+∑t=1px∑i=1n∗g−1ln1−1−e−φxigtβ−∑t=1px∑i=1n∗φxigt+M+N∗lnφ+∑a=1py∑j=1mγ−1ln1−wj−∑a=1py∑j=1mφyηja+Mlnγ+∑a=1py∑j=1m+1Bjln1−wjγ−1−wj−1γ.The partial derivative ofγ is derived in (9). The partial derivatives of φ and β are as follows:(12)∂lnL2∂φ=N∗+Mφ+∑t=1px∑i=1n∗βg−1xigteφxi(g)t−1−xigt+∑a=1py∑j=1mγ−1yηjawj−1−1−yηja−∑t=1px∑i=1n∗βg−11−e−φxigtβ−1e−φxi(g)txigt1−1−e−φxigtβ+∑a=1py∑j=1m+1γBj1−wjγ−1wjyηja−1−wj−1γ−1wj−1yηj−1a1−wjγ−1−wj−1γ,(13)∂lnL2∂β=N∗β+∑t=1px∑i=1n∗gln1−e−φxigt−g−1ln1−e−φxigt1−e−φxigt−β−1.The ML estimators ofγ,φ, and β are derived by equating (9), (12), and (13) by zero. Despite the fact that the suggested estimators cannot be stated in closed forms, they may be easily produced using a numerical method. As a result, we can obtain the SSR estimator from (3) using the invariance property.


### 4.2. ML Estimator ofR=PXNRSS<YMRSSO

Suppose thatXηit,i=1,2,…,n;t=1,2,…,px, is a NRSS observed from GED β,φ, with sample size N = npx. Let Yjka,j=1,2,…,m∗,a=1,2,…,py;k=m∗+1/2th, be a MRSSO observed from GED γ,φ, with size M∗=m∗py. The log LF of observed data, say L3, would be as described as follows:(14)L3=∏t=1px∏i=1nC1fxηit∏i=1n+1Fxηit−Fxηi−1tAi∏a=1py∏j=1m∗C4Fyjkak−11−Fyjkak−1fyjka,where C4=m∗!/k−1!2. The log LF of the X and Y distributions that depend on NRSS and MRSSO schemes is as follows:(15)lnL3∝Nlnφ+lnβ+∑t=1px∑i=1nβ−1ln1−e−φxηit−φxηit+∑t=1px∑i=1n+1Ailn1−ziβ−1−zi−1β+M∗lnφ+lnγ+∑a=1py∑j=1m∗γk−1ln1−e−φyjka+∑a=1py∑j=1m∗k−1ln1−1−e−φyjkaγ−φyjka.The partial derivative with respect toβ is derived in (8). The partial derivatives with respect to φ and γ are as follows:(16)∂lnL3∂φ=N+M∗φ+∑t=1px∑i=1nβ−1xηiteφxηit−1−xηit+∑t=1px∑i=1n+1βAi1−ziβ−1(zixη(i)t−1−zi−1β−1zi−1xηi−1t1−ziβ−1−zi−1β+∑a=1py∑j=1m∗γk−1yjkaeφyjka−1−yjka−∑a=1py∑j=1m∗γk−11−e−φyjkaγ−1e−φyjkayjka1−1−e−φyjkaγ,(17)∂lnL3∂γ=M∗γ+∑a=1py∑j=1m∗kln1−e−φyjka−∑a=1py∑j=1m∗k−1ln1−e−φyjka1−e−φyjka−γ−1.Equating (8), (16), and (17) by zero, the ML estimators of φ,β, and γ are obtained. Although the suggested estimators are not in closed forms, they may be easily produced using a numerical approach. Hence, we obtain the SSR estimator by inserting the ML estimators of β and γ in (3).


### 4.3. ML Estimator ofR=PXMRSSE<YNRSS

Suppose thatXiut,i=1,2,…,n∗;t=1,2,…,px;u=n∗/2th, is a MRSSE observed from GED β,φ, with sample size N∗=n∗px. The observed MRSSE is illustrated in Table 3.Table 3
MRSS with even set size scheme.
ObservationSelected MRSSEX11…X1uX1u+1…X1n∗X1u⋮…⋮⋮…⋮⋮Xu1…XuuXuu+1…Xun∗XuuXu+11…Xu+1uXu+1u+1…Xu+1n∗Xu+1u+1⋮…⋮⋮…⋮⋮Xn∗1…Xn∗uXn∗u+1…Xn∗n∗Xn∗u+1LetYηja,j=1,2,…,m,a=1,2,…,py, be a NRSS observed from GED γ,φ distribution, with sample size M = mpy. The LF of sample, denoted by L4, would be as described as follows:(18)L4=∏t=1px∏i=1uC5Fxiutu−11−Fxiutufxiut∏t=1px∏i=u+1n∗C5Fxiu+1tu1−Fxiu+1t)u−1fxiu+1t∏a=1py∏j=1mC2fyη(ja∏j=1m+1Fyηja−Fyηj−1aBj],where C5=n∗!/u−1!u!. The log LF of the GED β,φ and GED γ,φ based on MRSSE and NRSS schemes is as follows:(19)lnL4∝N∗lnφ+lnβ+∑t=1px∑i=1uβu−1ln1−e−φxiut+∑t=1px∑i=1uuln1−1−e−φxiutβ−∑t=1px∑i=1uφxiut+∑i=u+1n∗φxiu+1t+∑t=1∑i=u+1n∗βu−1+βln1−e−φxiu+1t+∑t=1px∑i=u+1n∗u−1ln1−1−e−φxiu+1tβ+Mlnφ+lnγ+∑a=1py∑j=1mγ−1ln1−wj−φyηj)a+∑a=1py∑j=1m+1Bjln1−wjγ−1−wj−1γ.The partial derivative ofγ is derived in (9). The partial derivatives of φ and β are as follows:(20)∂lnL4∂φ=N∗+Mφ+∑t=1px∑i=1uβu−1xiuteφxiut−1−xiut−βue−φxiutxiut1−e−φxiutβ−11−1−e−φxiutβ+∑t=1px∑i=u+1n∗βu+β−1xiu+1teφxiu+1t−1−xiu+1t+∑a=1py∑j=1mγ−1yηjaeφyηja−1−∑t=1px∑i=u+1n∗βu−1e−φxiu+1txiu+1t1−e−φxiu+1tβ−11−1−e−φxiu+1tβ−∑a=1py∑j=1myηja+∑a=1py∑j=1m+1γBj1−wjγ−1wjyηja−1−wj−1γ−1wj−1yηj−1a1−wjγ−1−wj−1γ,(21)∂lnL4∂β=N∗β+∑t=1px∑i=1uuln1−e−φxiut+∑t=1px∑i=u+1n∗u+1ln1−e−φxiu+1t−∑t=1px∑i=1uuln1−e−φxi0t1−e−φxiut−β−1−∑t=1px∑i=u+1n∗u−1ln1−e−φxiu+1t1−e−φxiu+1t−β−1.Equating (9), (20), and (21) by zero, the ML estimators of γ, φ, and β are obtained. Despite the fact that the suggested estimators cannot be stated in closed forms, they may be easily produced using a numerical method. Consequently, we can get the reliability estimator using (3).


### 4.4. ML Estimator ofR=PXNRSS<YMRSSE

Suppose thatXηit,i=1,2,...,n;t=1,2,...,px, is a NRSS observed from GED β,φ, with sample size N = npx. Let Yjva,j=1,2,…,m∗,a=1,2,…,py;v=m∗/2th, be a MRSSE observed from GED γ,φ, with sample size M∗=m∗py. The observed LF, say L5, would be as described as follows:(22)L5=∏t=1px∏i=1nC1fxηit∏i=1n+1Fxηit−Fxηi−1tAi]∏a=1py∏j=1vC6Fyjvav−11−Fyjvavfyjva,∏a=1py∏j=v+1m∗C6Fyjv+1av1−Fyjv+1av−1fyjv+1a.where C6=m∗!/v−1!v!. The log LF of X and Y distributions based on NRSS and MRSSE is as follows:(23)lnL5∝Nlnφ+lnβ+∑t=1px∑i=1nβ−1ln1−e−φxηit−∑t=1px∑i=1nφxηit+∑t=1px∑i=1n+1Ai[ln1−ziβ−1−zi−1β+∑a=1py∑j=1vγv−1ln1−e−φyjva+∑a=1py∑j=1vvln1−1−e−φyjvaγ−∑a=1py∑j=1vφyjva−∑j=v+1m∗φyjv+1a+M∗lnφ+lnγ+∑a=1py∑j=v+1m∗γv−1+γln1−e−φyjv+1a+∑a=1py∑j=v+1m∗v−1ln1−1−e−φyjv−1aγ.The partial derivative ofβ is given in (8). The partial derivatives of φ and γ are as follows:(24)∂lnL5∂φ=N+M∗φ+∑t=1px∑i=1nβ−1xηiteφxηit−1−xηit−∑a=1py∑j=1vyjva+∑j=v+1m∗yjv+1a+∑t=1px∑i=1n+1βAi1−ziβ−1zixη(i)t−1−zi−1β−1zi−1xηi−1t1−ziβ−1−zi−1β+∑a=1py∑j=1vγv−1yjvaeφyjva−1−∑j=v+1m∗γv−11−e−φyjv+1aγ−1e−φyjv+1ayjv+1a1−1−e−φyjv+1aγ−∑a=1py∑j=1vγv1−e−φyjvaγ−1e−φyjvayjva1−1−e−φyjvaγ+∑j=v+1m∗γv+γ−1yjv+1aeφyjv+1a−1,(25)∂lnL5∂γ=M∗γ+∑a=1py∑j=1vvln1−e−φyjva+∑a=1py∑j=v+1m∗v+1ln1−e−φyjv+1a−∑a=1py∑j=1vvln1−e−φyjva1−e−φyjva−γ−1+∑j=v+1m∗v−1ln1−e−φyjv+1a1−e−φyjv+1a−γ−1.After setting (8), (24), and (25) to zero and solving them numerically, we get the ML estimators of φ,β, and γ. It is straightforward to create a reliability estimator of R using the invariance property.


## 4.1. ML Estimators ofR=PXMRSSO<YNRSS

Suppose thatXigt,i=1,2,…,n∗;t=1,2,…,px;g=n∗+1/2th, is a MRSSO observed from GED β,φ distribution, with sample size N∗=n∗px. The observed MRSSO is illustrated in Table 2.Table 2
MRSS for odd set size scheme.
ObservationsSelected MRSSOX11…X1g…X1n∗X1g⋮…⋮…⋮⋮Xg−11…Xg−1g…Xg−1n∗Xg−1gXg1…Xgg…Xgn∗Xgg⋮…⋮…⋮⋮Xn∗−11…Xn∗−1g…Xn∗−1n∗Xn∗−1gXn∗1…Xn∗g…Xn∗n∗Xn∗gLetYηja,j=1,…,m,a=1,…,py, be a NRSS observed from GED γ,φ distribution, with sample size M = mpy. The LF of observed data would be as described as follows:(10)L2=∏t=1px∏i=1n∗C3Fxigtg−11−Fxigtg−1fxigt,∏a=1py∏j=1mC2f(yηja)∏j=1m+1Fyηja)−F(yη(j−1)aηj−η(j−1)−1,where C3=n∗!/g−1!2. The log LF of the GED β,φ and GED γ,φ distributions based on NRSS and MRSSO schemes is as follows:(11)lnL2∝N∗lnβ+∑t=1px∑i=1n∗βg−1ln1−e−φxigt+∑t=1px∑i=1n∗g−1ln1−1−e−φxigtβ−∑t=1px∑i=1n∗φxigt+M+N∗lnφ+∑a=1py∑j=1mγ−1ln1−wj−∑a=1py∑j=1mφyηja+Mlnγ+∑a=1py∑j=1m+1Bjln1−wjγ−1−wj−1γ.The partial derivative ofγ is derived in (9). The partial derivatives of φ and β are as follows:(12)∂lnL2∂φ=N∗+Mφ+∑t=1px∑i=1n∗βg−1xigteφxi(g)t−1−xigt+∑a=1py∑j=1mγ−1yηjawj−1−1−yηja−∑t=1px∑i=1n∗βg−11−e−φxigtβ−1e−φxi(g)txigt1−1−e−φxigtβ+∑a=1py∑j=1m+1γBj1−wjγ−1wjyηja−1−wj−1γ−1wj−1yηj−1a1−wjγ−1−wj−1γ,(13)∂lnL2∂β=N∗β+∑t=1px∑i=1n∗gln1−e−φxigt−g−1ln1−e−φxigt1−e−φxigt−β−1.The ML estimators ofγ,φ, and β are derived by equating (9), (12), and (13) by zero. Despite the fact that the suggested estimators cannot be stated in closed forms, they may be easily produced using a numerical method. As a result, we can obtain the SSR estimator from (3) using the invariance property.


## 4.2. ML Estimator ofR=PXNRSS<YMRSSO

Suppose thatXηit,i=1,2,…,n;t=1,2,…,px, is a NRSS observed from GED β,φ, with sample size N = npx. Let Yjka,j=1,2,…,m∗,a=1,2,…,py;k=m∗+1/2th, be a MRSSO observed from GED γ,φ, with size M∗=m∗py. The log LF of observed data, say L3, would be as described as follows:(14)L3=∏t=1px∏i=1nC1fxηit∏i=1n+1Fxηit−Fxηi−1tAi∏a=1py∏j=1m∗C4Fyjkak−11−Fyjkak−1fyjka,where C4=m∗!/k−1!2. The log LF of the X and Y distributions that depend on NRSS and MRSSO schemes is as follows:(15)lnL3∝Nlnφ+lnβ+∑t=1px∑i=1nβ−1ln1−e−φxηit−φxηit+∑t=1px∑i=1n+1Ailn1−ziβ−1−zi−1β+M∗lnφ+lnγ+∑a=1py∑j=1m∗γk−1ln1−e−φyjka+∑a=1py∑j=1m∗k−1ln1−1−e−φyjkaγ−φyjka.The partial derivative with respect toβ is derived in (8). The partial derivatives with respect to φ and γ are as follows:(16)∂lnL3∂φ=N+M∗φ+∑t=1px∑i=1nβ−1xηiteφxηit−1−xηit+∑t=1px∑i=1n+1βAi1−ziβ−1(zixη(i)t−1−zi−1β−1zi−1xηi−1t1−ziβ−1−zi−1β+∑a=1py∑j=1m∗γk−1yjkaeφyjka−1−yjka−∑a=1py∑j=1m∗γk−11−e−φyjkaγ−1e−φyjkayjka1−1−e−φyjkaγ,(17)∂lnL3∂γ=M∗γ+∑a=1py∑j=1m∗kln1−e−φyjka−∑a=1py∑j=1m∗k−1ln1−e−φyjka1−e−φyjka−γ−1.Equating (8), (16), and (17) by zero, the ML estimators of φ,β, and γ are obtained. Although the suggested estimators are not in closed forms, they may be easily produced using a numerical approach. Hence, we obtain the SSR estimator by inserting the ML estimators of β and γ in (3).


## 4.3. ML Estimator ofR=PXMRSSE<YNRSS

Suppose thatXiut,i=1,2,…,n∗;t=1,2,…,px;u=n∗/2th, is a MRSSE observed from GED β,φ, with sample size N∗=n∗px. The observed MRSSE is illustrated in Table 3.Table 3
MRSS with even set size scheme.
ObservationSelected MRSSEX11…X1uX1u+1…X1n∗X1u⋮…⋮⋮…⋮⋮Xu1…XuuXuu+1…Xun∗XuuXu+11…Xu+1uXu+1u+1…Xu+1n∗Xu+1u+1⋮…⋮⋮…⋮⋮Xn∗1…Xn∗uXn∗u+1…Xn∗n∗Xn∗u+1LetYηja,j=1,2,…,m,a=1,2,…,py, be a NRSS observed from GED γ,φ distribution, with sample size M = mpy. The LF of sample, denoted by L4, would be as described as follows:(18)L4=∏t=1px∏i=1uC5Fxiutu−11−Fxiutufxiut∏t=1px∏i=u+1n∗C5Fxiu+1tu1−Fxiu+1t)u−1fxiu+1t∏a=1py∏j=1mC2fyη(ja∏j=1m+1Fyηja−Fyηj−1aBj],where C5=n∗!/u−1!u!. The log LF of the GED β,φ and GED γ,φ based on MRSSE and NRSS schemes is as follows:(19)lnL4∝N∗lnφ+lnβ+∑t=1px∑i=1uβu−1ln1−e−φxiut+∑t=1px∑i=1uuln1−1−e−φxiutβ−∑t=1px∑i=1uφxiut+∑i=u+1n∗φxiu+1t+∑t=1∑i=u+1n∗βu−1+βln1−e−φxiu+1t+∑t=1px∑i=u+1n∗u−1ln1−1−e−φxiu+1tβ+Mlnφ+lnγ+∑a=1py∑j=1mγ−1ln1−wj−φyηj)a+∑a=1py∑j=1m+1Bjln1−wjγ−1−wj−1γ.The partial derivative ofγ is derived in (9). The partial derivatives of φ and β are as follows:(20)∂lnL4∂φ=N∗+Mφ+∑t=1px∑i=1uβu−1xiuteφxiut−1−xiut−βue−φxiutxiut1−e−φxiutβ−11−1−e−φxiutβ+∑t=1px∑i=u+1n∗βu+β−1xiu+1teφxiu+1t−1−xiu+1t+∑a=1py∑j=1mγ−1yηjaeφyηja−1−∑t=1px∑i=u+1n∗βu−1e−φxiu+1txiu+1t1−e−φxiu+1tβ−11−1−e−φxiu+1tβ−∑a=1py∑j=1myηja+∑a=1py∑j=1m+1γBj1−wjγ−1wjyηja−1−wj−1γ−1wj−1yηj−1a1−wjγ−1−wj−1γ,(21)∂lnL4∂β=N∗β+∑t=1px∑i=1uuln1−e−φxiut+∑t=1px∑i=u+1n∗u+1ln1−e−φxiu+1t−∑t=1px∑i=1uuln1−e−φxi0t1−e−φxiut−β−1−∑t=1px∑i=u+1n∗u−1ln1−e−φxiu+1t1−e−φxiu+1t−β−1.Equating (9), (20), and (21) by zero, the ML estimators of γ, φ, and β are obtained. Despite the fact that the suggested estimators cannot be stated in closed forms, they may be easily produced using a numerical method. Consequently, we can get the reliability estimator using (3).


## 4.4. ML Estimator ofR=PXNRSS<YMRSSE

Suppose thatXηit,i=1,2,...,n;t=1,2,...,px, is a NRSS observed from GED β,φ, with sample size N = npx. Let Yjva,j=1,2,…,m∗,a=1,2,…,py;v=m∗/2th, be a MRSSE observed from GED γ,φ, with sample size M∗=m∗py. The observed LF, say L5, would be as described as follows:(22)L5=∏t=1px∏i=1nC1fxηit∏i=1n+1Fxηit−Fxηi−1tAi]∏a=1py∏j=1vC6Fyjvav−11−Fyjvavfyjva,∏a=1py∏j=v+1m∗C6Fyjv+1av1−Fyjv+1av−1fyjv+1a.where C6=m∗!/v−1!v!. The log LF of X and Y distributions based on NRSS and MRSSE is as follows:(23)lnL5∝Nlnφ+lnβ+∑t=1px∑i=1nβ−1ln1−e−φxηit−∑t=1px∑i=1nφxηit+∑t=1px∑i=1n+1Ai[ln1−ziβ−1−zi−1β+∑a=1py∑j=1vγv−1ln1−e−φyjva+∑a=1py∑j=1vvln1−1−e−φyjvaγ−∑a=1py∑j=1vφyjva−∑j=v+1m∗φyjv+1a+M∗lnφ+lnγ+∑a=1py∑j=v+1m∗γv−1+γln1−e−φyjv+1a+∑a=1py∑j=v+1m∗v−1ln1−1−e−φyjv−1aγ.The partial derivative ofβ is given in (8). The partial derivatives of φ and γ are as follows:(24)∂lnL5∂φ=N+M∗φ+∑t=1px∑i=1nβ−1xηiteφxηit−1−xηit−∑a=1py∑j=1vyjva+∑j=v+1m∗yjv+1a+∑t=1px∑i=1n+1βAi1−ziβ−1zixη(i)t−1−zi−1β−1zi−1xηi−1t1−ziβ−1−zi−1β+∑a=1py∑j=1vγv−1yjvaeφyjva−1−∑j=v+1m∗γv−11−e−φyjv+1aγ−1e−φyjv+1ayjv+1a1−1−e−φyjv+1aγ−∑a=1py∑j=1vγv1−e−φyjvaγ−1e−φyjvayjva1−1−e−φyjvaγ+∑j=v+1m∗γv+γ−1yjv+1aeφyjv+1a−1,(25)∂lnL5∂γ=M∗γ+∑a=1py∑j=1vvln1−e−φyjva+∑a=1py∑j=v+1m∗v+1ln1−e−φyjv+1a−∑a=1py∑j=1vvln1−e−φyjva1−e−φyjva−γ−1+∑j=v+1m∗v−1ln1−e−φyjv+1a1−e−φyjv+1a−γ−1.After setting (8), (24), and (25) to zero and solving them numerically, we get the ML estimators of φ,β, and γ. It is straightforward to create a reliability estimator of R using the invariance property.


## 5. Numerical Study

We devised a simulation scenario to investigate and compare the behavior of SSR estimates derived from various sampling techniques. Measures such as the absolute bias (AB=R^−R), the mean squared error (MSE), and the relative efficiency (RE) criteria are utilized to check the validity of estimates. The set sizes of X and Y are taken as (n, m) = (n∗,m∗) = (2, 2), (4, 4), (6, 6), (3, 3), (5, 5), and (7, 7), and number of cycles are chosen as px = py = 5. Consequently, the sizes of the samples are N=npx,M=mpy = N∗=n∗px,M∗=m∗py = (10, 10), (20, 20), (30, 30), (15, 15), (25, 25), and (35, 35). The values of parameters are selected as β,γ = (0.5, 0.4), (4, 0.5), and (5, 0.4) and φ = 1, and therefore, the exact value of R is equal to 0.5556, 0.8889, and 0.925. 1000 random samples from X ∼ GED β,φ and Y ∼ GED γ,φ are generated. The relative efficiency of SSR estimates of the RSS with respect to NRSS is defined as follows:(26)RE1=MSER^XMRSS<YNRSSMSER^XNRSS<YNRSS,RE2=MSER^XNRSS<YMRSSMSER^XNRSS<YNRSS.The values of ABs, MSEs, and REs of the SSR estimate are listed in Tables4 and 5 for specific values of (n, m), n∗,m∗, and distribution parameters.Table 4
Some measures of SSR estimates based on MRSSO and NRSS.
R = 0.5556(n, m) n∗,m∗XNRSS<YNRSSXMRSS<YNRSSRE1XNRSS<YMRSSRE2ABMSEABMSEABMSE(3, 3)0.05090.00530.06180.00751.420.06760.00991.87(5, 5)0.05000.00330.05960.00511.550.07880.00772.33(7, 7)0.04870.00290.05590.00401.380.05810.00411.41R = 0.8889(3, 3)0.06500.00430.06910.00491.140.08450.00721.67(5, 5)0.05450.00300.06420.00431.430.07810.00612.03(7, 7)0.04760.00230.05920.00371.610.06260.00391.70R = 0.925(3, 3)0.05230.00270.00370.00361.330.06680.00451.67(5, 5)0.03870.00150.01860.00261.730.06410.00412.73(7, 7)0.02690.00090.04160.00192.110.05380.00293.22Table 5
Some measures of SSR estimates based on MRSSE and NRSS.
R = 0.5556(n, m) n∗,m∗XNRSS<YNRSSXMRSS<YNRSSRE1XNRSS<YMRSSRE2ABMSEABMSEABMSE(4, 4)0.04710.00380.06710.00631.660.14300.02275.97(6, 6)0.04850.00300.05300.00571.900.12640.01705.67R = 0.8889(2, 2)0.0710.00540.07650.00631.170.09510.00911.69(4, 4)0.05570.00320.07530.00591.840.08330.00692.16(6, 6)0.05160.00270.06830.00521.930.07570.00572.11R = 0.925(2, 2)0.00050.00350.00250.00481.370.06340.00631.80(4, 4)0.04050.00170.04410.00342.000.07260.00563.29(6, 6)0.03360.00120.05440.0032.500.06130.00383.17Some conclusions, based on Tables4-5 and Figures 1−9, are summarized as follows:(i)
The SSR estimates whenX and Y random variables are selected from NRSS are more efficient than the others under different ranking schemes, that is, XMRSS,YNRSS and (XNRSS,YMRSS) for chosen values of sample sizes and true value of R (see, e.g., Figures 1−2 and Tables 4−5).(ii)
The MSE of reliability estimates decreases with increasing set size, in all schemes (see Figures1−2 and Tables 4−5).(iii)
The MSE of SSR estimates decreases with increasedR values in all ranked schemes (see Figure 3 and Tables 4−5).(iv)
The MSE of SSR estimates takes the smallest values based on NRSS for all set sizes (see Figures4−3 and Tables 4−5).(v)
The MSE ofR^=PXMRSS<YNRSS is smaller than the corresponding for R^=PXNRSS<YMRSS at all set sizes (see Figures 5−6 and Tables 4−5).(vi)
R^=PXMRSS<YNRSS is more efficient than R^=PXNRSS<YMRSS at all set sizes (see Figures 7−8 and Tables 4−5).(vii)
The MSE of theR^=PXMRSS<YNRSS in case of odd set size is smaller than the corresponding in case of even set size at all cases (see Figure 9, Tables 4 and 5).(viii)
The values of AB decrease as the sample size increases in most sampling schemes (see Tables4 and 5).(ix)
For all even set sizes, the AB ofR^=PXNRSS<YMRSS decreases as the value of R grows (see Table 5).(x)
At set size (5, 5), the AB ofR^=PXNRSS<YNRSS decreases as the value of R increases (see Table 4).(xi)
The AB ofR^=PXNRSS<YNRSS takes the smallest values in comparison with the others, i.e., R^=PXMRSS<YNRSS,R^=PXNRSS<YMRSS for odd set size (see Table 4).(xii)
For all true values ofR at (2, 2), (4, 4), and (6, 6), the AB of R^=PXNRSS<YNRSS receives the lowest values compared with the other sampling designs (see Table 5).Figure 1
MSE ofR^ at (R) = 0.5556.Figure 2
MSE ofR^ at R = 0.8889.Figure 3
MSE ofR^ at R = 0.925.Figure 4
MSE ofR^ at different values of R.Figure 5
RE of SSR estimates atR = 0.5556.Figure 6
RE of SSR estimates atR = 0.8889.Figure 7
MSE of SSR estimate based on MRSS and NRSS atR = 0.556.Figure 8
MSE of SSR estimate based on NRSS and MRSS atR = 0.925.Figure 9
MSE of SSR estimate forR^=PXMRSS<YNRSS at R = 0.556.


## 6. Data Analysis

In this section, two applications were utilized to verify the proposed estimators explored in the simulation investigation.


### 6.1. First Application

For illustration purposes, we take two data sets and illustrate all of the information. The two data sets were first published via Xia et al. [48], which represent the breaking strength of jute fibers at 5 mm, 10 mm, 15 mm, and 20 mm gauge lengths. We used the data sets about the (Tables 6, 7) breaking strengths of jute fiber at gauge lengths of 5 and 15 mm with sizes of the two samples equal to 30. The data sets are scheduled as follows.Table 6
Data set 1 (5 mm) MPa.
566.31270.79516.48823.03226.53367.7185.42441.87618.57546.11268.2315.33809.23218.86583.97304.84129.08537.45496.28167.87306.99178.25370.02168.2554.61360.8260.97254.29495.51187.68Table 7
Data set 2 (15 mm) MPa.
594.4202.75168.37574.86225.6576.38156.67127.81813.87562.39468.47135.0972.24497.94355.56569.07640.48200.76550.42748.75489.66678.06457.71106.73716.342.6680.4339.2270.09193.42We verify the effectiveness of the fitted model using the Kolmogorov–Smirnov (KS) goodness-of-fit test and itsP value (PV). The KS distance between the empirical distribution and the fitted distribution is 0.1375 with PV = 0.622 for the first data set and 0.1828 with PV = 0.269 for the second data set, indicating satisfactory fits. The estimated densities, estimated CDF, PP plots, and empirical survival for the considered data appear in Figures 10 and 11.(1)
LetX∼GEDβ,φ1 and Y∼GEDγ,φ2, the estimates of β,φ1,γ, and φ2 are as follows: β^ = 29.464, φ^1 = 0.987, γ^ = 2.102, and φ^2 = 0.246, and the log-likelihood value is lnℓ1 = 402.381.(2)
LetX∼GEDβ,φ and Y∼GEDγ,φ, the estimates of β,γ, and φ are as follows: β^= 11.91, γ^= 3.651, and φ^= 0.990, and the associated log-likelihood value is lnℓ2 = 402.6. The following hypotheses are examined.(27)H0:φ1=φ2vs.H1:φ1≠φ2,and in this case, −2ℓ2−ℓ1 = 0.238, and hence, the null hypothesis is accepted; i.e., the assumption of φ1=φ2 is justified.Figure 10
Plots of density, CDF, and survival of the GED for data set 1.Figure 11
Plots of density, CDF, and survival of the GED for data set 2.The NRSS and MRSS designs are applied to chosen data sets based on previous theoretical outcomes. Table8 shows SSR estimates from the GED using suggested sampling techniques for some selected values of n,n∗,m∗, and m using 5 repeated cycles. The MATHEMATICA program is used to create NRSS and MRSS samples from the data sets that have been evaluated.Table 8
SSR estimate of the breaking strength of jute fiber data.
(n, m) n∗,m∗XNRSS<YNRSSXMRSS<YNRSSXNRSS<YMRSS(2, 2)0.75540.69710.6409(3, 3)0.69320.61390.5560(4, 4)0.62310.57850.5101(5, 5)0.57050.52040.4733


### 6.2. Second Application

Here, we examine two real data sets and show how the recommended approaches may be applied in practice. Lawless [49] provided (Table 9) the first data set, while the second was provided (Table 10) by Linhart and Zucchini [50]. For different lifetime distributions, both are analyzed and fitted. Separately, we fitted the GED to each data set. The first data set reported here is from deep groove ball bearing endurance testing. For each of the 23 ball bearings in the life test, the numbers shown are the number of million revolutions before failure.Table 9
First data set represents the number of revolutions to failure (in millions) of 23 deep-groove ball-bearings.
17.8828.9233.0041.5242.1245.6048.8051.8451.9654.1255.5667.8068.6468.6468.8884.1293.1298.64105.12105.84127.92128.04173.40Table 10
Second data set represents 30 failure times of the air conditioning system of an airplane (in hours).
2326187712014624722571246214220512120113147111141116901165295We use the KS goodness-of-fit test and its PV to validate the effectiveness of the fitted model. For the first data, the KS distance between the empirical and fitted distributions is 0.10583, with PV = 0.959. The KS distance in the second data set is 0.17197, with PV = 0.338. Figures12 and 13 show the estimated densities, estimated CDF, PP plots, and empirical survival for both the data under consideration.Figure 12
Plots of density, CDF, and survival of the GED for data set 1.Figure 13
Plots of density, CDF, and survival of the GED for data set 2.LetX∼GEDβ,φ1 and Y∼GEDγ,φ2, the estimates of β,φ1,γ, and φ2 are as follows: β^ = 22.879, φ^1 = 0.9364, γ^ = 0.1675, and φ^2=0.2356, and the log-likelihood value is lnℓ1=−265.177.LetX∼GEDβ,φ and Y∼GEDγ,φ, the estimates of β,γ, and φ are as follows: β^ = 5.327, γ^ = 0.3681, and φ^ = 0.9584, and the associated log-likelihood value is lnℓ2 = −265.235. The following hypotheses are examined:(28)H0:φ1=φ2vs.H1:φ1≠φ2,and in this case, −2ℓ2−ℓ1 = 0.116, and hence, the null hypothesis is accepted; i.e., the assumption of φ1=φ2 is justified.Based on earlier theoretical results, the NRSS and MRSS designs are applied to selected data sets. Table11 gives the SSR estimates from the GED via proposed sampling methods for some selected values of n,n∗,m∗, and m using 5 replicated cycles. The MATHEMATICA program is used to generate NRSS and MRSS samples from considered data sets.Table 11
ML estimator ofR based on different sampling designs for the second application.
(n, m) n∗,m∗XNRSS<YNRSSXMRSS<YNRSSXNRSS<YMRSS(2, 2)0.7964540.6434750.6145(3, 3)0.7154960.6054680.5985(4, 4)0.6264580.58430.5537(5, 5)0.5969010.5302220.5093


## 6.1. First Application

For illustration purposes, we take two data sets and illustrate all of the information. The two data sets were first published via Xia et al. [48], which represent the breaking strength of jute fibers at 5 mm, 10 mm, 15 mm, and 20 mm gauge lengths. We used the data sets about the (Tables 6, 7) breaking strengths of jute fiber at gauge lengths of 5 and 15 mm with sizes of the two samples equal to 30. The data sets are scheduled as follows.Table 6
Data set 1 (5 mm) MPa.
566.31270.79516.48823.03226.53367.7185.42441.87618.57546.11268.2315.33809.23218.86583.97304.84129.08537.45496.28167.87306.99178.25370.02168.2554.61360.8260.97254.29495.51187.68Table 7
Data set 2 (15 mm) MPa.
594.4202.75168.37574.86225.6576.38156.67127.81813.87562.39468.47135.0972.24497.94355.56569.07640.48200.76550.42748.75489.66678.06457.71106.73716.342.6680.4339.2270.09193.42We verify the effectiveness of the fitted model using the Kolmogorov–Smirnov (KS) goodness-of-fit test and itsP value (PV). The KS distance between the empirical distribution and the fitted distribution is 0.1375 with PV = 0.622 for the first data set and 0.1828 with PV = 0.269 for the second data set, indicating satisfactory fits. The estimated densities, estimated CDF, PP plots, and empirical survival for the considered data appear in Figures 10 and 11.(1)
LetX∼GEDβ,φ1 and Y∼GEDγ,φ2, the estimates of β,φ1,γ, and φ2 are as follows: β^ = 29.464, φ^1 = 0.987, γ^ = 2.102, and φ^2 = 0.246, and the log-likelihood value is lnℓ1 = 402.381.(2)
LetX∼GEDβ,φ and Y∼GEDγ,φ, the estimates of β,γ, and φ are as follows: β^= 11.91, γ^= 3.651, and φ^= 0.990, and the associated log-likelihood value is lnℓ2 = 402.6. The following hypotheses are examined.(27)H0:φ1=φ2vs.H1:φ1≠φ2,and in this case, −2ℓ2−ℓ1 = 0.238, and hence, the null hypothesis is accepted; i.e., the assumption of φ1=φ2 is justified.Figure 10
Plots of density, CDF, and survival of the GED for data set 1.Figure 11
Plots of density, CDF, and survival of the GED for data set 2.The NRSS and MRSS designs are applied to chosen data sets based on previous theoretical outcomes. Table8 shows SSR estimates from the GED using suggested sampling techniques for some selected values of n,n∗,m∗, and m using 5 repeated cycles. The MATHEMATICA program is used to create NRSS and MRSS samples from the data sets that have been evaluated.Table 8
SSR estimate of the breaking strength of jute fiber data.
(n, m) n∗,m∗XNRSS<YNRSSXMRSS<YNRSSXNRSS<YMRSS(2, 2)0.75540.69710.6409(3, 3)0.69320.61390.5560(4, 4)0.62310.57850.5101(5, 5)0.57050.52040.4733


## 6.2. Second Application

Here, we examine two real data sets and show how the recommended approaches may be applied in practice. Lawless [49] provided (Table 9) the first data set, while the second was provided (Table 10) by Linhart and Zucchini [50]. For different lifetime distributions, both are analyzed and fitted. Separately, we fitted the GED to each data set. The first data set reported here is from deep groove ball bearing endurance testing. For each of the 23 ball bearings in the life test, the numbers shown are the number of million revolutions before failure.Table 9
First data set represents the number of revolutions to failure (in millions) of 23 deep-groove ball-bearings.
17.8828.9233.0041.5242.1245.6048.8051.8451.9654.1255.5667.8068.6468.6468.8884.1293.1298.64105.12105.84127.92128.04173.40Table 10
Second data set represents 30 failure times of the air conditioning system of an airplane (in hours).
2326187712014624722571246214220512120113147111141116901165295We use the KS goodness-of-fit test and its PV to validate the effectiveness of the fitted model. For the first data, the KS distance between the empirical and fitted distributions is 0.10583, with PV = 0.959. The KS distance in the second data set is 0.17197, with PV = 0.338. Figures12 and 13 show the estimated densities, estimated CDF, PP plots, and empirical survival for both the data under consideration.Figure 12
Plots of density, CDF, and survival of the GED for data set 1.Figure 13
Plots of density, CDF, and survival of the GED for data set 2.LetX∼GEDβ,φ1 and Y∼GEDγ,φ2, the estimates of β,φ1,γ, and φ2 are as follows: β^ = 22.879, φ^1 = 0.9364, γ^ = 0.1675, and φ^2=0.2356, and the log-likelihood value is lnℓ1=−265.177.LetX∼GEDβ,φ and Y∼GEDγ,φ, the estimates of β,γ, and φ are as follows: β^ = 5.327, γ^ = 0.3681, and φ^ = 0.9584, and the associated log-likelihood value is lnℓ2 = −265.235. The following hypotheses are examined:(28)H0:φ1=φ2vs.H1:φ1≠φ2,and in this case, −2ℓ2−ℓ1 = 0.116, and hence, the null hypothesis is accepted; i.e., the assumption of φ1=φ2 is justified.Based on earlier theoretical results, the NRSS and MRSS designs are applied to selected data sets. Table11 gives the SSR estimates from the GED via proposed sampling methods for some selected values of n,n∗,m∗, and m using 5 replicated cycles. The MATHEMATICA program is used to generate NRSS and MRSS samples from considered data sets.Table 11
ML estimator ofR based on different sampling designs for the second application.
(n, m) n∗,m∗XNRSS<YNRSSXMRSS<YNRSSXNRSS<YMRSS(2, 2)0.7964540.6434750.6145(3, 3)0.7154960.6054680.5985(4, 4)0.6264580.58430.5537(5, 5)0.5969010.5302220.5093


## 7. Concluding Remarks

In this study, we look atR = P[Y < X], for SSR estimation, assuming that strength X and stress Y are independent generalized exponential random variables. Under the MRSS and NRSS systems, the SSR estimators are considered in five cases. When both the stress and strength random variables have the same sample design NRSS, one of them is considered, whereas the other is used when the X and Y distributions have dissimilar sampling designs. We use the simulation research to evaluate and compare various behavior estimates using a set of criteria. In all issues, the MSE of SSR estimates using NRSS yielded the least values compared with the others based on MRSS data, according to the study’s outcomes. In general, SSR estimates that use both stress and strength data from the NRSS are more efficient than those that use strength and stress data from the MRSS. Furthermore, the SSR estimates when X is MRSS and Y is NRSS data are superior to the other in the reversed case (when X is NRSS and Y is MRSS data). Real data applications illustrate these results.



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*Source: 2608656-2022-03-31.xml*