# Statistical Inference in Dependent Component Hybrid Systems with Masked Data

Authors: Naijun Sha; Ronghua Wang; Ping Hu; Xiaoling Xu
Journal: Advances in Statistics
 (2015)

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## Abstract

Complex systems are usually composed of simple hybrid systems. In this paper,
we consider statistical inference for two fundamental hybrid systems: series-parallel
and parallel-series systems based on masked data. Assuming dependent lifetimes of
components modelled by Marshall and Olkin’s bivariate exponential distribution in
the system, we present maximum likelihood and interval estimation of parameters of
interest. Intensive simulation studies are performed to demonstrate the efficiency of
the methods.


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## Body

## 1. Introduction

In a system consisting of several components, the reliability analyses are usually made by analyzing lifetime data. The system data includes two parts: (i) the system’s lifetime and (ii) the failure reason, that is, which component causes system failure. In real situations, however, some things may prevent systems from revealing the failure reason such as shortage of funds, limit of time, error of records, lack of diagnostic tools, and destructive consequences caused by the failure of some components. For example, in the reliability problems of computers and integrated circuits, the reason for the system failure is often attributed to a module containing several components, but one could not determine exactly which component causes the system failure. Therefore, the observable data from the test includes the failure time and failure reason related to a subset of components. In these cases, the reason for the failure of the system is masked and the lifetime data is called masked data.The statistical analysis of masked data has a long history. Usher and Hodgson [1] initially proposed the parameter estimation under masked data. Since then, a significant amount of literature has emerged on various models. In the series system with constant, linear and polynomial failure components in the presence of masked data, the maximum likelihood (ML) and other estimation methods were studied among many researchers (e.g., [2–6]). Sarhan and El-Bassiouny [7] considered a parallel system using masked data. Bayes methods with various priors were also used for the estimation of parameters in series and parallel systems (see, e.g., Sarhan [8–10], Jiang and Zhang [11]). El-Gohary [12] discussed a series system with two dependent components in a Bayesian approach. So far, most researches of masked data focused on a system with either series or parallel only and assumed independent and identical component lifetime in the system. In many real situations, however, a “hybrid” system is often seen in which the working components are connected in a way of joining together with series and parallel. For example, currently, air supply systems generally are modular designed, where the power system consists of a number of semiconductor units combined in a series or hybrid method [13, 14].Complex systems are usually composed of simple subsystems such as three-component series-parallel and parallel-series systems illustrated in Figure1. In this paper, we mainly focus on statistical inferences of the two fundamental hybrid systems, in which the component lifetimes are nonindependent and nonidentically distributed. For the two systems, first we note that the system failure occurrence is attributed to one of the four failures consisting of components 1, 2, 3, and 12, where 12 denotes the occurrence of components 1 and 2 failure simultaneously. Let S be the set of all nine events causing the system failure; that is, (1)S=1,2,3,12,1,2,1,3,2,3,12,3,1,2,3.If s∈S consists of more than one element, then the reason of the system failure is not exact and the life data is masked. Notice that here we differentiate two occurrences {1,2} and {12} by assuming different independent processes damaging component 1 only, component 2 only, and both components in the next section. We make statistical inference of parameters on likelihood-based methods in the presence of masked data. Section 2 presents the life distribution and reliability for the hybrid systems. Section 3 concentrates on the parameter estimation for the series-parallel and parallel-series systems, respectively. In Section 4, we assess the performance of the methods on simulation studies. Lastly, we conclude the paper with a brief discussion in Section 5.Figure 1
Hybrid systems of three components.
(a)
Series-parallel system
(b)
Parallel-series system


## 2. Model Specification

For the three-component hybrid system in Figure1, there is a subsystem consisting of components 1 and 2. From a practical viewpoint, the lifetimes of the components in the subsystem are usually dependent on each other and independent of component 3 outside the subsystem. The unit lifetime model is addressed in the following.


### 2.1. Life Distribution

A bivariate model is developed by Marshall and Olkin [15] to describe the correlated lifetimes of two units and is widely used in two-component system. Basically, it was assumed that two-component system is affected by “fatal shocks” governed by three different independent Poisson processes with parameters λ1, λ2, and λ12, according to the shocks damage component 1 only, component 2 only, and both components, respectively. Particularly, in the hybrid system of Figure 1, the lifetimes T1 and T2 of the units 1 and 2 are constructed through Ti=min(Zi,Z12), i=1,2, where Z1, Z2, and Z12 are mutually independent random variables with Zi~exp(λi), i=1,2, and Z12~exp(λ12). Then, (T1,T2) follows a bivariate exponential distribution Biexp(λ1,λ2,λ12), whose joint reliability function is (2)F¯12t1,t2=exp⁡-λ1t1-λ2t2-λ12max⁡t1,t2,t1>0,t2>0and the joint density function is (3)f12t1,t2=λ2λ1+λ12exp⁡-λ1+λ12t1-λ2t2,t1>t2λ1λ2+λ12exp⁡-λ2+λ12t2-λ1t1,t1<t2.The probability of both components failure at time t corresponds to the mass probability of singular part p12(t)=P(T1=T2=t)=P(Z1≥t,Z2≥t,Z12=t)=λ12e-(λ1+λ2+λ12)t, t>0. The component 3 is shocked by another independent Poisson process with parameter λ, and so its lifetime is exponentially distributed with the density f3(t3)=λe-λt3, t3>0, λ>0.


### 2.2. Reliability and Density of Hybrid System

First we briefly introduce the concept of masked probability. Assume that there is a masked events∈S with the exact failure case K in the hybrid system; then the probability of failure due to the masked occurrence s at time t is (4)Pt<T≤t+dt,S=s=∑j∈sPt<T≤t+dt,S=s,K=j=∑j∈sPt<T≤t+dt,K=j·PS=s∣t<T≤t+dt,K=j,where P(S=s∣t<T≤t+dt,K=j) is called masked probability and P(t<T≤t+dt,K=j) is the probability of system failure caused by component(s) j at the time t, j=1,2,3,12. In statistical analysis of masked data, it is usually assumed that the masked occurrence is independent of the cause and failure time; that is, (5)PS=s∣t<T≤t+dt,K=j=PS=s=c.The lifetime of the series-parallel system in Figure 1(a) is T=min(max(T1,T2),T3). With the assumption that (T1,T2)~Biexp(λ1,λ2,λ12) and an independent T3~exp(λ), the reliability at time t is (6)PT>t=Pmax⁡T1,T2>tPT3>t=1-PT1≤t,T2≤tF¯3t=1-F12t,tF¯3t=1-∬0tf12t1,t2dt1dt2-∫0tp12sdsF¯3t=e-λ+λ1+λ2+λ12teλ1t+eλ2t-1,t>0and the probability densities of failure at time t due to each event are (7)P1=Pt<T≤t+dt,K=1=Pt<T1≤t+dt,T2<tPT3>t=∫0tf12t,t2dt2×F¯3t=λ1+λ12e-λ+λ1+λ12t1-e-λ2t,P2=Pt<T≤t+dt,K=2=PT1<t,t<T2≤t+dtPT3>t=∫0tf12t1,tdt1×F¯3t=λ2+λ12e-λ+λ2+λ12t1-e-λ1t,P12=Pt<T≤t+dt,K=12=Pt<T1=T2≤t+dtPT3>t=p12tF¯3t=λ12e-λ+λ1+λ2+λ12t,P3=Pt<T≤t+dt,K=3=1-PT1≤t,T2≤tf3t=1-F12t,tf3t=λe-λ+λ1+λ2+λ12teλ1t+eλ2t-1.Likewise, for the parallel-series system with three components as shown in Figure 1(b), the system life becomes T=max⁡(min⁡(T1,T2),T3). Therefore, the reliability is (8)PT>t=1-Pmin⁡T1,T2≤t,T3≤t=1-1-PT1>t,T2>tF3t=1-1-F¯12t,tF3t=e-λt+e-λ1+λ2+λ12t-e-λ+λ1+λ2+λ12t,t>0and the probability densities of failure at time t due to each case are (9)P1=Pt<T≤t+dt,K=1=Pt<T1≤t+dt,T2>tPT3<t=∫t∞f12t,t2dt2×F3t=λ1e-λ1+λ2+λ12t1-e-λt,P2=Pt<T≤t+dt,K=2=PT1>t,t<T2≤t+dtPT3<t=∫t∞f12t1,tdt1×F3t=λ2e-λ1+λ2+λ12t1-e-λt,P12=Pt<T≤t+dt,K=12=Pt<T1=T2≤t+dtPT3<t=p12tF3t=λ12e-λ1+λ2+λ12t1-e-λt,P3=Pt<T≤t+dt,K=3=1-PT1≥t,T2≥tf3t=1-PZ1≥t,Z2≥t,Z12≥tf3t=λe-λt1-e-λ1+λ2+λ12t.Finally, the density function for the system at t due to the masked occurrence s can be expressed as P(t<T≤t+dt,S=s)=∑j∈scPj. The likelihood-based parameter inference for the two hybrid systems is presented in the following.


## 2.1. Life Distribution

A bivariate model is developed by Marshall and Olkin [15] to describe the correlated lifetimes of two units and is widely used in two-component system. Basically, it was assumed that two-component system is affected by “fatal shocks” governed by three different independent Poisson processes with parameters λ1, λ2, and λ12, according to the shocks damage component 1 only, component 2 only, and both components, respectively. Particularly, in the hybrid system of Figure 1, the lifetimes T1 and T2 of the units 1 and 2 are constructed through Ti=min(Zi,Z12), i=1,2, where Z1, Z2, and Z12 are mutually independent random variables with Zi~exp(λi), i=1,2, and Z12~exp(λ12). Then, (T1,T2) follows a bivariate exponential distribution Biexp(λ1,λ2,λ12), whose joint reliability function is (2)F¯12t1,t2=exp⁡-λ1t1-λ2t2-λ12max⁡t1,t2,t1>0,t2>0and the joint density function is (3)f12t1,t2=λ2λ1+λ12exp⁡-λ1+λ12t1-λ2t2,t1>t2λ1λ2+λ12exp⁡-λ2+λ12t2-λ1t1,t1<t2.The probability of both components failure at time t corresponds to the mass probability of singular part p12(t)=P(T1=T2=t)=P(Z1≥t,Z2≥t,Z12=t)=λ12e-(λ1+λ2+λ12)t, t>0. The component 3 is shocked by another independent Poisson process with parameter λ, and so its lifetime is exponentially distributed with the density f3(t3)=λe-λt3, t3>0, λ>0.


## 2.2. Reliability and Density of Hybrid System

First we briefly introduce the concept of masked probability. Assume that there is a masked events∈S with the exact failure case K in the hybrid system; then the probability of failure due to the masked occurrence s at time t is (4)Pt<T≤t+dt,S=s=∑j∈sPt<T≤t+dt,S=s,K=j=∑j∈sPt<T≤t+dt,K=j·PS=s∣t<T≤t+dt,K=j,where P(S=s∣t<T≤t+dt,K=j) is called masked probability and P(t<T≤t+dt,K=j) is the probability of system failure caused by component(s) j at the time t, j=1,2,3,12. In statistical analysis of masked data, it is usually assumed that the masked occurrence is independent of the cause and failure time; that is, (5)PS=s∣t<T≤t+dt,K=j=PS=s=c.The lifetime of the series-parallel system in Figure 1(a) is T=min(max(T1,T2),T3). With the assumption that (T1,T2)~Biexp(λ1,λ2,λ12) and an independent T3~exp(λ), the reliability at time t is (6)PT>t=Pmax⁡T1,T2>tPT3>t=1-PT1≤t,T2≤tF¯3t=1-F12t,tF¯3t=1-∬0tf12t1,t2dt1dt2-∫0tp12sdsF¯3t=e-λ+λ1+λ2+λ12teλ1t+eλ2t-1,t>0and the probability densities of failure at time t due to each event are (7)P1=Pt<T≤t+dt,K=1=Pt<T1≤t+dt,T2<tPT3>t=∫0tf12t,t2dt2×F¯3t=λ1+λ12e-λ+λ1+λ12t1-e-λ2t,P2=Pt<T≤t+dt,K=2=PT1<t,t<T2≤t+dtPT3>t=∫0tf12t1,tdt1×F¯3t=λ2+λ12e-λ+λ2+λ12t1-e-λ1t,P12=Pt<T≤t+dt,K=12=Pt<T1=T2≤t+dtPT3>t=p12tF¯3t=λ12e-λ+λ1+λ2+λ12t,P3=Pt<T≤t+dt,K=3=1-PT1≤t,T2≤tf3t=1-F12t,tf3t=λe-λ+λ1+λ2+λ12teλ1t+eλ2t-1.Likewise, for the parallel-series system with three components as shown in Figure 1(b), the system life becomes T=max⁡(min⁡(T1,T2),T3). Therefore, the reliability is (8)PT>t=1-Pmin⁡T1,T2≤t,T3≤t=1-1-PT1>t,T2>tF3t=1-1-F¯12t,tF3t=e-λt+e-λ1+λ2+λ12t-e-λ+λ1+λ2+λ12t,t>0and the probability densities of failure at time t due to each case are (9)P1=Pt<T≤t+dt,K=1=Pt<T1≤t+dt,T2>tPT3<t=∫t∞f12t,t2dt2×F3t=λ1e-λ1+λ2+λ12t1-e-λt,P2=Pt<T≤t+dt,K=2=PT1>t,t<T2≤t+dtPT3<t=∫t∞f12t1,tdt1×F3t=λ2e-λ1+λ2+λ12t1-e-λt,P12=Pt<T≤t+dt,K=12=Pt<T1=T2≤t+dtPT3<t=p12tF3t=λ12e-λ1+λ2+λ12t1-e-λt,P3=Pt<T≤t+dt,K=3=1-PT1≥t,T2≥tf3t=1-PZ1≥t,Z2≥t,Z12≥tf3t=λe-λt1-e-λ1+λ2+λ12t.Finally, the density function for the system at t due to the masked occurrence s can be expressed as P(t<T≤t+dt,S=s)=∑j∈scPj. The likelihood-based parameter inference for the two hybrid systems is presented in the following.


## 3. Parameter Estimation

In our statistical inference, two common censoring schemes are considered: type-I and type-II. Forn tested systems, through reordering the failure times, we assume that there are rk systems failures due to the kth mechanism in S with the failure times tr¯k-1+1,tr¯k-1+2,…,tr¯k, where r¯k=∑l=1k‍rl, r0=0, k=1,2,…,9. Obviously there are totally r=∑l=19‍rl=r¯9 observed failure times and n-r censored observations. For type-I censoring, the test is continuing until a prespecified time τ is reached and we observed r systems failed; whereas for type-II censoring, the test is carried out until the prespecified rk systems failures for the kth mechanism, and so the test terminated time τ=max⁡(t1,…,tr¯1,tr¯1+1,…,tr¯2,…,tr¯9). For both cases, we express the observed life data D={t1,…,tr¯1,tr¯1+1,…,tr¯2,…,tr¯9,τ}. The corresponding masked failure event is si∈S for the system i and masked probability ci=P(Si=si), i=1,2,…,r. The probability density of system i at ti for each case in (7) and (9) is expressed as Pij=Pj(ti), i=1,2,…,r, j=1,2,3,12, indicating the failure due to the components 1, 2, and 3 and both components 1 and 2, respectively. Finally, the density function for system i at ti becomes ∑j∈siciPij. Therefore, the applicable unified likelihood function for both hybrid systems and censoring schemes is (10)LD=∏i=1r∑j∈siciPijPT>τn-r=C∏k=19∏i=r¯k-1+1r¯k∑j∈siPijPT>τn-r=C∏i=1r¯1Pi1∏i=r¯1+1r¯2Pi2∏i=r¯2+1r¯3Pi3∏i=r¯3+1r¯4Pi12∏i=r¯4+1r¯5Pi1+Pi2·∏i=r¯5+1r¯6Pi1+Pi3∏i=r¯6+1r¯7Pi2+Pi3∏i=r¯7+1r¯8Pi12+Pi3·∏i=r¯8+1rPi1+Pi2+Pi3PT>τn-r,where the constant C=∏i=1rci does not contain the parameters of interest in Pij.For the purpose of simplicity, we only consider two special cases of failure rates: (1) the components were shocked by independent Poisson processes with same parameters; that is,λ1=λ2=λ12=λ; (2) the Poisson processes affecting the three components individually have the same parameters but different from that of the Poisson process applying on components 1 and 2 simultaneously; that is, λ1=λ2=λ≠λ12. The maximum likelihood estimation (MLE) approach will be implemented for the inference. To make notation simpler, we denote the log-likelihood function as l(θ)=log⁡L(θ∣D), where θ is the parameter of failure rates included in the life densities. We also apply the approximated chi-squared likelihood ratio statistic [16] to numerically obtain the confidence intervals of parameters. Particularly, for our case, the likelihood ratio statistic for the parameter Λ=-2logL(θ)/L(θ^) approximately follows χν2 where θ=λ or θ=(λ,λ12) and its MLE θ^, and ν is the dimension of θ. In general, this method works well even for the situation of small sample size; that is, the coverage probability of the constructed interval is very close to the nominal confidence level.


### 3.1. Series-Parallel System

(1)
λ
1
=
λ
2
=
λ
12
=
λ. Based on the reliability in (6) and the densities in (7), the likelihood function (10) becomes (11)Lλ∣D=C∏i=1r¯22λe-3λti-2λe-4λti·∏i=r¯2+1r¯32λe-3λti-λe-4λti·∏i=+¯r3+1r¯4λe-4λti·∏i=r¯4+1r¯54λe-3λti-4λe-4λti·∏i=r¯5+1r¯74λe-3λti-3λe-4λti·∏i=r¯7+1r¯82λe-3λti·∏i=r¯8+1r6λe-3λti-5λe-4λti×e-4n-rλτ2eλτ-1n-r.So, the log-likelihood can be simplified as (12)lλ=log⁡C-4n-rλτ+n-rlog⁡2eλτ-1+∑i=1rlog⁡λ-∑i=1r4λti+∑i=1r¯2log⁡2eλti-2+∑i=r¯2+1r¯3log⁡2eλti-1+∑i=r¯4+1r¯5log⁡4eλti-4+∑i=r¯5+1r¯7log⁡4eλti-3+∑i=r¯7+1r¯8log⁡2eλti+∑i=r¯8+1rlog⁡6eλti-5and its derivative with respect to λ is (13)l′λ=-4n-rτ+2n-rτ2-e-λτ+rλ-∑i=1r4ti+∑i=1r¯2ti1-e-λti+∑i=r¯2+1r¯32ti2-e-λti+∑i=r¯4+1r¯5ti1-e-λti+∑i=r¯5+1r¯74ti4-3e-λti+∑i=r¯7+1r¯8ti+∑i=r¯8+1r6ti6-5e-λti.Since no analytical form of MLE λ^ can be obtained from the equation l′(λ)=0, a numerical method has to be implemented for specific data observations. The uniqueness of MLE can be justified in the following way: the terms involving exponent in l′(λ) can be expressed as a unified functional form g(λ)=a/(b-ce-λt) with positive constants a,b, and c. Since g′(λ)=-acte-λt/(b-ce-λt)2<0, we have (14)l′′λ=-2n-rτ2e-λτ2-e-λτ2-rλ2-∑i=1r¯2ti2e-λti1-e-λti2-∑i=r¯2+1r¯32ti2e-λti2-e-λti2-∑i=r¯4+1r¯5ti2e-λti1-e-λti2-∑i=r¯5+1r¯712ti2e-λti4-3e-λti2-∑i=r¯8+1r30ti2e-λti6-5e-λti2<0.Hence, the log-likelihood function l(λ) is strictly concave and therefore l′(λ)=0 implies a unique MLE λ^. Additionally, limλ→0⁡l′(λ)=∞, limλ→∞⁡l′(λ)=-3(n-r)τ-3∑i=1rti<0, and so the MLE λ^ is a positive value.(2)
λ
1
=
λ
2
=
λ
≠
λ
12. Under this case, the likelihood function (10) reduces to (15)Lλ,λ12∣D=C∏i=1r¯2λ+λ12e-2λ+λ12ti1-e-λti·∏i=r¯2+1r¯3λe-2λ+λ12ti2-e-λti·∏i=r¯3+1r¯4λ12e-3λ+λ12ti·∏i=r¯4+1r¯52λ+λ12e-2λ+λ12ti1-e-λti·∏i=r¯5+1r¯7λ+λ12e-2λ+λ12ti1-e-λti+λe-2λ+λ12ti2-e-λti∏i=r¯7+1r¯8λ12e-3λ+λ12ti+λe-2λ+λ12ti2-e-λti·∏i=r¯8+1r2λ+λ12e-2λ+λ12ti1-e-λti+λe-2λ+λ12ti2-e-λtie-n-r2λ+λ12τ2-e-λτn-rand so the log-likelihood function is (16)lλ,λ12=log⁡C-n-r2λ+λ12τ+n-rlog⁡2-e-λτ-∑i=1r2λ+λ12ti+∑i=1r¯2log⁡λ+λ12+log⁡1-e-λti+∑i=r¯2+1r¯3log⁡λ+log⁡2-e-λti+∑i=r¯3+1r¯4log⁡λ12-λti+∑i=r¯4+1r¯5log⁡2+log⁡λ+λ12+log⁡1-e-λti+∑i=r¯5+1r¯7log⁡3λ+λ12-2λ+λ12e-λti+∑i=r¯7+1r¯8log⁡2λ-λ-λ12e-λti+∑i=r¯8+1rlog⁡4λ+2λ12-3λ+2λ12e-λti.The MLEs λ^, λ^12 can be obtained numerically in the equations ∂l(λ,λ12)/∂λ=∂l(λ,λ12)/∂λ12=0. The existence of MLE is provided in the Appendix.


### 3.2. Parallel-Series System

(1)
λ
1
=
λ
2
=
λ
12
=
λ. Based on the reliability in (8) and the densities in (9), the likelihood function (10) becomes (17)Lλ∣D=C∏i=1r¯2λe-3λti-λe-4λti·∏i=r¯2+1r¯3λe-λti-λe-4λti·∏i=r¯3+1r¯4λe-3λti-λe-4λti·∏i=r¯4+1r¯52λe-3λti-2λe-4λti·∏i=r¯5+1r¯8λe-λti+λe-3λti-2λe-4λti·∏i=r¯8+1rλe-λti+2λe-3λti-3λe-4λti×e-λτ+e-3λτ-e-4λτn-rand then the log-likelihood function is (18)lλ=log⁡C-4n-rλτ+n-rlog⁡e3λτ+eλτ-1+∑i=1rlog⁡λ-∑i=1r4λti+∑i=1r¯2log⁡eλti-1+∑i=r¯2+1r¯3log⁡e3λti-1+∑i=r¯3+1r¯4log⁡eλti-1+∑i=r¯4+1r¯5log⁡2eλti-2+∑i=r¯5+1r¯8log⁡e3λti+eλti-2+∑i=r¯8+1rlog⁡e3λti+2eλti-3.Taking derivative with respect to λ, we obtain (19)l′λ=-4n-rτ+n-r3τ+τe-2λτ1+e-2λτ-e-3λτ+rλ-∑i=1r4ti+∑i=1r¯2ti1-e-λti+∑i=r¯2+1r¯33ti1-e-3λti+∑i=r¯3+1r¯5ti1-e-λti+∑i=r¯5+1r¯83ti+tie-2λti1+e-2λti-2e-3λti+∑i=r¯8+1r3ti+2tie-2λti1+2e-2λti-3e-3λti.Since limλ→0⁡l′(λ)=∞ and limλ→∞⁡l′(λ)=-(n-r)τ-∑i=1r4ti+∑i=1r¯2ti+∑i=r¯2+1r¯33ti+∑i=r¯3+1r¯5ti+∑i=r¯5+1r3ti<0, l′(λ)=0 has a positive root λ^.(2)
λ
1
=
λ
2
=
λ
≠
λ
12. Under this special case, the likelihood function (10) then reduces to (20)Lλ,λ12∣D=C∏i=1r¯2λe-2λ+λ12ti1-e-λti·∏i=r¯2+1r¯3λe-λti1-e-2λ+λ12ti·∏i=r¯3+1r¯4λ12e-2λ+λ12ti1-e-λti·∏i=r¯4+1r¯52λe-2λ+λ12ti1-e-λti·∏i=r¯5+1r¯7λe-2λ+λ12ti1-e-λti+λe-λti1-e-2λ+λ12ti·∏i=r¯7+1r¯8λ12e-2λ+λ12ti1-e-λti+λe-λti1-e-2λ+λ12ti·∏i=r¯8+1r2λe-2λ+λ12ti1-e-λti+λe-λti1-e-2λ+λ12tie-λτ+e-2λ+λ12τ-e-3λ+λ12τn-rand so the log-likelihood function is (21)lλ,λ12=log⁡C-n-rλτ+n-rlog⁡1+e-λ+λ12τ-e-2λ+λ12τ-∑i=1rλti+∑i=1r¯2log⁡λ-λ+λ12ti+log⁡1-e-λti+∑i=r¯2+1r¯3log⁡λ+log⁡1-e-2λ+λ12ti+∑i=r¯3+1r¯4log⁡λ12-λ+λ12·ti+log⁡1-e-λti+∑i=r¯4+1r¯5log⁡2-λ+λ12ti+log⁡λ+log⁡1-e-λti+∑i=r¯5+1r¯7log⁡λe-λ+λ12ti1-e-λti+λ1-e-2λ+λ12ti+∑i=r¯7+1r¯8log⁡λ12e-λ+λ12ti1-e-λti+λ1-e-2λ+λ12ti+∑i=r¯8+1rlog⁡2λe-λ+λ12ti1-e-λti+λ1-e-2λ+λ12ti.The MLEs λ^, λ^12 will be obtained numerically in the equations ∂l(λ,λ12)/∂λ=∂l(λ,λ12)/∂λ12=0. The prove of the existence of MLE is given in the Appendix.


## 3.1. Series-Parallel System

(1)
λ
1
=
λ
2
=
λ
12
=
λ. Based on the reliability in (6) and the densities in (7), the likelihood function (10) becomes (11)Lλ∣D=C∏i=1r¯22λe-3λti-2λe-4λti·∏i=r¯2+1r¯32λe-3λti-λe-4λti·∏i=+¯r3+1r¯4λe-4λti·∏i=r¯4+1r¯54λe-3λti-4λe-4λti·∏i=r¯5+1r¯74λe-3λti-3λe-4λti·∏i=r¯7+1r¯82λe-3λti·∏i=r¯8+1r6λe-3λti-5λe-4λti×e-4n-rλτ2eλτ-1n-r.So, the log-likelihood can be simplified as (12)lλ=log⁡C-4n-rλτ+n-rlog⁡2eλτ-1+∑i=1rlog⁡λ-∑i=1r4λti+∑i=1r¯2log⁡2eλti-2+∑i=r¯2+1r¯3log⁡2eλti-1+∑i=r¯4+1r¯5log⁡4eλti-4+∑i=r¯5+1r¯7log⁡4eλti-3+∑i=r¯7+1r¯8log⁡2eλti+∑i=r¯8+1rlog⁡6eλti-5and its derivative with respect to λ is (13)l′λ=-4n-rτ+2n-rτ2-e-λτ+rλ-∑i=1r4ti+∑i=1r¯2ti1-e-λti+∑i=r¯2+1r¯32ti2-e-λti+∑i=r¯4+1r¯5ti1-e-λti+∑i=r¯5+1r¯74ti4-3e-λti+∑i=r¯7+1r¯8ti+∑i=r¯8+1r6ti6-5e-λti.Since no analytical form of MLE λ^ can be obtained from the equation l′(λ)=0, a numerical method has to be implemented for specific data observations. The uniqueness of MLE can be justified in the following way: the terms involving exponent in l′(λ) can be expressed as a unified functional form g(λ)=a/(b-ce-λt) with positive constants a,b, and c. Since g′(λ)=-acte-λt/(b-ce-λt)2<0, we have (14)l′′λ=-2n-rτ2e-λτ2-e-λτ2-rλ2-∑i=1r¯2ti2e-λti1-e-λti2-∑i=r¯2+1r¯32ti2e-λti2-e-λti2-∑i=r¯4+1r¯5ti2e-λti1-e-λti2-∑i=r¯5+1r¯712ti2e-λti4-3e-λti2-∑i=r¯8+1r30ti2e-λti6-5e-λti2<0.Hence, the log-likelihood function l(λ) is strictly concave and therefore l′(λ)=0 implies a unique MLE λ^. Additionally, limλ→0⁡l′(λ)=∞, limλ→∞⁡l′(λ)=-3(n-r)τ-3∑i=1rti<0, and so the MLE λ^ is a positive value.(2)
λ
1
=
λ
2
=
λ
≠
λ
12. Under this case, the likelihood function (10) reduces to (15)Lλ,λ12∣D=C∏i=1r¯2λ+λ12e-2λ+λ12ti1-e-λti·∏i=r¯2+1r¯3λe-2λ+λ12ti2-e-λti·∏i=r¯3+1r¯4λ12e-3λ+λ12ti·∏i=r¯4+1r¯52λ+λ12e-2λ+λ12ti1-e-λti·∏i=r¯5+1r¯7λ+λ12e-2λ+λ12ti1-e-λti+λe-2λ+λ12ti2-e-λti∏i=r¯7+1r¯8λ12e-3λ+λ12ti+λe-2λ+λ12ti2-e-λti·∏i=r¯8+1r2λ+λ12e-2λ+λ12ti1-e-λti+λe-2λ+λ12ti2-e-λtie-n-r2λ+λ12τ2-e-λτn-rand so the log-likelihood function is (16)lλ,λ12=log⁡C-n-r2λ+λ12τ+n-rlog⁡2-e-λτ-∑i=1r2λ+λ12ti+∑i=1r¯2log⁡λ+λ12+log⁡1-e-λti+∑i=r¯2+1r¯3log⁡λ+log⁡2-e-λti+∑i=r¯3+1r¯4log⁡λ12-λti+∑i=r¯4+1r¯5log⁡2+log⁡λ+λ12+log⁡1-e-λti+∑i=r¯5+1r¯7log⁡3λ+λ12-2λ+λ12e-λti+∑i=r¯7+1r¯8log⁡2λ-λ-λ12e-λti+∑i=r¯8+1rlog⁡4λ+2λ12-3λ+2λ12e-λti.The MLEs λ^, λ^12 can be obtained numerically in the equations ∂l(λ,λ12)/∂λ=∂l(λ,λ12)/∂λ12=0. The existence of MLE is provided in the Appendix.


## 3.2. Parallel-Series System

(1)
λ
1
=
λ
2
=
λ
12
=
λ. Based on the reliability in (8) and the densities in (9), the likelihood function (10) becomes (17)Lλ∣D=C∏i=1r¯2λe-3λti-λe-4λti·∏i=r¯2+1r¯3λe-λti-λe-4λti·∏i=r¯3+1r¯4λe-3λti-λe-4λti·∏i=r¯4+1r¯52λe-3λti-2λe-4λti·∏i=r¯5+1r¯8λe-λti+λe-3λti-2λe-4λti·∏i=r¯8+1rλe-λti+2λe-3λti-3λe-4λti×e-λτ+e-3λτ-e-4λτn-rand then the log-likelihood function is (18)lλ=log⁡C-4n-rλτ+n-rlog⁡e3λτ+eλτ-1+∑i=1rlog⁡λ-∑i=1r4λti+∑i=1r¯2log⁡eλti-1+∑i=r¯2+1r¯3log⁡e3λti-1+∑i=r¯3+1r¯4log⁡eλti-1+∑i=r¯4+1r¯5log⁡2eλti-2+∑i=r¯5+1r¯8log⁡e3λti+eλti-2+∑i=r¯8+1rlog⁡e3λti+2eλti-3.Taking derivative with respect to λ, we obtain (19)l′λ=-4n-rτ+n-r3τ+τe-2λτ1+e-2λτ-e-3λτ+rλ-∑i=1r4ti+∑i=1r¯2ti1-e-λti+∑i=r¯2+1r¯33ti1-e-3λti+∑i=r¯3+1r¯5ti1-e-λti+∑i=r¯5+1r¯83ti+tie-2λti1+e-2λti-2e-3λti+∑i=r¯8+1r3ti+2tie-2λti1+2e-2λti-3e-3λti.Since limλ→0⁡l′(λ)=∞ and limλ→∞⁡l′(λ)=-(n-r)τ-∑i=1r4ti+∑i=1r¯2ti+∑i=r¯2+1r¯33ti+∑i=r¯3+1r¯5ti+∑i=r¯5+1r3ti<0, l′(λ)=0 has a positive root λ^.(2)
λ
1
=
λ
2
=
λ
≠
λ
12. Under this special case, the likelihood function (10) then reduces to (20)Lλ,λ12∣D=C∏i=1r¯2λe-2λ+λ12ti1-e-λti·∏i=r¯2+1r¯3λe-λti1-e-2λ+λ12ti·∏i=r¯3+1r¯4λ12e-2λ+λ12ti1-e-λti·∏i=r¯4+1r¯52λe-2λ+λ12ti1-e-λti·∏i=r¯5+1r¯7λe-2λ+λ12ti1-e-λti+λe-λti1-e-2λ+λ12ti·∏i=r¯7+1r¯8λ12e-2λ+λ12ti1-e-λti+λe-λti1-e-2λ+λ12ti·∏i=r¯8+1r2λe-2λ+λ12ti1-e-λti+λe-λti1-e-2λ+λ12tie-λτ+e-2λ+λ12τ-e-3λ+λ12τn-rand so the log-likelihood function is (21)lλ,λ12=log⁡C-n-rλτ+n-rlog⁡1+e-λ+λ12τ-e-2λ+λ12τ-∑i=1rλti+∑i=1r¯2log⁡λ-λ+λ12ti+log⁡1-e-λti+∑i=r¯2+1r¯3log⁡λ+log⁡1-e-2λ+λ12ti+∑i=r¯3+1r¯4log⁡λ12-λ+λ12·ti+log⁡1-e-λti+∑i=r¯4+1r¯5log⁡2-λ+λ12ti+log⁡λ+log⁡1-e-λti+∑i=r¯5+1r¯7log⁡λe-λ+λ12ti1-e-λti+λ1-e-2λ+λ12ti+∑i=r¯7+1r¯8log⁡λ12e-λ+λ12ti1-e-λti+λ1-e-2λ+λ12ti+∑i=r¯8+1rlog⁡2λe-λ+λ12ti1-e-λti+λ1-e-2λ+λ12ti.The MLEs λ^, λ^12 will be obtained numerically in the equations ∂l(λ,λ12)/∂λ=∂l(λ,λ12)/∂λ12=0. The prove of the existence of MLE is given in the Appendix.


## 4. Simulation Study

In this section, we conduct a simulation study to investigate the performance of our methodology. We choose two parameter values of failure rates for each case in the two hybrid systems; that is,λ=1.0,0.8 in the case of same failure rates, and (λ,λ12)=(1.0,0.5),(0.8,0.4) for the case of different failure rates. Under each setting of parameter values, we carry out simulation study to generate the lifetimes Ti, i=1,2,3, following the construction described in Section 2.1 under two sample sizes n=24,30, for each of which two complete samples (n=r=∑k=19rk) with two settings of failure numbers rk and two censored samples with two failure numbers (r=20,21 for n=24 and r=26,28 for n=30) are considered to determine the sample size n and rk variation effects for the estimation precision. We conduct 10,000 Monte-Carlo simulations for each setting of parameter value, sample size, and failure number. The averaged MLE, mean squared error (MSE), length of 95% confidence interval, and coverage probability are displayed in Tables 1 and 2 for the series-parallel system and Tables 3 and 4 for the parallel-series system.Table 1
Series-parallel system:λ1=λ2=λ12=λ (CP = coverage probability).
n
(
=
r
)
r
k
,
k
=
1,2
,
…
,
9
λ
λ
^
MSE(λ^)
95% CI length
CP (%)
24
4, 4, 4, 2, 2, 2, 2, 2, 2
1.0
1.0481
0.0267
0.7164
98.3
5, 4, 3, 3, 3, 2, 2, 1, 1
1.0
1.0214
0.0226
0.6926
98.1
4, 4, 4, 2, 2, 2, 2, 2, 2
0.8
0.8308
0.0177
0.5664
98.4
5, 4, 3, 3, 3, 2, 2, 1, 1
0.8
0.8297
0.0172
0.5652
98.8
30
5, 5, 5, 3, 3, 3, 2, 2, 2
1.0
1.0371
0.0226
0.6366
97.2
6, 6, 5, 3, 3, 2, 2, 2, 2
1.0
1.0288
0.0212
0.6196
96.2
5, 5, 5, 3, 3, 3, 2, 2, 2
0.8
0.8357
0.0153
0.5116
97.8
6, 6, 5, 3, 3, 2, 2, 2, 1
0.8
0.8255
0.0142
0.4956
97.2
n
(
>
r
)
r
k
,
k
=
1,2
,
…
,
9
λ
λ
^
MSE(λ^)
95% CI length
CP (%)
24
3, 3, 3, 2, 2, 2, 2, 2, 1
1.0
1.0370
0.0298
0.7645
98.2
4, 4, 3, 2, 2, 2, 2, 1, 1
1.0
1.0323
0.0283
0.7290
98.1
3, 3, 3, 2, 2, 2, 2, 2, 1
0.8
0.8307
0.0175
0.6117
98.5
4, 4, 3, 2, 2, 2, 2, 1, 1
0.8
0.8264
0.0172
0.5879
99.4
30
5, 4, 5, 3, 2, 2, 2, 2, 1
1.0
1.0407
0.0271
0.6749
97.4
5, 5, 4, 3, 3, 2, 2, 2, 2
1.0
1.0380
0.0245
0.6491
98.3
5, 4, 5, 3, 2, 2, 2, 2, 1
0.8
0.8325
0.0157
0.5368
97.5
5, 5, 4, 3, 3, 2, 2, 2, 2
0.8
0.8245
0.0157
0.5193
98.3Table 2
Series-parallel system:λ1=λ2=λ≠λ12 (CP = coverage probability).
n
(
=
r
)
r
k
,
k
=
1,2
,
…
,
9
λ
,
λ
12
λ
^
,
λ
^
12
MSE(λ^,λ^12)
95% CI length (λ,λ12)
CP (%)
24
3, 4, 3, 3, 2, 2, 2, 2, 3
1.0, 0.5
0.9912, 0.5285
0.0247, 0.0168
0.8265, 0.5748
97.4
4, 4, 4, 2, 2, 2, 2, 2, 2
1.0, 0.5
1.0434, 0.4604
0.0238, 0.0161
0.8236, 0.5597
98.5
3, 4, 3, 3, 2, 2, 2, 2, 3
0.8, 0.4
0.7880, 0.4231
0.0166, 0.0153
0.6673, 0.4716
97.2
4, 4, 4, 2, 2, 2, 2, 2, 2
0.8, 0.4
0.8347, 0.3685
0.0155, 0.0145
0.6654, 0.4668
98.4
30
4, 4, 4, 3, 3, 3, 3, 3, 3
1.0, 0.5
1.0328, 0.5270
0.0189, 0.0062
0.7190, 0.5542
97.6
5, 5, 5, 3, 3, 3, 2, 2, 2
1.0, 0.5
1.0161, 0.5317
0.0187, 0.0058
0.7164, 0.5502
98.5
4, 4, 4, 3, 3, 3, 3, 3, 3
0.8, 0.4
0.8362, 0.4276
0.0172, 0.0055
0.5835, 0.4487
98.0
5, 5, 5, 3, 3, 3, 2, 2, 2
0.8, 0.4
0.8273, 0.4344
0.0151, 0.0052
0.5762, 0.4452
98.7
n
(
>
r
)
r
k
,
k
=
1,2
,
…
,
9
λ
,
λ
12
λ
^
,
λ
^
12
MSE(λ^,λ^12)
95% CI length (λ,λ12)
CP (%)
24
3, 3, 2, 2, 2, 2, 2, 2, 2
1.0, 0.5
1.0196, 0.4803
0.0260, 0.0188
0.8344, 0.5909
98.2
3, 3, 3, 2, 2, 2, 2, 2, 2
1.0, 0.5
1.0388, 0.4554
0.0257, 0.0182
0.8288, 0.5792
97.6
3, 3, 2, 2, 2, 2, 2, 2, 2
0.8, 0.4
0.8446, 0.3985
0.0304, 0.0162
0.6913, 0.4898
96.4
3, 3, 3, 2, 2, 2, 2, 2, 2
0.8, 0.4
0.8621, 0.3788
0.0279, 0.0161
0.6882, 0.4730
95.6
30
4, 4, 4, 3, 3, 2, 2, 2, 2
1.0, 0.5
1.0180, 0.5496
0.0202, 0.0087
0.7425, 0.5616
97.6
5, 4, 4, 3, 3, 3, 2, 2, 2
1.0, 0.5
1.0247, 0.5476
0.0196, 0.0082
0.7216, 0.5545
97.2
4, 4, 4, 3, 3, 2, 2, 2, 2
0.8, 0.4
0.8300, 0.4477
0.0188, 0.0079
0.6053, 0.4529
97.6
5, 4, 4, 3, 3, 3, 2, 2, 2
0.8, 0.4
0.8302, 0.4432
0.0170, 0.0067
0.5845, 0.4490
98.0Table 3
Parallel-series system:λ1=λ2=λ12=λ (CP = coverage probability).
n
(
=
r
)
r
k
,
k
=
1,2
,
…
,
9
λ
λ
^
MSE
95% CI length
CP (%)
24
4, 4, 4, 2, 2, 2, 2, 2, 2
1.0
1.0181
0.0257
0.6311
96.4
5, 4, 3, 3, 3, 2, 2, 1, 1
1.0
1.0062
0.0219
0.6092
96.6
4, 4, 4, 2, 2, 2, 2, 2, 2
0.8
0.8115
0.0149
0.5004
96.4
5, 4, 3, 2, 2, 2, 2, 1, 1
0.8
0.8045
0.0138
0.4865
96.6
30
5, 4, 5, 3, 3, 3, 2, 2, 3
1.0
1.0049
0.0194
0.5546
95.7
5, 5, 5, 3, 3, 3, 2, 2, 2
1.0
1.0022
0.0185
0.5543
96.1
5, 4, 5, 3, 3, 3, 2, 2, 3
0.8
0.8038
0.0119
0.4412
95.7
5, 5, 5, 3, 3, 3, 2, 2, 2
0.8
0.8002
0.0118
0.4406
96.1
n
(
>
r
)
r
k
,
k
=
1,2
,
…
,
9
λ
λ
^
MSE
95% CI length
CP (%)
24
4, 3, 2, 2, 2, 2, 2, 1, 2
1.0
1.0050
0.0266
0.6681
96.9
3, 3, 3, 2, 2, 2, 2, 2, 2
1.0
1.0087
0.0259
0.6619
95.7
4, 3, 2, 2, 2, 2, 2, 1, 2
0.8
0.8085
0.0166
0.5369
97.8
3, 3, 3, 2, 2, 2, 2, 2, 2
0.8
0.8093
0.0163
0.5313
96.5
30
4, 4, 3, 3, 3, 3, 2, 2, 2
1.0
1.0065
0.0210
0.5874
97.2
4, 4, 4, 4, 3, 3, 2, 2, 2
1.0
1.0102
0.0204
0.5733
96.2
4, 4, 3, 3, 3, 3, 2, 2, 2
0.8
0.8052
0.0130
0.4700
97.2
4, 4, 4, 4, 3, 3, 2, 2, 2
0.8
0.7991
0.0126
0.4514
95.1Table 4
Parallel-series system:λ1=λ2=λ≠λ12 (CP = coverage probability).
n
(
=
r
)
r
k
,
k
=
1,2
,
…
,
9
λ
,
λ
12
λ
^
,
λ
^
12
MSE(λ^,λ^12)
95% CI length (λ,λ12)
CP (%)
24
2, 2, 2, 3, 3, 3, 3, 3, 3
1.0, 0.5
1.0082, 0.4556
0.0214, 0.0142
0.6364, 0.5473
98.4
3, 3, 3, 3, 2, 2, 2, 2, 4
1.0, 0.5
1.0319, 0.4560
0.0196, 0.0113
0.6350, 0.5439
98.2
2, 2, 2, 3, 3, 3, 3, 3, 3
0.8, 0.4
0.8258, 0.3637
0.0173, 0.0104
0.5097, 0.4172
97.0
3, 3, 3, 3, 2, 2, 2, 2, 4
0.8, 0.4
0.8227, 0.3625
0.0181, 0.0113
0.5081, 0.4162
97.6
30
2, 2, 3, 3, 4, 4, 4, 4, 4
1.0, 0.5
0.9921, 0.4522
0.0191, 0.0191
0.5488, 0.4903
97.8
3, 2, 2, 3, 4, 4, 4, 4, 4
1.0, 0.5
0.9880, 0.4602
0.0152, 0.0121
0.5474, 0.4811
98.4
2, 2, 3, 3, 4, 4, 4, 4, 4
0.8, 0.4
0.8150, 0.3691
0.0125, 0.0107
0.4498, 0.4295
98.2
3, 2, 2, 3, 4, 4, 4, 4, 4
0.8, 0.4
0.7900, 0.3710
0.0125, 0.0077
0.4488, 0.4261
98.0
n
(
>
r
)
r
k
,
k
=
1,2
,
…
,
9
λ
,
λ
12
λ
^
,
λ
^
12
MSE(λ^,λ^12)
95% CI length (λ,λ12)
CP (%)
24
2, 2, 2, 2, 2, 2, 2, 3, 3
1.0, 0.5
0.9731, 0.4589
0.0236, 0.0180
0.6981, 0.6887
97.6
2, 2, 2, 2, 2, 2, 3, 3, 3
1.0, 0.5
0.9890, 0.4624
0.0172, 0.0151
0.6628, 0.6728
97.2
2, 2, 2, 2, 2, 2, 2, 3, 3
0.8, 0.4
0.8265, 0.3638
0.0187, 0.0086
0.5837, 0.5771
98.6
2, 2, 2, 2, 2, 2, 3, 3, 3
0.8, 0.4
0.8301, 0.3635
0.0179, 0.0069
0.5644, 0.5655
98.4
30
2, 2, 3, 3, 3, 3, 3, 3, 4
1.0, 0.5
0.9551, 0.4645
0.0169, 0.0049
0.5982, 0.6193
96.6
2, 2, 3, 3, 3, 3, 4, 4, 4
1.0, 0.5
0.9587, 0.4627
0.0119, 0.0039
0.5852, 0.6047
97.4
2, 2, 3, 3, 3, 3, 3, 3, 4
0.8, 0.4
0.7884, 0.3882
0.0100, 0.0031
0.4937, 0.5152
98.0
2, 2, 3, 3, 3, 3, 4, 4, 4
0.8, 0.4
0.7967, 0.3869
0.0086, 0.0031
0.4865, 0.5087
97.8In each table, the estimation results in the upper panel correspond to the complete sample and lower panel to the censored sample. It seems that the estimations are reasonably good under these relative small sample sizes, and all the coverage probabilities of confidence intervals exceed the nominal confidence level, indicating that it is a conservative method for interval estimation by chi-squared likelihood ratio statistics. As expected, under the same sample sizen, the MSEs and interval lengths are smaller in complete samples than these in censored samples. Due to the scale of the true parameter values, we noticed that given the same sample size n and failure numbers rk’s, the MSE and interval length of estimates under larger true parameter values are consistently larger than these under smaller true values. In Table 1, for example, given n=24, r=24, the MSE = 0.0267 and 95% confidence interval length = 0.7164 when λ=1, whereas MSE = 0.0177 and the length = 0.5664 when λ=0.8. However, it is common that, for a fair comparison between estimates variability with different units or different parameter values, one should use a relative variability measure such as coefficient of variation instead of a measure of dispersion like MSE or interval length. In our case, we propose a “normalized” measure of dispersion R=length/estimate to remove the scale effect for the comparison. As a result, the estimation results mentioned above give us R=0.7164/1.0481=0.6818 and 0.5664/0.8308=0.6817, respectively, which are very close to each other. Similar outcomes are obtained for other estimation results across the tables, indicating a consistent precision for the estimation procedure.Additionally, other findings can be seen from the estimation results. (i) For the complete samples, the upper panels in the tables interestingly show that given the same sizen, the MSEs and interval lengths are consistently smaller in the setting of larger variation of rk’s than those in the setting of less variation of rk’s. In other words, the estimations are more efficient under “unbalanced” failure numbers (rk’s vary largely) than “balanced” failure numbers (rk’s are close to each other). The possible reason is that the likelihood function with “unbalanced” failure numbers is less dispersed so that it accommodates more amount of information of parameters. (ii) For the censored samples, the MSE and interval length are getting smaller as the sample size n and failure number r are getting larger. For example, for the true parameter values (λ,λ12)=(1.0,0.5) in the lower panel of Table 2, when n=30, r=28, the MSE(λ^,λ^12)=(0.0196,0.0082) and the interval lengths for λ,λ12: 0.7216 and 0.5545, respectively, while the corresponding MSE(λ^,λ^12)=(0.0257,0.0182) and interval lengths for λ,λ12: 0.8288 and 0.5792 under n=24, r=21. Furthermore, given the sample size n=24, the MSE and interval length under r=21 are smaller than these under n=20, where the MSE(λ^,λ^12)=(0.0260,0.0188) and the interval lengths of λ,λ12: 0.8344 and 0.5909. In summary, the results indicate that it is more accurate for the estimates if more failures are observed.


## 5. Conclusions and Discussions

In this paper, we have studied statistical inference for three-component hybrid systems based on masked data, for which the lifetimes of units are nonindependent and nonidentical distributed. Two commonly censored schemes type-I and type-II were considered in the analysis. We have presented the maximum likelihood estimates of parameters when the failure rates of three components in the hybrid system were assumed to be the same and different, respectively. In addition, we obtained the approximate interval estimation of parameters by using likelihood ratio statistic. We have assessed the performance of estimation methods by simulation studies. The results have demonstrated that the procedure can achieve good estimation performances under small and moderate sample sizes, and the estimates are more accurate if more failures are observed, indicating the efficiency of the estimation method. While the method can be extended to more complex systems in the presence of masked data, the representation and evaluation of the likelihood function would become cumbersome for large systems. There is an alternative method based on signature that explores component topology. The system signature is the probability vector whose element is the probability of each component failure resulting in the system failure, and it provides an elegantly simple representation of a system [17]. Some advances and various applications of the signature are discussed in [18–20]. Recently, using the system signature, a Bayesian inference to the system with masked lifetime data was proposed by Aslett [21]. The generic likelihood function for complex systems can be easily expressed by data augmentation method; the parameter inference is relied on the samples from an iterative Markov chain Monte-Carlo simulation of all the component failure times and parameters. This intensive computing method provides an alternative to the traditional likelihood-based approach to deal with general systems.



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*Source: 525136-2015-06-07.xml*