Abstract
In reliability engineering and lifetime analysis, many units of the product fail with different causes of failure, and some tests require stress higher than normal stress. Also, we need to design the life experiments which present methodology for formulating scientific and engineering problems using statistical models. So, in this paper, we adopted a partially constant stress accelerated life test model to present times to failure in a small period of time for Gompertz life products. Also, considering that, units are failing with the only two independent causes of failure and tested under type-I generalized hybrid censoring scheme the data built. Obtained data are analyzed with two methods of estimations, maximum likelihood and Bayes methods. These two methods are used to construct the point and interval estimators with the help of the MCMC method. The developed results are measured and compared under Monte Carlo studying. Also, a data set is analyzed for illustration purposes. Finally, some comments are presented to describe the numerical results.
1. Introduction
The data under life-testing experiments may be complete or censoring data; the words of complete data set are used when the time to failure for all units under the test is obtained. But, the word censoring data is used when some but not all data about tested units is obtained. Type-I and type-II censoring schemes are from the oldest censoring schemes in life testing experiments. In the type-I censoring scheme, tested time is prefixed and the number of failure units is random. But, in type-II censoring scheme, the tested time is random and the number of failure units is prefixed. The two types of censoring have the lack of memory where the number of failure units may be very small or zero in type-I censoring scheme, but the total time of the test may be very large in type-II censoring scheme. And, the joint censoring scheme of type-I and type-II is called hybrid censoring scheme.
In the plan of type-I hybrid censoring scheme (type-I HCS), units are randomly selected from the product. The ideal test time and a suitable number of failure units that need statistical inference are proposed to be and , respectively. The experimenter terminates the test at the min (, ). Type-I HCS is exposed and studied by different authors, [1, 2] and recently by [3]. But, in the plan of type-II hybrid censoring scheme (type-II HCS), also, identical units are randomly selected from the product. And, the ideal test time and a suitable number of failure units that need statistical inference are proposed to be and , respectively. The experimenter terminates the test at the max (, ); for more details, see [4]. The types of censoring, type-I HCS and type-II HCS also, satisfy the property that a smaller number of failures may be zero and there is a large test time, respectively; see [5]. Then, this problem has been treated with a generalized form of two types of censoring schemes known as a generalized hybrid censoring scheme (GHCS) [6].
For type-I GHCS, identical units selected from product to put to the test and two prefixed integers and satisfying that and prefixed time are proposed. The plan of type-I GHCS can be described as follows. When the experiment is running, the failure time is recorded until number of failures is observed. If , then the test is terminated at a minimum time of (, ). In another case, if , the test is terminated at . Then, in type-I GHCS, the minimum number of failure must be satisfied and the data is summarized as where is defined by
For type-II GHCS, identical units selected from the product to put to the test and prefixed integers as well as two prefixed times satisfying are proposed. The plan of type-II GHCS is described as follows. After the experiment is running, the failure time is recorded until the time is reached. If , then the test is terminated at . In another case, if , the test is terminated at . But, if , then, the test is terminated at . Then, we say that the minimum time must be observed and the maximum time cannot be beyond it, and the random time data where is defined by
In life testing experiments, the problem of obtaining sufficient information about the life of a product under recent technology is more difficult for a long-life product. Then, the related statistical inferences became more difficult. This problem can be solved with a good choice of the type of censoring scheme. Another solution to this problem is exposing the test unit to stress higher than normal stress conditions which is known as accelerated life tests (ALTs). Studies [7, 8] presented the key reference of ALTs. Recently, this problem is handled in [9, 10]. Different forms of ALTs are available. The first is known as constant stress ALTs, in which the experiment is loaded under constant stress until the final point of the experiments. The second type is called step stress ALTs, in which the experiment is running at different stress levels and changing at a prefixed time or number [11]. The last type is progressive stress ALTs, in which the stress is kept with a continuous increase at all experiment steps [12]. In some tests, units are running under normal stress and accelerated stress; then, this type of acceleration is called partially accelerated life test. The model of partially constant ALTs can be built as follows. For identical tested units randomly chosen from population, and () units are selected randomly to test under used and accelerated conditions, respectively. Then, the failure times at each stage are recorded under a determined censoring scheme. Units under test can fail with different fetal risks; one of these risks is caused by the failure and the problem of measuring the risk of one cause of the failure known as competing risks. This problem is discussed by different authors [13–16]. Recently, this problem is handled for the accelerated model in [17].
The main objective in this paper is adopting the type-I GHCS with a partially constant ALT model when test units fail with only two independent causes of failure and the failure time is distributed with Gompertz distribution (GD). The Gompertz lifetime population with random variable has probability density function (PDF) given byand cumulative distribution function (CDF) is presented bywhere and are shape parameters. Then, we describe the mechanism of the model and formulate the likelihood function. Also, we present under observed data the point and interval estimators of model parameters with maximum likelihood and Bayes estimations. The theoretical results are measured and compared with Monte Carlo simulation and data analysis.
The paper is organized as follows. Section 2 presents some abbreviations and the model description. Section 3 gives the classical estimation with the MLE method. Section 4 presents the Bayes estimation with the MCMC method. Section 5 reported the results of the Monte Carlo studying. Section 6 presents lifetime data analysis for illustrating purpose. In Section 7, we give a report about the numerical results obtained from the simulation study and data analysis.
2. The Model
In this section, we present the list of abbreviations that are used in the paper as well as a complete description of the model mechanism and the corresponding likelihood function.
2.1. Abbreviations
GD : Gompertz distribution MLE : maximum likelihood estimation ME : mean PC : probability coverage CDF : cumulative distribution function HRF : hazard failure rate function : failure time under stress and cause MH : Metropolis–Hastings algorithm CI : credible intervals MCMC : Markov chain Monte Carlo MSE : mean squared error ML : mean interval length PDF : probability density function SF : survival function SEL : squared error loss : the indicator value expressed to cause ACI : approximate confidence interval CDF : cumulative distribution function
2.2. The Model under Type-I GHCS
Suppose that identical units are randomly chosen from the population to put to the test, and let be randomly selected to test under used condition and units to test under stress condition. Previously, two integers and such that and with ideal test time are determined. Firstly, when the test is running, the times to failure , and , denoting used and accelerated conditions, respectively, are recorded. The values of are defined in (1), and the mechanism is described as follows. If , then, the test is terminated at the minimum time of (, ). In another case, if , the test is terminated at . Considering only the two causes of failure, the time to failure and the corresponding cause of failure are recorded. Then, type-I GHCS under the competing risks model is described bywhereand means cause one or cause two, . The joint likelihood function of observed data {} with the CDF and PDF of random variables given by and , , denotes cause 1 and cause 2, respectively; then, it is given bywhere and
Considering that, tested units with CDF given by (4) for used condition and for independent two causes of failure that are reduced to the distribution have PDFs given by
The Gompertz lifetime distribution with common shape parameters and different shape parameters , and also the CDFs and SFs are given by
Consider that the proportional hazard model (also named Cox model) is relevant to handle the effects of the environment or stress on the lifetime distribution. Thus, the survival function under used stress is GD but the survival function under the higher stress takes the form
Therefore, , CDFs, and PDFs under accelerated condition are given, respectively, by
3. Estimation under ML Method
The results of the point and asymptotic confidence intervals of model parameters are discussed in this section with MLE for two independent causes of failure.
Let be the sample of type-I GHC competing risks data from GD; for distribution (10) and (14), the joint function (7) is reduced towhere and denote the number of unit failures under used and accelerated conditions, respectively. Also, and are the number of unit failures under the first and second causes, respectively. Then, the natural logarithm of the likelihood function (16) is reduced to
3.1. Point Estimators
The point MLE of model parameters can be obtained after taking the partial derivatives of (17) with respect to vector . Then, the derivative with respect to and is reduced towhere
Then, the likelihood equations are reduced to one linear equation of obtained after replacing , , and by (18)–(20) in (21). The initial value of any iteration can be obtained from the profile likelihood function obtained from (16) after replacing , , and . Then, the estimates are obtained , and .
3.2. Interval Estimation
The Fisher information matrix is defined as the minus expectation of second derivatives from the log-likelihood function with respect to model parameters. In practice, the expectation problem of the second derivative is more difficult to practice; then, the approximate Fisher information presents a suitable approximation that is used to build interval estimation as follows. Let present the second derivative of parameters vector given by
Then, the approximate information matrix at the values of the MLE of the parameters vector is denoted by where . Then, the estimate is distributed as a normal distribution with mean and variance-covariance matrix described by
Then, approximate interval estimate of is given bywhere presents the standard normal values with probability tailed and the values , , , and are the diagonal of the matrix .
4. Bayes Estimation
In this section, we present the Bayes point and interval estimation for the unknown model parameters to formulate the available information in the form of statistical distribution. The available information is exposed in the prior information and information exposed in the data. So, we consider the independent gamma priors for Gompertz parameters {} and noninformative prior for accelerated factor as follows:
Then, the joint prior density is given by
The joint posterior distribution can be formulated by
The Bayes estimate of model parameters depends on the posterior distribution and the choice of the loss function. Without loss of generality, considering squared error loss function (SEL), the Bayes estimators for any function () are formulated by
Integration in (28) and (29) is generally more difficult; hence, we need some approximation to compute these integrations. Different methods are available to approximate the integral such as numerical integration from the important ones applied in the Bayes context called MCMC method described as follows.
4.1. MCMC Approach
The problem estimation with the Bayesian approach with the help of the MCMC method is needed to build the posterior conditional distributions of model parameters as follows:
Therefore, the problem of building conditional distributions of (28) given data is presented by
The conditional distribution (31) to (34) shows that the conditional posterior distribution of , and takes gamma distribution. But, the conditional distribution of is more similar to the normal distribution. Then, the suitable scheme of the MCMC method is Metropolis–Hastings (MH) under Gibbs algorithms [18] described as follows.
4.2. MCMC Algorithm
Step 1: begin with initial parameter values put . Step 2: gamma distribution is used to generate from (29), from (30), and from (31) with the Gibbs technique. Step 3: normal proposal distribution is used to generate from the conditional distribution (32). Step 4: report the value of the parameters vector . Step 5: put to be . Step 6: repeat steps (2) to (5) times. Step 7: determine the iteration number that is needed for the stationary state (burn-in); then, for any function , Bayes estimators are presented by and the corresponding variance is defined by where and . Step 8: the ordered value of the vector is denoted by , and then, the corresponding credible interval of is given by5. Monte Carlo Simulation Study
In this section, we assess the developed results in classical MLE or Bayesian approach for different combinations of sample size and different combinations of and . Also, the study reported different effect sizes , and different test times . We adopt one parameter set to be . For prior information, we adopt noninformative prior0 (posterior is proportional with likelihood function) and informative prior1 . Mathematics Vr 10 is used, and iteration is reported for 1000 samples of type-I GHC data generated from the Gompertz distribution. For the Bayesian approach with MCMC methods, we generate and discard the first . For the point estimate, we compute the mean value of parameter estimates (MEs) and mean squared error (MSE). But, interval estimation is measured with probability coverage and the mean interval length. The results of the simulation study are reported in Tables 1 and 2 as follows.
6. Data Analysis Simulation
In this section, we choose the set of data generated from GD with respect to type-I GHCS and accelerated under partially constant stress ALTs. The random sample is generated from two GDs over the following algorithms. Step 1: let the total sample take the size and and let and . Step 2: suppose that the parameter vectors are randomly chosen to be ; then, a suitable is chosen to be 2.5. Step 3: the prior information is almost selected to satisfy ; then . Step 4: generate two samples of size from the two populations (10) with causes of failure. The two samples are put in ordered pairs to choose the minimum from each pair. Then, the minimum values are put in ascending order and determine the sample with used conditions, random values, and its cause of failure. If , then we terminated time min (, ). In another case, if , the terminated time is . Then, the value is observed. Step 5: step 4 is repeated for distribution (13) to determine . Step 6: compute and .
The data under used and accelerated conditions with its cause of failure is obtained from Tables 3 and 4 with and . The point MLE and corresponding confidence interval are summarized in Table 5. Also, for the Bayesian approach, we run the chain 11000 with the first 1000 values as burn-in, and the point and interval estimates are summarized in Table 5. Figures 1–8 describe the generated MCMC sample which describes the convergence satisfied by the MCMC method. The results obtained from each figure have shown that the MCMC method serves very well.
7. Conclusions
Modern technology products have a long period of time, and information about the life product is more difficult. Then, to overcome this problem, the experimenter determined the censoring scheme that serves this problem. In this paper, we choose type-I GHCS which keeps the minimum number needed in statistical inference in a small period of time. Also, this type of censoring is applied with the concept of partially constant ALTs for units that fail under two independent causes of failure and units that have Gompertz lifetime distribution. Moreover, we have seen that the proposed model can be easily extended for different populations. Also. we can mention that we can use a more general class of prior information such as priors with log-concave density functions. The simulation study is conducted, and the results are reported in Tables 1 and 2 which show the following:(1)The proposed model is more acceptable(2)The results for the large value of are more acceptable in terms of MSEs, PCs, and AL(3)The results are getting better for increasing values of increasing (, )(4)The results are getting better for closed values of (, )(5)The results under MLEs and noninformative priors are closed(6)Bayesian estimation under informative prior is better than MLEs(7)The results for the selected set of parameters are more acceptable
Data Availability
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.