Inference for four-unit hybrid system with masked data under partially acceleration life test

ABSTRACT In this paper, we investigate parameter inference for four-unit hybrid systems based on Type-II progressively hybrid censored and masked data from constant-stress partially accelerated life test. It is assumed that the unit lifetime in systems has exponential hazard rate. The maximum likelihood estimations (MLEs) for unknown parameters and acceleration factor are obtained. The observed Fisher information matrix, as well as the asymptotic variance-covariance matrix of the MLEs are derived. Approximate confidence intervals (CIs) and studentized-t bootstrap CIs for the parameters are presented based on normal approximation to the asymptotic distribution of MLEs and bootstrap method, respectively. A Monte Carlo simulation study is carried out to investigate the performance of the MLEs and to compare the effects of two different CIs in terms of interval length and coverage percentage for the parameters.


Introduction
The lifetime data from multi-unit systems plays an important role in system reliability analysis. The lifetime data of system generally includes the failure time and the information on the exact unit causing system failure. In some cases, however, due to lack of proper diagnostic equipment or cost and time constraints, the exact unit causing the system failure cannot be identified, and the failure cause is isolated to a subset of system units. In these cases, the cause for system failure is masked, and the lifetime data are called masked data. In recent years, research on masked data has attracted the attention of many authors. Usher and Hodgson (1988) presented the maximum likelihood estimations (MLE) for unknown parameter of unit life distribution based on the masked data from a series system. Sarhan and El-Bassiouny (2003), Zhang and Shi (2011) presented the MLE and Bayesian estimation (BE) of unknown parameters in the parallel system with masked data. Meena and Vasanthi (2013) focused on series systems with two units, each of which has a Pareto lifetime. Under multiple type-I censoring, Fan and Hsu (2015) discussed statistical Inference for a two-unit series system with correlated log-normal lifetime distribution. Liu, Shi, Zhang, and Bai (2017) studied reliability nonparametric Bayesian estimation for the masked data of parallel systems in step-stress accelerated life tests. Cai, Shi, and CONTACT Xiaolin Shi linda20016@163.com Liu (2015a) discussed the Bayesian analysis for Burr-XII masked system in step-stress partially accelerated life test under type-I progressive hybrid censoring. Under constant and linear hazard rate for system unit life, the MLE and other estimation methods were studied among many researchers, see Lin, Usher, and Guess (1993), Usher (1996) and Sarhan (2004).
In the existing literature, most researches of masked data focused on a system that is either series or parallel only. In many real situations, however, it is often seen that a system function in a way better described by a combination of series and parallel constructions, such as computer network transmission systems, power transmission systems in power stations, and complex electric burst networks (Zhang, Yuan, & Zhang, 2001). Statistical analysis on hybrid system with masked data was very rare. Assume that the hazard rate function of unit in system are h(t) = c and h(t) = ct, (c > 0), respectively. Wang, Sha, Gu, and Xu (2015) investigated statistical inference for two basic hybrid systems based on masked data, and they presented the MLE and interval estimation of parameters of interest. Sha, Wang, Hu, and Xu (2015) assumed dependent lifetimes of units modelled by Marshall and Olkin's bivariate exponential distribution in two basic hybrid systems based on masked data, and they presented MLE and interval estimation of parameters of interest. Unfortunately, the literature  and  neither involved accelerated life testing nor related to other types of hazard functions for unit life.
The constant and linear hazard rate model can describe some units in reliability analysis. However, they cannot describe some other units because their hazard rate are not constant or linear. For example, when the lifetime of unit follows Gompertz distribution, the hazard rate of unit is an exponential function h(t) = λe t , (λ > 0), see Ghitany, Alqallaf, and Balakrishnan (2014). This exponential hazard rate model has played an important role in survival analysis and reliability engineering. Recently, many authors have contributed to the statistical methodology and characterization of this hazard rate model. For example, Mohie El-Din, Abdel-Aty, and Abu-Moussa (2017), Ghitany et al. (2014), Chang and Tsai (2003), and Pollard and Valkovics (1992). Jiang and Zhang (2006) assumed that hazard rate of the unit in series system is an exponential function h(t) = λe t , (λ > 0). They derived the MEL and BE of model parameters based on complete samples. However, in life testing and reliability studies, the experimenter may not always obtain complete information on failure times for all experimental items. Data obtained from such experiments are called censored data. Saving the total time on test and the cost associated with it is one of the major merits for censoring. A censoring scheme (CS), which can balance among total times spent for the experiment, number of items used in the experiment and the efficiency of statistical inference based on the results of the experiment, is desirable. The most common CSs are Type-I (time) censoring, and Type-II (item) censoring. The conventional Type-I and Type-II CSs do not have the flexibility of allowing removal of units at points other than the terminal point of the experiment. For this reason, a more general CS called Type-II progressive hybrid censoring (PHC) has been used in this paper. Type-II progressive hybrid censoring, initially proposed by Kundu and Joarder (2006), is a new censoring scheme. It cannot only remove the sample to study properties during the test, but also effectively control the test time and reduce test costs. Ordinary Type-II censoring and complete scheme are the special cases of PHC. Therefore, the Type-II PHC has attracted wide attention. Some related literature see Tomer, Gupta, and Kumar (2015), Cramer and Balakrishnan (2013), Sun and Shi (2014) and Soliman, Ellah, Abou-Elheggag, and El-Sagheer (2015).
For some products with high reliability and long lifetime, it is difficult to get the failure information under use stress level. Accelerated life test (ALT) is a strategy to solve this problem. Based on the failure data of accelerated life test to study the life characteristics of product, we need to use the relationship between the reliability index and stress levels, that is, acceleration model. In many cases, however, the accelerated model does not exist or is difficult to be found so that ALT is not available. Therefore, Partially accelerated life test (PALT) is a better way in practical applications (Abushal & Soliman, 2015). The stress can be applied in two different ways: constant-stress and step-stress. In the step-stress PALT (SSPALT), a test product runs at the normal use operating conditions first and if it does not fail in a specific time point, and then it runs at accelerated condition until failures or the observation censored. Several authors have dealt with SSPALT, for example, see Abd-Elfattah, Hassan, and Nassr (2008), Alaa and Essam (2014), Ismail (2012) and Cai, Shi, and Yue (2015b). Moreover, the constant-stress PALT (CSPALT) runs each item at either use stress level or accelerated stress level only, i.e. each item is run at a constant-stress level until the test is terminated. CSPALT have been studied by several authors, see Ahmad, Soliman, and Yousef (2015), Abushal and Soliman (2015), Zhao, Shi, and Yan (2014) and Jaheen (2014).
Four-unit hybrid systems shown in Figures 1 and 2 have been used in engineering practice. For example, a dual electric explosion network is generally a seriesparallel or parallel-series system consisting of four units (Zhang et al., 2001). This paper mainly focuses on statistical inferences for two basic hybrid systems under CSPALT using progressively hybrid censoring scheme. Based on Type-II progressively hybrid censored and masked data from CSPALT, we will derive the MLE and the approximate confidence intervals (ACIs) and bootstrap confidence intervals (BCIs) for unknown parameters of unit  lifetime distribution and acceleration factor. The performances of the obtained MLE and confidence interval are investigated in terms of mean square error, interval length and coverage percentage for the parameters, respectively. The main novelties of this paper are: (1) Extend life test on hybrid system to partially accelerated life test in the presence of masked data. (2) Extend Type-I and Type-II censoring scheme to progressive hybrid censoring scheme, and apply it to CSPALT.
The rest of this paper is organized as follows. A brief description for CSPALT model and the basic assumptions are elaborated in Section 2. The derivation of the MLEs of the unknown system parameters is derived in Section 3. The ACIs and BCIs of system parameters are established in Section 4. In Section 5, a Monte Carlo simulation study is carried out to illustrate the performance of the MLEs, ACIs and BCIs, and simulation results are presented. Section 6 concludes the paper and suggests some future ideas.

Model description and basic assumptions
In order to investigate statistical inference for hybrid systems under CSPALT, we first introduce the test model and the acquisition of observed data.

Model description
In CSPALT, n 1 systems are randomly chosen among N test systems which are allocated to use stress level (use condition) X 1 , and n 2 = N − n 1 remaining systems are subjected to an accelerated stress level (accelerated condition) X 2 . Under stress level X k (k = 1, 2), the Type-II progressive hybrid censoring can be applied as follows. At the first failure time t k1 , r k1 systems are randomly removed from the remaining (n k − 1) systems. At the second failure time t k2 , r k2 systems are randomly removed from the remaining (n k − r k1 − 2) systems and so on. Finally, if the m-th failure t km occurs before the time point T 0 , then test terminates at the time t km and all remaining r km = n k − m − r k1 − r k2 − . . . − r k(m−1) are removed. Otherwise, if the m-th failure t km does not occur before the time T 0 , and only J k failure occurs before the time T 0 , then test terminates at the time T 0 , and all remaining R * J k systems are removed, where R * J k = n k − r k1 − r k2 − . . . − r kJ k − J k , J k < m, and the integer m < n k (k = 1, 2). In our study, the r ki , m and T 0 are fixed in advance, i = 1, 2, . . . , m, k = 1, 2. Note that the exact cause of system failure may be masked and it is isolated to a Minimum Random Subset (MRS) S, S ⊆ {1, 2, 3, 4}. Let s ki denote the observed value of the MRS corresponding to the failure time t ki , i = 1, 2, . . . , m, k = 1, 2. If s ki includes more than one unit, then the cause for the system failure is not identified and the lifetime data is masked. Thus, the observed data are: Case 1: {(t 11 , s 11 ), (t 12 , s 12 ), . . . , (t 1m , s 1m ); (t 21 , s 21 ),

Basic assumptions
Statistical inference for the hybrid system with masked data under CSPALT usually depends on the following assumptions: A1: Under use condition, the lifetime of each unit in the system has exponential hazard rate (HR), which is given by h 1 (t) = λe t , t ≥ 0, λ > 0. The probability density function (PDF), cumulative distribution function (CDF) and reliability function (RF) are respectively given by A2: Under accelerated condition, the hazard rate function (HRF) of each unit in the system is h 2 (t) = βh 1 (t) = βλe t , t ≥ 0, λ > 0, where β is called acceleration factor and β > 1. Therefore, in accelerated condition, the PDF, CDF and RF for a unit in the system are given, respectively, by A3: The lifetime of tested systems is independent and identically distributed, under both use condition and accelerated conditions. A4: The masked occurrence is statistically independent of the failure cause of system and failure time.

Maximum likelihood estimations
Maximum likelihood and Bayesian methods are two most important and widely used methods in statistics. The MLEs are consistent and asymptotically normally distributed. The Bayesian method needs to select the prior knowledge of unknown parameters, but it is usually difficult to select the appropriate prior knowledge in practice.
In addition, the calculation of Bayesian estimation generally involves complex integral operations. This paper adopts a progressive hybrid censoring scheme in the accelerated life test of the hybrid system. The obtained likelihood function is naturally complex. If the Bayesian method is used to estimate unknown system parameters, the calculation amount would be much greater. Hence, MLEs for the parameters of unit distribution and acceleration factor are evaluated by using ML method in this section. And we will further investigate the application of Bayesian method in parameter inference of the hybrid system in the future.
Here, we first briefly introduce the concept of masking probability. Let T ki be the lifetime of the system i under stress level X k , k = 1, 2. Assume that there is a masked event S ki ⊆ {1, 2, 3, 4} with the observed value s ki , and K ki is the exact unit causing the system i failure, i = 1, 2, . . . , n k , k = 1, 2. Then, the probability of the system i failure due to the masked occurrence s ki at time t ki is where P(S ki = s ki |t ki < T ki < t ki + dt ki , K ki = j) is called masking probability, and P(t ki < T ki < t ki + dt ki , K ki = j) is the probability of system i failure caused by the unit j at the time t ki , k = 1, 2, i = 1, 2, . . . , n k , j = 1, 2, 3, 4.
Next, we derive the RF and the PDF of the hybrid system. Assume that the lifetimes of four unit in the systems are statistically independent of each other, and T kij is the lifetime of the j-th unit of the system i and t kij is the observed value of T kij , where k = 1, 2, and i = 1, 2, . . . , n k , j = 1, 2, 3, 4.
For parallel-series system in Figure 1, the lifetime of the system i is , and the RF of the system i is For j = 1, 2, 3, 4, the probabilities of the system i failure caused by the unit j at the time t ki are presented as follows.
From basic assumption A4, we obtain P(S ki = s ki |t ki < T ki < t ki + dt ki , K ki = j) = P(S ki = s ki ) = c ki . Thus, the PDF of the system i at time t ki is j∈s ki c ki f kij .
For parallel-series system in Figure 1, the likelihood function based on Type-II PHC sample in CSPALT is given by For Case 1, R k = m, R * k = 0, for Case 2, According to assumptions A1 and A2, under stress level X k , the hazard rate of each unit in system can be given by h k (t) = β k−1 λe t , t ≥ 0, λ > 0, k = 1, 2. Therefore, the RF of system i is derived as follows By substituting (1) and (3) into (2), the likelihood function in Equation (2) is written as From Equation (4), the log-likelihood function l 1 = ln L 1 (λ, β) may be written as where Setting the partial derivatives of l 1 with respect to λ and β to zero, we can obtain Using an iterative method such as Newton-Raphson to solve the above equations, we can derive the MLEs of the parameters λ and β.
For the series-parallel system in Figure 2, under stress level X k , the lifetime of the system i is given by T ki = min[max(T ki1 , T ki2 ), max(T ki3 , T ki4 )], k = 1, 2, i = 1, 2, . . . , n k . The probabilities of the system i failure caused by the unit j(j = 1, 2, 3, 4) at the time t ki are presented as follows Then, the likelihood function based on Type-II PHC sample in CSPALT can be simplified as . Now, we consider the RF of series-parallel system in Figure 2, under stress level X k , the lifetime of the system By substituting (7) and (9) into (8), we can get likelihood functions as follows According to Equation (10), the log-likelihood function l 2 = ln L 2 (λ, β) can be written as where Setting the partial derivatives of l 2 with respect to λ and β to zero, we can obtain Using an iterative method such as Newton-Raphson to solve the Equation (12), we can derive the MLEs of the parameters λ andβ. Based on the invariance of MLE, the MLE of RF for hybrid systems in Figures 1 and 2 under use condition is respectively given bŷ whereλ is the MLE of parameter λ.

Confidence intervals
In this section, we construct two confidence intervals of parameters λ, β: approximate confidence intervals (ACIs) and bootstrap confidence intervals (BCIs).

Approximate confidence intervals
In order to construct the confidence intervals of parameters λ, β, we use an approximate method based on asymptotic likelihood theory. It is known that the asymptotic distribution of the MLEs for parameters λ, β is given by where F −1 (λ, β) is the inverse of matrix F(λ, β) and F(λ, β) is the Fisher information matrix of unknown parameters λ, β. Like Nelson (1990), F(λ, β) is the 2 × 2 symmetric matrix of negative second partial derivatives of the loglikelihood function with respect to λ, β. For parallel-series system in Figure 1, the elements of the matrix F(λ, β), F ij (λ, β), can be calculated as follows: The elements F ij (λ, β) can be approximated by F ij (λ,β), whereλ,β are the MLEs of λ, β. Thus, we can get the MLE of matrix F(λ, β), it is denoted as F(λ,β). The inverse of matrix F(λ,β) is calculated by The approximate 100(1 − γ )% confidence intervals for the parameters λ and β can be given by where I 11 (λ,β) and I 22 (λ,β) are the elements on the main diagonal of the asymptotic variance-covariance matrix F −1 (λ,β), and z γ /2 is the upper γ /2 percentile of the standard normal distribution.
Similarly, for series-parallel system in Figure 2, we can get

Bootstrap confidence intervals
Studentized-t (Stud-t) bootstrap CI suggested by Hall (1988) is used to construct CIs for the unknown parameter (λ, β). Hall (1988) showed that the Stud-t bootstrap CI is better than the percentile bootstrap CI (Efron, 1982) from an asymptotic point of view, although the finite sample properties are not yet known. It was observed by Kundu, Kannan, and Balakrishnan (2004) that the nonparametric bootstrap method does not work well. Therefore, the Stud-t bootstrap CI is used in this paper. In the following paper, we letψ = (φ 1 ,φ 2 ) be the MLE of the parameter ψ = (ϕ 1 , ϕ 2 ), whereφ 1 =λ andφ 2 = β,λ,β are the MLE of λ, β, respectively. The following steps are followed to obtain the Stud-t bootstrap CI of parameter (λ, β): (1) For given the original type-II PHC sample and masked data in CSPALT, the MLEψ of the parameter ψ can be obtained by using the algorithm proposed in the section 3.
(2) Using the MLEψ to generate two random samples {(t * ki , s * ki )}, i = 1, . . . , R k ; k = 1, 2, which represent Type-II PHC bootstrap samples with masked data in CSPALT and exponential hazard rate for unit life in system.

Simulation studies
Since the performance of the different methods or censoring schemes cannot be compared theoretically, the Monte Carlo simulation study is conducted to evaluate the performance of the resulting estimators in section 3 and confidence intervals in section 4. Without loss of generality, we only consider the parallel-series system, and the simulation studies for the series-parallel system can be done similarly. We take parameter values λ = 1.2, β = 1.5, and choose n 1 = n 2 = n, r ki = r i , k = 1, 2, i = 1, 2, . . . , m. For given T 0 , n and m, we consider three different progressive censoring schemes (CS): The simulation study is conducted according to the following steps: (1) For given n(n = n 1 = n 2 ), two independent random samples of sizes n, which are denoted as (u k1 , u k2 , . . . , u kn ), k = 1, 2, are generated from uniform (0, 1) distribution.
For different n, m, T 0 and censoring schemes, we replicate steps 1-5 3000 times in each case. The average of MLEs (A-MLE) and mean square error (MSE) for the parameter λ and the acceleration factor β are calculated, and these results are listed in Tables 1 and 2. The average values of 95% approximate CIs and Stud-t bootstrap CIs, and the corresponding interval length (CIL) and coverage percentage (CP) of the parameter λ and β are calculated. These results are tabulated in Tables 3 and 4. From Tables 1 to 4, it can be observed that: (1) For the fixed T 0 and sample size n, the MSE of the MLE decreases as m increases, and A-MLE is close to the real value.
(2) For the fixed T 0 and m, the MSE of the MLE decreases as the sample size n increases, and A-MLE is close to the real value. (3) For fixed sample size n and m, the MSE of the MLE decreases as T 0 increases. (4) Under the same progressive censoring scheme, as sample size n, m and T 0 are fixed, the lengths of Studt bootstrap CIs are smaller than that of approximate CIs, and coverage percentage for the parameters in Stud-t bootstrap CIs are greater than that of approximate CIs. (5) For the fixed sample size n and T 0 , the length of Studt bootstrap CIs and approximate CIs become smaller as m increases. This shows that when there are more failures, the estimation effect is better.
In the parameter interval estimation theory, when the number of tested products and failure products are fixed, the smaller the length of the confidence interval is, the higher the estimation accuracy is. The greater the coverage percentage of the confidence interval on the parameters, the higher the estimation accuracy. The simulation results have demonstrated that the MLE procedure achieves good estimation performance, and the estimates are more accurate if more failures are observed. Under the same conditions, the bootstrap CIs can achieve better results than approximate CIs. Table 4. Average values of 95% CI for parameters and corresponding CIL and CP (λ = 1.2, β = 1.5, T 0 = 1.5).

Conclusion
In this paper, we have studied statistical inferences for two hybrid systems under CSPALT with progressive hybrid censored and masked data. The MLEs and CIs of unknown parameters and acceleration factor have been presented when the hazard rate of unit are exponential function. In addition, we obtained the ACIs and BCIs of model parameters based on normal approximation and bootstrap method, respectively. The performance of estimation methods has been assessed by simulated studies. The results have demonstrated that the maximum likelihood procedure can achieve good estimation performance. For confidence intervals of model parameter, the bootstrap procedure can achieve better results than approximate normal procedure under the same conditions. The method proposed in this paper can be extended to more complex hybrid systems with masked data. As a future work, Bayesian statistical inference and reliability analysis for hybrid system under CSPALT assuming the same progressively hybrid censoring schemes proposed in this paper would be considered. Another important aspect, namely optimal design on CSPALT for hybrid system with masked data, would be addressed.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work is supported by the National Natural Science