1. Introduction
The system of a stress-strength model has been used for various purposes across many disciplines, such as engineering, medicine, psychology, and others. Due to the diverse spread of possible application, the system reliability has become a popular topic in research. The concept of stress and strength has been well-defined in the past few decades. Today, numerous researchers have considered the reliability assessment of the stress-strength system by using various inference methods and assumptions for different distributions in literature. The system with strength, X, subjected to stress, Y, is defined as failure when the applied stress is greater than the strength of system. The performance of system is referred to reliability, which can be measured byδ=P(Y<X). Numerous authors have studied the point and interval estimation methods ofδ . For example, Kundu and Gupta [1] studiedδ based on random samples from two independent Weibull distributions. Rezaei et al. [2] investigatedδ utilizing random samples from two independent generalized Pareto (GP) distributions. Kohansal and Rezakhah [3] worked onδ with progressively censored samples from the independent two-parameter Rayleigh distribution, and Akgül and Şenğlu [4] consideredδbased on two type II right censoring ranked set samples from two independent Weibull distributions.
Meanwhile the aforementioned stress-strength model can be extended to a multicomponent stress-strength (MSS) system that consists of k independent and identical strength components and all components are subject to common stress. One example of the MSS systems is the suspension bridge, and its reliability can be measured by the probability that the strengths of a certain number of cables outweigh the stress. The MSS system will continue to survive if at least s(1≤s≤k)of the components survive. It is also referred to as the s-out-of-k system. The reliability of the MSS system is evaluated by the probability,δs,k= P (at least s out of k components’ strength exceed the common stress). Various researchers have studied the inference ofδs,k for the MSS system using different distributions. For example, Rao [5] studiedδs,k for the generalized exponential distribution; Rao et al. [6] investigatedδs,k for the Burr-XII distribution; Hassan and Alohali [7] studiedδs,k for the generalized linear failure rate distribution; and Pak et al. [8] investigatedδs,k for the power Lindley model. Readers can refer Kotz, et al. [9] for a comprehensive review of the MSS.
Progressive Type II Censoring Scheme
In order to collect lifetime data, a life test is completed once a specified time has been reached or all components have failed. Depending on the exact type of life test, censoring schemes may be considered to save the testing time and cost. Censoring refers to the value of an observation is only partially known. Type I censoring is used to terminate the life test when the preassigned schedule time is up and type II censoring is used to terminate the life test when a specified number of lifetimes of components are observed. If components are removed during the life test, the specific censoring scheme is called progressive censoring, see Mann et al. [10].
In this paper, the progressive type II censoring scheme is considered for evaluating the reliability of an MSS system. The progressive type II censoring scheme can be implemented as follows: n identical units are placed on a life test at the initial time. Once one unit fails,R1units are removed from the surviving components. After the next unit fails,R2units are removed from the surviving components. This process continues until the m-th unit failure is observed and then allRmunits in the life test are removed. It can be shown thatn=m+R1+R2+…+Rm. The progressive type II censoring scheme is denoted by(R1,R2,…,Rm)hear and after. The stress-strength parameter, denoted byδ=P(Y<X) , of a single-component system has been well studied. Kohansal and Rezakhah [3] investigatedδ based on progressively type II censored samples. Lio and Tsai [11] studiedδ based on progressive first failure-censoring samples from the Burr-XII distributions. Rezaei et al. [12] investigatedδbased on progressively censored samples from the GP distributions. Based on our best knowledge, no research working onδs,kfor an MSS system based on progressively type II censored samples is found in literature. In this paper, the strength of the components and the common stress applied to the MSS system are assumed to follow GP distributions, which have different shape parameters and the same scale parameter.
The rest of this paper is organized as follows: the analytic form ofδs,k for an MSS system based on progressively type II censored samples from the GP distributions is presented in Section 2. Section 3 provides the maximum likelihood estimates (MLEs) of the model parameters andδs,k . In Section 4, two procedures for computing the confidence interval of the MSS system reliability are addressed via using the delta method with Fisher information matrix and a bootstrap procedure, respectively. Section 5 presents the design of Monte Carlo simulation and numerical results. Moreover, comprehensive discussions are presented to evaluate the performance of the proposed inference procedures. Conclusions and remarks are given in Section 6.
2. The MSS System Based on Pareto Distribution
Vilfredo Pareto introduced the Pareto distribution in 1897 in an economics text after observing the trend in income data. Since then, the Pareto distribution has been used to model population sizes, environmental extrema, and insurance claims. In the past decades, the Pareto distribution also has been used to characterize lifetime data. It is chiefly suited for modeling events with left-skewed, non-negative data. Pickands introduced the GP distribution in 1975, which was a slightly modified version of the Pareto distribution, see Arnold [13]. The probability density function (PDF) of the GP distribution is defined by
f(x;σ,η)=1σ1−ηxσ1−ηη,x>0,
whereσ>0andη∈(−∞,0). Considering the re-parameterized transformation process by lettingσ=1λαandη=−1α , the PDF of the GP distribution in Equation (1) can be represented by
fX(x;α,λ)=αλ(1+λx)−α−1,x>0,
and the cumulative density function (CDF) of the GP distribution can be presented by
FX(x;α,λ)=1−(1+λx)−α,x>0,
whereα>0is the shape parameter andλ>0 is the scale parameter. The GP distribution with Equations (2) and (3) has been used to investigateδ for a single-component system based on random samples (see Rezaei et al. [2]) and progressively type II censored samples (see Rezaei et al. [12]). In this study, the reliability of an MSS system is investigated based on the GP distribution defined by Equations (2) and (3).
Reliability of MSS
LetX1,X2,…,Xk∼GP(α1,λ)denote the respective strengths of k components in an MSS system andY∼GP(α2,λ)denote the common stress. The reliability of the MSS system is defined by
δs,k=P(atleastsoutofkcomponents′strengthexceedY)=∫−∞∞P(atleastsoutofkcomponents′strengthexceedy|Y=y)dFY(y)=∫−∞∞∑i=skki(1−FX(y))i (FX(y))k−idFY(y)=∑i=skki∫−∞∞ (1−FX(y))i (FX(y))k−idFY(y)=∑i=skki∫0∞ (1−(1−(1+λy)−α1))i (1−(1+λy)−α1)k−i α2λ(1+λy)−α2−1dy=∑i=skki∫0∞ (1+λy)−α1i (1−(1+λy)−α1)k−i α2λ(1+λy)−α2−1dy=α2∑i=skki∫0∞ (1+λy)−α1i−α2−1 (1−(1+λy))−α1(k−i)λdy
Equation (4) can be also represented as the follows,
δs,k=α2 α1∑i=skki∫01 u(−1/α1)(−α1i−α2−1) (1−u)k−i u−1α1−1du=α2 α1∑i=skkiBi+α2 α1,k−i+1
whereBi+α2 α1,k−i+1=∫01 ui+α2 α1−1 (1−u)k−idu. Whenk=1,δ1,1=P(Y<X)=δ. Note here thatδs,kdepends onα1andα2but is free ofλ . Equation (5) can be used to derive the maximum likelihood estimator ofδs,k based on the progressively type II censored samples in Section 3.
3. Maximum Likelihood Estimation
Let{x(1),x(2),…,x(r1)}be the realization of a progressively type II censored sample fromGP(α1,λ)with schemeRx={Rx,1,Rx,2,…,Rx,r1}and{y(1),y(2),…,y(r2)}be a progressively type II censored sample fromGP(α2,λ)with schemeRy={Ry,1,Ry,2,…,Ry,r2}. Then, the likelihood function based onGP(α1,λ),GP(α2,λ)and schemesRxandRyis given as follows,
L≡L(Θ)=C1 ∏i=1r1 fX(x(i);α1,λ)(1−FX(x(i);α1,λ))Rx,i ×C2 ∏j=1r2 fY(y(j);α2,λ)(1−FY(y(j);α2,λ))Ry,j
whereΘ=(α1,α2,λ),C1=n(n−Rx,1−1)(n−Rx,1−Rx,2−2)…(n−Rx,1−…−Rx,r1−1−r1+1)andC2=m(m−Ry,1−1)(m−Ry,1−Ry,2−2)…(m−Ry,1−...−Ry,r1−1−r2+1). Additionally, note that in this case,n=r1+∑i=1r1 Rx,iandm=r2+∑i=1r2 Ry,i . The likelihood function in Equation (6) can be represented as follows,
L=C1∏i=1r1λα1 (1+x(i)λ)−α1−1 (1−(1−(1+λx(i))−α1))Rx,i ×C2∏j=1r2λα2 (1+y(j)λ)−α2−1 (1−(1−(1+λy(j))−α2))Ry,j
and log-likelihood function, ℓ, is given as
ℓ≡ℓ(Θ)=log(C1)+log(C2)+(r1+r2)log(λ)+r1log(α1)+r2log(α2)−∑i=1r1(1+α1(Rx,i+1))log(1+λx(i))−∑j=1r2(1+α2(Ry,j+1))log(1+λy(j))
The MLEs ofα1,α2, andλare denoted asα^1,α^2, andλ^, respectively, and can be obtained as the solution to the following system,
∂ℓ∂α1=r1 α1−∑i=1r1(Rx,i+1)log(1+λx(i))=0
∂ℓ∂α2=r2 α2−∑j=1r2(Ry,j+1)log(1+λy(j))=0
∂ℓ∂λ=r1+r2λ−∑i=1r1(1+α1(Rx,i+1))1+λx(i)−∑j=1r2(1+α2(Ry,j+1))1+λy(j)=0.
From Equations (9)–(11), we obtain
α^1=r1∑i=1r1 (Rx,i+1)log(1+λ^x(i)),
α^2=r2∑j=1r2 (1+Ry,j)log(1+λ^y(j)).
It is important to mention thatλ^is needed to calculate bothα^1andα^2 . For this purpose, Equations (12) and (13) are used to transform Equation (11) into the following equation
r1+r2λ=∑i=1r11+r1∑i=1r1 (Rx,i+1)log(1+λx(i))(Rx,i+1)1+λx(i)+∑j=1r21+r2∑j=1r2 (1+Ry,j)log(1+λy(j))(Ry,j+1)1+λy(j).
Let
1g(λ)=∑i=1r11+r1∑i=1r1 (Rx,i+1)log(1+λx(i))(Rx,i+1)(1+λx(i))(r1+r2)+∑j=1r21+r2∑j=1r2 (1+Ry,j)log(1+λy(j))(Ry,j+1)(1+λy(j))(r1+r2)
Therefore,λ^can be obtained through using the following nonlinear equation
g(λ)=λ.
Becauseλ^ is a fixed point solution to Equation (16), it can be obtained by using the following iterative procedure:
g(λj)=λj+1,
whereλjis the obtained value ofλ^at the jth iteration. The iteration procedure will stop when the difference betweenλjandλj+1is sufficiently small. Then, the average ofλjandλj+1is used as the value ofλ^. An alternative procedure to solveλ^can be obtained through using R procedure uniroot to search the unit root for equationg(λ)−λ=0. The MLE ofδs,k, denoted asδ^s,k, is then obtained by replacingα1andα2 with their respective MLEs into Equation (5):
δ^s,k=α^2 α^1∑i=skkiBi+α^2 α^1,k−i+1
4. Confidence Intervals 4.1. Confidence Interval Based on Fisher Information
Rezaei et al. [12] presented the asymptotic distribution ofα^1,α^2,λ^andδ^1,1based on the progressively type II censored samples. The procedure can be extended to derive an approximate confidence interval ofδs,k. Denote the Fisher information matrix by
I(Θ)=I11I12I13I21I22I23I31I32I33.
Given a progressively type II censored sample,x(1)<x(2)<x(3)<⋯<x(r), fromGP(α,λ)with censoring schemeR={R1,R2,…,Rr} , Kamps and Cramer [14] had provided the PDF ofX(i)as
fX(i) (x)=ci−1∑l=1ial,i α1λ(1+λx)−(α1 γl+1),
whereγl=n−l+1+∑j=lr Rj,ci−1=∏j=1i γjandal,i=∏j=1;j≠li1γj−γifori=1,2,3,⋯,r . Following the same procedure presented by Rezaei et al. [12],Iijfori,j=1,2,3can be obtained by
I11=−E∂2ℓ∂α12=r1 α12
I12=I21=−E∂2ℓ∂α1 α2=0
I22=−E∂2ℓ∂α22=r2 α22
I13=I31=−E∂2ℓ∂α1∂λ=α1λ∑i=1r1(1+Rx,i)∑l=1ial,i ci−1B(2,α1 γl)
I23=I32=−E∂2ℓ∂α2∂λ=α2λ∑j=1r2(1+Ry,j)∑l=1jal,j cj−1B(2,α2 γl)
I33=−E∂2ℓ∂λ2=1λ2{(n+m)−α1∑i=1n[α1(1+ri)+1]ci−1∑k=1iak,iB(3,α1 γk)−α2∑j=1m[α2(1+rj)+1]cj−1∑k=1jak,jB(3,α2 γk)}
Using the delta method, the variance ofδ^s,kcan be approximated byσδ2=∇Ψs,kTI(Θ)−1∇Ψs,k, where∇Ψs,kis the gradient ofδs,kwith respective toΘ. A symmetric1−pconfidence interval can be approximated by
(δ^s,k−zp/2 σ^δ,δ^s,k+zp/2 σ^δ),
whereσ^δ2is the value ofσδ2but replacingΘbyΘ^=(α^1,α^2,λ^).zp/2is the(p/2)-th quantile of the standard normal distribution,N(0,1); that isP(Z>zp/2)=1−Φ(zp/2)=p/2for0<p<1, whereΦ(.)is the CDF ofN(0,1). The computed confidence interval ofδs,kusing the delta method with Fisher information will be denoted by CI-D.
4.2. Bootstrap Confidence Interval
Since the exact sampling distribution ofδ^s,kis unavailable, bootstrapping methodology can be an alternative to develop an approximated confidence interval ofδs,k besides the aforementioned delta method approximation. Bootstrap methodology is implemented based on the resampling procedure. Dr. Bradley Efron is the pioneer to connect the basic statistical concepts and ideas of the bootstrap method in 1979 [15]. The bootstrap procedure can be a nonparametric or parametric procedure. In this study, the parametric percentile bootstrap method is applied to find the confidence interval ofδs,k . The readers can refer to Efron and Tibshirani [16] for more information about the use of bootstrap methods. In this study, progressively type II censored samples from the GP distributions are generated based on the algorithm established by Balakrishnan and Sandhu [17]. The following algorithm is used to implement the parametric percentile bootstrap method:
Step 1
Given a progressively type II censored sample{x(1),x(2),⋯,x(r1)}from theGP(α1,λ)with the censoring scheme ofRx={Rx,1,Rx,2,…,Rx,r1}and a progressively type II censored sample{y(1),y(2),⋯,y(r2)}from theGP(α2,λ)with the censoring scheme ofRy={Ry,1,Ry,2,…,Ry,r2}. Obtain MLEsα^1,α^2andλ^ using the procedure described in Section 3.
Step 2
A bootstrap progressively type II censored sample, denoted by{x(j)*,j=1,2,⋯,r1}, is generated from theGP(α^1,λ^)based on the censoring scheme ofRx={Rx,1,Rx,2,…,Rx,r1}. A bootstrap progressively type II censored sample, denoted by{y(j)*,j=1,2,⋯,r2}, is generated from theGP(α^2,λ^)based on the censoring scheme ofRy={Ry,1,Ry,2,…,Ry,r2}.
Step 3
The bootstrap estimates,α^1*,α^2*andλ^*, forα1,α2andλ are obtained by using the procedure described in Section 3 with the generated bootstrap progressively type II censored samples in Step 2. Then, the bootstrap estimate ofδs,k is obtained by using Equation (5) and replacingα1andα2byα^1*andα^2*, respectively. Denote the obtained bootstrap estimate byδ^s,k*.
Step 4
Repeat steps 2 and 3 N times, where N is a given huge number. The bootstrap sample,{δ^s,kj*,j=1,2,⋯,N}, is collected.
Step 5
The empirical distribution function, denoted byG^, based on the bootstrap sample,{δ^s,kj*,j=1,2,⋯,N}, is obtained. Letδ^s,kBp (x)=G^−1(x)for0<x<1. The100(1−p)%confidence interval ofδs,kis given by
(δ^s,kBp (p/2),δ^s,kBp (1−p/2)).
We denote the obtained confidence interval ofδs,kvia using the parametric percentile bootstrap method by CI-B.
5. Simulation Study
An intensive simulation study is conducted in this section to evaluate the performance of the MLE and two aforementioned confidence interval procedures forδs,k,1≤s≤kand1≤k≤5 , under different progressive type II censoring schemes, respectively. In order to generate progressively type II censored samples, some progressive censoring schemes are adopted. Those selected progressive censoring schemes are similar to those considered by Wu and Kuş [18] and Lio and Tsai [11]. The surviving items can be removed at the initial or end stages, or the removal can be done during the life test. A pair of the GP distributions withΘ=(α1,α2,λ)=(2.5,2.5,1.0)are considered to implement the simulation study. The other parameters in the simulation study are set byn1=n2=n=20,30,50,60andr1=r2=m=5,15,20,30. In the simulation study, we considerRx=Ry=(0,0,0,0,15),(15,0,0,0,0),(3,3,3,3,3)form=5;Rx=Ry=(0,0,…,0,15),(15,0,0,…,0),(3,0,0,3,0,0,…,3,0,0)form=15;Rx=Ry=(0,0,…,0,30),(30,0,…,0,0),(3,0,3,0,…,3,0)form=20; andRx=Ry=(0,0,…,0,20),(20,0,0,…,0),(2,0,0,2,0,0,…,2,0,0)form=30 . The algorithm proposed by Balakrishnan and Sandhu [17] is used to generate progressively type II censored samples forN=10,000simulation runs. During the ith (1≤i≤N) simulation run, the MLE,δ^s,ki, and the correspondent 95% CI-D using Fisher information matrix are obtained, and additionally,N=10,000bootstrap sample observations,δ^s,kj*,j=1,2,⋯,N are generated to develop a 95% bootstrap approximate CI-B. Refer to Section 4.2 for more details regarding Bootstrap approximate CI-B.
Based onδ^s,ki,i=1,2,⋯N, the bias and mean square error (MSE) ofδ^s,kare evaluated by
Bias=110,000∑i=110,000(δ^s,ki −δs,k)
and
MSE=110,000∑i=110,000(δ^s,ki −δs,k)2,
respectively. To evaluate the performance of each confidence interval procedure, the coverage probability (CP) that is defined as the percentage among 10,000 simulated confidence intervals covering the true parameterδs,kis used. Moreover, the average of lower limit and average of upper limit are respectively calculated from the simulated 10,000 lower confidence limits and upper confidence limits for each confidence interval procedure.
Because the exact forms of the MLEs,α^1,α^2andλ^, do not exist, R procedure uniroot is used to search the solution, labeled byλ^, ofλ to Equation (16). Then the MLEs,α^1andα^2can be obtained by pluggingλ^ into Equations (12) and (13), respectively. After the values ofα^1andα^2are obtained, the MLE,δ^s,k, ofδs,kcan be obtained by pluggingα^1andα^2 into Equation (5). Using the delta method with the Fisher information matrix described in Section 4.1, the CI-D ofδs,kcan be obtained for each combination of(n,m)=(20,5), (30,15), (50,30) and (63,30) and different censoring schemes. The above procedure has been implemented for 10,000 simulation runs. The CP of the CI-D is obtained as the percentage of these 10,000 repetitions of simulated confidence intervals that cover the trueδs,k.
The parts of the simulation results from using the delta method are reported in Table 1, which displays the general pattern of all the simulated CI-D. In view of Table 1, we can find that the delta method is too conservative and produces almost all the CPs of CI-D uniformly larger than the nominal level. All the simulated values of CP in Table 1 were higher than 0.99 except for the cell with(n,m)=(50,30) and the removal scheme of (0,0,0,…,0,30). The simulation results in Table 1 indicate that the delta method cannot perform well to obtain a reliable confidence interval ofδs,k . Actually, the proposed maximum likelihood estimation method in Section 3 produces a good point estimate ofδs,k in terms of small bias and MSE. The proposed parametric percentile bootstrap method described in Section 4.2 can replace the delta method to obtain a reliable confidence interval ofδs,k . We will use the simulation results provided in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 to show good quality of the proposed maximum likelihood estimation method and the parametric percentile bootstrap method.
All the simulation results for CI-B show similar numerical behavior. Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 only report a portion of simulation results of the bias, MSE and CI-B for discussion. First, we would like to evaluate the quality of the proposed MLEs. In view of Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, we find that the bias of ofδ^s,kis close to 0 and the MSE ofδ^s,kare also small for all simulation settings. The findings indicate the proposed maximum likelihood estimation method is reliable to obtain the point estimate of the model parameters; and then, the reliable point estimate,δ^s,k, can be obtained by pluggingα^1andα^2 into Equation (5).
Second, we would like to check if the proposed parametric percentile bootstrap method produces a more reliable confidence interval than the delta method does? Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 show that the values of CP in all simulation cases are uniformly closer to the nominal value of 0.95 than that in Table 1. Overall, the simulation results of CI-B for all cases under investigation show that the width of the approximated confidence interval ofδs,kis shorter as the number of components in the life testing increases and the MSE ofδ^s,kdecreases as the number of components in the life testing increases. Hence, using more sample resources for live testing can result in a more reliable point and interval estimation results.
In summary, the proposed maximum likelihood estimation procedure is reliable to obtain the MLEs of the model parameters andδs,k. The delta method is easy to be used to obtain a confidence interval ofδs,k with less computation time than using the parametric percentile bootstrap method. However, the CP of the CI-D could seriously overestimate the nominal confidence level. This fact indicates that the confidence interval based on the delta method is too conservative. The simulation results in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 show that the CP of the parametric percentile bootstrap confidence interval ofδs,kis close to its nominal confidence level. Hence, the parametric percentile bootstrap method outperforms the delta method and can provide a reliable confidence interval forδs,k. On the basis of our findings, we recommend obtaining an approximated confidence interval ofδs,kfor the MSS system by using the parametric percentile bootstrap method based on progressively type II censored samples from the GP distributions.
6. Conclusions
In this paper, the reliability of an MSS system with k identical components is investigated based on progressively type II censored samples from GP distributions. The MSS system is functional if at leasts(1≤s≤k)components’ strength in the system exceed the common stress. The stress distribution and the strength distribution of components are assumed to have different shape parameters and same scale parameter. A maximum likelihood estimation procedure is analytically studied and the Fisher information matrix is obtained. The proposed maximum likelihood estimation procedure is reliable to obtain the MLEs of the model parameters and the stress-strength parameter.
In order to obtain a reliable confidence interval for the stress-strength parameter, the interval inference procedures based on the delta method and the parametric percentile bootstrap method are investigated, and the obtained confidence intervals are denoted by CI-D and CI-B, respectively. The quality of the CI-D and CI-B of the stress-strength parameter are evaluated via using Monte Carlo simulations. Simulation results indicate that the CI-B method outperforms the CI-D method in terms of the coverage probability.
The strength of components and the stress applied to the MSS system are assumed to follow GP distributions, which have different shape parameters and the same scale parameter. The mathematical derivation is very difficult if two GP distributions have different shapes and scale parameters in this study. This topic is beyond the goal of this paper and can be an open question for future study. Moreover, this paper can be a valuable addition to the works of Rezaei, et al. [12] and Rezaei, et al. [2], who considered the Pareto distribution, as well as Pak et al. [8]; Mokhlis and Khames [19]; Rao [5]; and many others who considered reliability inference under various working assumptions. A comprehensive review of all published information regarding this topic should be compiled in the future in hopes of inferring generalized conclusions regarding the reliability of stress-strength systems.
| n | m | Removal Scheme | CP * | Confidence Interval | |
|---|---|---|---|---|---|
| Average Lower Limit | Average Upper Limit | ||||
| (0, 0, 0, 0, 15) | 0.9924 | 0.0091 | 0.9504 | ||
| 20 | 5 | (15, 0, 0, 0, 0) | 0.9931 | 0.0097 | 0.9479 |
| (3, 3, 3, 3, 3) | 0.9927 | 0.0091 | 0.9503 | ||
| (0, 0,…,15) | 0.9988 | 0.0364 | 0.9292 | ||
| 30 | 15 | (15, 0,…,0) | 0.9987 | 0.0389 | 0.9236 |
| (3, 0,0,…,3,0, 0) | 0.9987 | 0.0376 | 0.9263 | ||
| (0, 0, 0,…,0, 30) | 0.9335 | 0.1939 | 0.7815 | ||
| 50 | 30 | (30, 0, 0,…,0, 0) | 0.9995 | 0.0852 | 0.8832 |
| (3, 0, 3,0,…,3, 0) | 0.9995 | 0.0832 | 0.8874 | ||
| 60 | 30 | (3,0,0,…,3, 0, 0) | 0.9995 | 0.0826 | 0.8884 |
Note: * The coverage probability of 95% confidence interval based on the normal approximation.
| n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
|---|---|---|---|---|---|---|---|
| Average | Average | ||||||
| Lower Limit | Upper Limit | ||||||
| (0, 0, 0, 0, 15) | −0.0037 | 0.0072 | 0.9635 | 0.3205 | 0.6588 | ||
| 20 | 5 | (15, 0, 0, 0, 0) | −0.0201 | 0.0197 | 0.9618 | 0.2045 | 0.7334 |
| (3, 3, 3, 3, 3) | −0.0091 | 0.0211 | 0.9573 | 0.2240 | 0.7498 | ||
| (0, 0,…,15) | −0.0068 | 0.0215 | 0.9551 | 0.2285 | 0.7541 | ||
| 30 | 15 | (15, 0,…,0) | −0.0122 | 0.0067 | 0.9663 | 0.3055 | 0.6482 |
| (3,0,0,…,3,0, 0) | −0.0078 | 0.0070 | 0.9651 | 0.3136 | 0.6542 | ||
| (0, 0, 0,…,0, 30) | −0.0041 | 0.0055 | 0.9584 | 0.3433 | 0.6400 | ||
| 50 | 20 | (30, 0, 0,…,0, 0) | −0.0131 | 0.0050 | 0.9585 | 0.3274 | 0.6285 |
| (3, 0, 3,0,…,3, 0) | −0.0071 | 0.0053 | 0.9602 | 0.3379 | 0.6364 | ||
| (0, 0, 0, …, 0, 20) | −0.0042 | 0.0035 | 0.9611 | 0.3677 | 0.6139 | ||
| 30 | (20, 0, 0, 0,…,0) | −0.0113 | 0.0034 | 0.9567 | 0.3574 | 0.6062 | |
| (2, 0, 0,…,2, 0,0) | −0.0078 | 0.0035 | 0.9595 | 0.3628 | 0.6105 | ||
| 60 | 30 | (3,0,0,…,3, 0, 0) | −0.0068 | 0.0035 | 0.9598 | 0.3644 | 0.6118 |
Note: * The coverage probability of 95% confidence interval based on the percentile bootstrap method.
| n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | ||
|---|---|---|---|---|---|---|---|---|
| Average | Average | |||||||
| Lower Limit | Upper Limit | |||||||
| 20 | 5 | (0, 0, 0, 0, 15) | −0.0205 | 0.0267 | 0.9551 | 0.3242 | 0.8912 | |
| (15, 0, 0, 0, 0) | −0.0338 | 0.0255 | 0.9618 | 0.2922 | 0.8772 | |||
| (3, 3, 3, 3, 3) | −0.0227 | 0.0264 | 0.9573 | 0.3184 | 0.8884 | |||
| 30 | 15 | (0, 0,…,15) | −0.0084 | 0.0090 | 0.9635 | 0.4477 | 0.8216 | |
| (15, 0,…,0) | −0.0174 | 0.0086 | 0.9663 | 0.4285 | 0.8124 | |||
| (3, 0,0,…,3,0, 0) | −0.0128 | 0.0089 | 0.9651 | 0.4389 | 0.8177 | |||
| 50 | 20 | (0, 0, 0,…,0, 30) | −0.0078 | 0.0069 | 0.9584 | 0.4773 | 0.8057 | |
| (30, 0, 0,…,0, 0) | −0.0175 | 0.0065 | 0.9585 | 0.4572 | 0.7953 | |||
| (3, 0, 3,0,…,3, 0) | −0.0110 | 0.0067 | 0.9602 | 0.4705 | 0.8025 | |||
| 30 | (0, 0, 0,…,0, 20) | −0.0068 | 0.0044 | 0.9611 | 0.5087 | 0.7825 | ||
| (20, 0, 0, 0,…,0) | −0.0145 | 0.0043 | 0.9567 | 0.4959 | 0.7750 | |||
| (2, 0, 0,…,2, 0,0) | −0.0107 | 0.0044 | 0.9595 | 0.5025 | 0.7791 | |||
| 60 | 30 | (3,0,0,…,3, 0, 0) | −0.0096 | 0.0045 | 0.9598 | 0.5046 | 0.7804 | |
Note: * The coverage probability of 95% confidence interval based on the percentile bootstrap method.
| n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | ||
|---|---|---|---|---|---|---|---|---|
| Average | Average | |||||||
| Lower Limit | Upper Limit | |||||||
| (0, 0, 0, 0, 15) | 0.0115 | 0.0137 | 0.9551 | 0.0938 | 0.5257 | |||
| 20 | 5 | (15, 0, 0, 0, 0) | −0.0005 | 0.0112 | 0.9618 | 0.0818 | 0.4962 | |
| (3, 3, 3, 3, 3) | 0.0093 | 0.0131 | 0.9573 | 0.0915 | 0.5192 | |||
| (0, 0,…,15) | 0.0271 | 0.0042 | 0.9635 | 0.1386 | 0.3988 | |||
| 30 | 15 | (15, 0,…,0) | −0.0042 | 0.0036 | 0.9663 | 0.1303 | 0.3873 | |
| (3, 0,0,…,3,0, 0) | −0.0006 | 0.0039 | 0.9651 | 0.1348 | 0.3938 | |||
| (0, 0, 0,…,0, 30) | 0.0010 | 0.0031 | 0.9584 | 0.1507 | 0.3776 | |||
| 50 | 20 | (30, 0, 0,…,0, 0) | −0.0062 | 0.0027 | 0.9585 | 0.1417 | 0.3658 | |
| (3, 0, 3,0,…,3, 0) | −0.0013 | 0.0030 | 0.9602 | 0.1470 | 0.3738 | |||
| (0, 0, 0,…,0, 20) | −0.0005 | 0.0020 | 0.9611 | 0.1641 | 0.3500 | |||
| 30 | (20, 0, 0, 0,…,0) | −0.0060 | 0.0018 | 0.9567 | 0.1580 | 0.3427 | ||
| (2, 0, 0,…,2, 0,0) | −0.0032 | 0.0019 | 0.9595 | 0.1611 | 0.3467 | |||
| 60 | 30 | (3,0,0,…,3, 0, 0) | −0.0024 | 0.0019 | 0.9598 | 0.1621 | 0.3480 | |
Note: * The coverage probability of 95% confidence interval based on the percentile Bootstrap method.
| n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
|---|---|---|---|---|---|---|---|
| Average | Average | ||||||
| Lower Limit | Upper Limit | ||||||
| (0, 0, 0, 0, 15) | −0.0125 | 0.0315 | 0.9551 | 0.2637 | 0.8750 | ||
| 20 | 5 | (15, 0, 0, 0, 0) | −0.0279 | 0.0294 | 0.9618 | 0.2347 | 0.8575 |
| (3, 3, 3, 3, 3) | −0.0151 | 0.0310 | 0.9573 | 0.258 | 0.8715 | ||
| (0, 0,…,15) | −0.0057 | 0.0111 | 0.9635 | 0.3754 | 0.786 | ||
| 30 | 15 | (15, 0,…,0) | −0.0161 | 0.0104 | 0.9663 | 0.3569 | 0.7754 |
| (3, 0,0,…,3,0, 0) | −0.0108 | 0.0108 | 0.9651 | 0.3669 | 0.7819 | ||
| (0, 0, 0,…,0, 30) | −0.0059 | 0.0085 | 0.9584 | 0.4035 | 0.7668 | ||
| 50 | 20 | (30, 0, 0,…,0, 0) | −0.0171 | 0.0078 | 0.9585 | 0.3838 | 0.7539 |
| (3, 0, 3,0,…,3, 0) | −0.0096 | 0.0083 | 0.9602 | 0.3968 | 0.7628 | ||
| (0, 0,0,…,0,20) | −0.0058 | 0.0055 | 0.9611 | 0.4338 | 0.7377 | ||
| 30 | (20, 0,0,0,…,0) | −0.0146 | 0.0053 | 0.9567 | 0.4209 | 0.7287 | |
| (2,0,0,…,2,0,0) | −0.0102 | 0.0054 | 0.9595 | 0.4276 | 0.7337 | ||
| 60 | 30 | (3,0,0,…,3,0,0) | −0.0090 | 0.0055 | 0.9598 | 0.4297 | 0.7352 |
Note: * The coverage probability of 95% confidence interval based on the percentile bootstrap method.
| n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
|---|---|---|---|---|---|---|---|
| Average | Average | ||||||
| Lower Limit | Upper Limit | ||||||
| (0, 0, 0, 0, 15) | 0.0131 | 0.0107 | 0.9551 | 0.0726 | 0.4595 | ||
| 20 | 5 | (15, 0, 0, 0, 0) | 0.0021 | 0.0085 | 0.9618 | 0.0630 | 0.4295 |
| (3, 3, 3, 3, 3) | 0.0100 | 0.0102 | 0.9573 | 0.0707 | 0.4529 | ||
| (0, 0,…,15) | 0.0032 | 0.0031 | 0.9635 | 0.1080 | 0.3339 | ||
| 30 | 15 | (15, 0,…,0) | −0.0027 | 0.0026 | 0.9663 | 0.1013 | 0.3231 |
| (3, 0,0,…,3,0, 0) | 0.0003 | 0.0029 | 0.9651 | 0.1050 | 0.3292 | ||
| (0, 0, 0,…,0, 30) | 0.0016 | 0.0023 | 0.9584 | 0.1178 | 0.3139 | ||
| 50 | 20 | (30, 0, 0,…,0, 0) | −0.0046 | 0.0019 | 0.9585 | 0.1104 | 0.3030 |
| (3, 0, 3,0,…,3, 0) | −0.0004 | 0.0022 | 0.9602 | 0.1153 | 0.3105 | ||
| (0, 0, 0,…,0, 20) | <0.0001 | 0.0014 | 0.9611 | 0.1286 | 0.2884 | ||
| 30 | (20, 0, 0, 0,…,0) | −0.0047 | 0.0013 | 0.9567 | 0.1236 | 0.2818 | |
| (2, 0, 0,…,2, 0,0) | −0.0023 | 0.0014 | 0.9595 | 0.1262 | 0.2855 | ||
| 60 | 30 | (3,0,0,…,3, 0, 0) | −0.0016 | 0.0014 | 0.9598 | 0.1270 | 0.2866 |
Note: * The coverage probability of 95% confidence interval based on the percentile bootstrap method.
| n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
|---|---|---|---|---|---|---|---|
| Average | Average | ||||||
| Lower Limit | Upper Limit | ||||||
| (0, 0, 0, 0, 15) | −0.0387 | 0.0242 | 0.9551 | 0.4494 | 0.9689 | ||
| 20 | 5 | (15, 0, 0, 0, 0) | −0.0488 | 0.0244 | 0.9618 | 0.4097 | 0.9635 |
| (3, 3, 3, 3, 3) | −0.0403 | 0.0241 | 0.9573 | 0.4423 | 0.9679 | ||
| (0, 0,…,15) | −0.0145 | 0.0074 | 0.9635 | 0.6034 | 0.9404 | ||
| 30 | 15 | (15, 0,…,0) | −0.0220 | 0.00742 | 0.966 | 0.5814 | 0.9352 |
| (3, 0,0,…,3,0, 0) | −0.0182 | 0.0074 | 0.9651 | 0.5934 | 0.9382 | ||
| (0, 0, 0,…,0, 30) | −0.0124 | 0.0056 | 0.9584 | 0.6382 | 0.9321 | ||
| 50 | 20 | 30, 0, 0,…,0, 0) | −0.0204 | 0.0055 | 0.9585 | 0.6158 | 0.9258 |
| (3, 0, 3,0,…,3, 0) | −0.0150 | 0.0056 | 0.9602 | 0.6308 | 0.9302 | ||
| (0, 0, 0,…,0, 20) | −0.0095 | 0.0036 | 0.9611 | 0.6744 | 0.9188 | ||
| 50 | 30 | (20, 0, 0, 0,…,0) | −0.0161 | 0.0036 | 0.9567 | 0.6605 | 0.9138 |
| (2, 0, 0,…,2, 0,0) | −0.0129 | 0.0036 | 0.9595 | 0.6677 | 0.9166 | ||
| 60 | 30 | (3,0,0,…,3, 0, 0) | −0.0120 | 0.0036 | 0.9598 | 0.6700 | 0.9174 |
Note: * The coverage probability of 95% confidence interval based on the percentile bootstrap method.
| n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
|---|---|---|---|---|---|---|---|
| Average | Average | ||||||
| Lower Limit | Upper Limit | ||||||
| 20 | 5 | (0, 0, 0, 0, 15) | −0.0006 | 0.0309 | 0.9551 | 0.2045 | 0.8179 |
| (15, 0, 0, 0, 0) | −0.0171 | 0.0277 | 0.9618 | 0.1804 | 0.7945 | ||
| (3, 3, 3, 3, 3) | −0.0035 | 0.03024 | 0.9573 | 0.2000 | 0.8131 | ||
| 30 | 15 | (0, 0,…,15) | −0.0016 | 0.0108 | 0.9635 | 0.2965 | 0.7036 |
| (15, 0,…,0) | −0.0122 | 0.0098 | 0.9663 | 0.2805 | 0.6900 | ||
| (3, 0, 0,…,3, 0,0) | −0.0067 | 0.0104 | 0.9651 | 0.2891 | 0.6978 | ||
| 50 | 20 | (0, 0, 0,…,0, 30) | −0.0028 | 0.0083 | 0.9584 | 0.3204 | 0.6794 |
| (30, 0, 0, 0,…,0) | −0.0141 | 0.0074 | 0.9585 | 0.3032 | 0.6647 | ||
| (3, 0, 3, 0,…,3, 0) | −0.0065 | 0.0079 | 0.9602 | 0.3146 | 0.6748 | ||
| 50 | 30 | (0, 0, 0,…,0, 20) | −0.0037 | 0.0053 | 0.9611 | 0.3465 | 0.6457 |
| (20, 0, 0, 0,…,0) | −0.0125 | 0.0050 | 0.9567 | 0.3351 | 0.6358 | ||
| (2, 0, 0,…,2, 0, 0) | −0.0085 | 0.0052 | 0.9595 | 0.3410 | 0.6413 | ||
| 60 | 30 | (3, 0, 0,…,3, 0, 0) | −0.0069 | 0.00532 | 0.9598 | 0.3429 | 0.6430 |
Note: * The coverage probability of 95% confidence interval based on the percentile bootstrap method.
| n | m | Removal Scheme | Bias | MSE | CP * | Confidence Interval | |
|---|---|---|---|---|---|---|---|
| Average | Average | ||||||
| Lower Limit | Upper Limit | ||||||
| 20 | 5 | (0, 0, 0, 0, 15) | 0.0134 | 0.0086 | 0.9551 | 0.0592 | 0.4089 |
| (15, 0, 0, 0, 0) | 0.0034 | 0.0067 | 0.9618 | 0.05128 | 0.3793 | ||
| (3, 3, 3, 3, 3) | 0.0115 | 0.0081 | 0.9573 | 0.05770 | 0.4023 | ||
| 30 | 15 | (0, 0,…,15) | 0.0034 | 0.0024 | 0.9635 | 0.0885 | 0.2873 |
| (15, 0,…,0) | −0.0019 | 0.0020 | 0.9663 | 0.0829 | 0.2773 | ||
| (3, 0,0,…,3,0, 0) | 0.0008 | 0.0022 | 0.9651 | 0.0859 | 0.2830 | ||
| 50 | 20 | (0, 0, 0,…,0, 30) | 0.0018 | 0.0017 | 0.9584 | 0.0967 | 0.2687 |
| (30, 0, 0,…,0, 0) | −0.0036 | 0.0014 | 0.9585 | 0.0905 | 0.2588 | ||
| (3, 0, 3,0,…,3, 0) | <0.0001 | 0.0016 | 0.9602 | 0.0946 | 0.2656 | ||
| 30 | (0, 0, 0,…,0, 20) | 0.0002 | 0.0011 | 0.9611 | 0.1057 | 0.2453 | |
| (20, 0, 0, 0,…,0) | −0.0038 | 0.0010 | 0.9567 | 0.1015 | 0.23942 | ||
| (2, 0, 0,…,2, 0,0) | −0.0017 | 0.0010 | 0.9595 | 0.1037 | 0.2427 | ||
| 60 | 30 | (3,0,0,…,3, 0, 0) | −0.0011 | 0.0010 | 0.9598 | 0.1043 | 0.2437 |
Note: * The coverage probability of 95% confidence interval based on the percentile bootstrap method.
Author Contributions
All authors equally contributed in writing this manuscript. All authors have read and agreed to the published version of the manuscript.
Funding
This study is supported by the grant of Ministry of Science and Technology, Taiwan MOST 108-2221-E-032-018-MY2.
Acknowledgments
The authors would also like to thank for the editing help provided by Dan Van Peursem to improve the paper.
Conflicts of Interest
The authors declare no conflict of interest.
1. Kundu, D.; Gupta, R.D. Estimation of P(Y < X) for Weibull Distributions. IEEE Trans. Reliab. 2006, 55, 270-280.
2. Rezaei, S.; Tahmasbi, R.; Mahmoodi, M. Estimation of P(Y < X) for generalized Pareto distribution. J. Stat. Plan. Inference 2010, 140, 480-494.
3. Kohansal, A.; Rezakhah, S. Inference of R = P(Y < X) for Two-parameter Rayleigh Distribution Based on Progressively Censored Samples. Statistics 2019, 53, 81-100.
4. Akgül, F.G.; Şenğlu, B. Estimation of Stress-Strength Reliability for Weibull Distribution Based on Type-II Right Censored Ranked Set Sampling Data. Pak. J. Stat. Oper. Res. 2018, 14, 781-806.
5. Rao, G.S. Estimation of Reliability in Multicomponent Stress-strength Based on Generalized Exponential Distribution. Rev. Colomb. Estadstíca 2012, 35, 67-76.
6. Rao, G.S.; Aslam, M.; Kundu, D. Burr-XII Distribution Parametric Estimation and Estimation of Reliability of Multicomponent Stress-Strength. Commun. Stat.-Theory Methods 2015, 44, 4953-4961.
7. Hassan, M.K.H.; Alohali, M.I. Estimation of Reliability in a Multicomponent Stress-Strength Model Based on Generalized Linear Failure Rate Distribution. Hacet. J. Math. Stat. 2018, 47, 1634-1651.
8. Pak, A.; Gupta, A.K.; Khoolenjani, N.B. On Reliability in a Multicomponent Stress-Strength Model with Power Lindley Distribution. Rev. Colomb. Estadstíca 2018, 41, 251-267.
9. Kotz, S.; Lumelskii, Y.; Pensky, M. The Stress-Strength Model and Its Generalizations: Theory and Applications; World Scientific: Singapore, 2003.
10. Mann, N.R.; Schafer, R.E.; Singpurwalla, N.D. Methods for Statistical Analysis of Reliability and Life Data; Wiley: New York, NY, USA, 1974.
11. Lio, Y.L.; Tsai, T.-R. Estimation of δ = P(X < Y) for Burr XII distribution based on the progressively first failure-censored samples. J. Appl. Stat. 2011, 39, 309-322.
12. Rezaei, S.; Noughabi, R.A.; Nadarajah, S. Estimation of Stress-Strength Reliability of the Generalized Pareto Distribution Based on Progressively Censored Samples. Ann. Data Sci. 2015, 2, 82-101.
13. Arnold, B.C. Pareto Distribution; Wiley: New York, NY, USA, 2015; pp. 1-10.
14. Kamps, U.; Cramer, E. On Distributions of Generalized Order Statistics. Statistics 2001, 35, 269-280.
15. Chernick, M.R. Bootstrap Methods: A Practitioner's Guide; John Wiley and Son: Hoboken, NJ, USA, 1999.
16. Efron, B.; Tibshirani, R.J. An Introduction to the Bootstrap; Chapman & Hall: London, UK, 1993.
17. Balakrishnan, N.; Sandhu, R.A. A Simple Simulational Algorithm for Generating Progressive Type-II Censored Samples. Am. Stat. 1995, 49, 229-230.
18. Wu, S.-J.; Kuş, C. On the Estimation Based on Progressive First Failure-censored Sample. Comput. Stat. Data Anal. 2009, 53, 3659-3670.
19. Mokhlis, N.A.; Khames, S.K. Reliability of Multi-component Stress-strength Models. J. Egypt. Math. Soc. 2011, 19, 106-111.
Lauren Sauer1, Yuhlong Lio1 and Tzong-Ru Tsai2,*
1Department of Mathematical Sciences, University of South Dakota, Vermillion, SD 57069, USA
2Department of Statistics, Tamkang University, Tamsui District, New Taipei City 251301, Taiwan
*Author to whom correspondence should be addressed.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
© 2020. This work is licensed under http://creativecommons.org/licenses/by/3.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Abstract
In this paper, the reliability of a k-component system, in which all components are subject to common stress, is considered. The multicomponent system will continue to survive if at least s out of k components’ strength exceed the common stress. The system reliability is investigated by utilizing the maximum likelihood estimator based on progressively type II censored samples from generalized Pareto distributions. The confidence interval of the system reliability can be obtained by using asymptotic normality with Fisher information matrix or bootstrap method approximation. An intensive simulation study is conducted to evaluate the performance of maximum likelihood estimators of the model parameters and system reliability for a variety of cases. For the confidence interval of the system reliability, simulation results indicate the bootstrap method approximation outperforms over the asymptotic normality approximation in terms of coverage probability.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Details
