Hybrid Censoring Schemes with Mixture Gamma Failure Distribution

Censoring schemes have received much attention over the past decades. Hybrid censoring schemes are censoring schemes mixed of type-I (T-1) and type-II (T-2) censoring schemes, a most popular area of study in life-testing or reliability experiments. More precisely, hybrid censoring can be described as a mixture of T-I and T-2 schemes. Gamma distribution is widely used, and its connection has more distributions. Mixture and single gamma distribution will be studied to estimate parameters, based on typeII hybrid censoring schemes (T-2HCS). We will apply algorithms to compute the maximum likelihood (ML) estimators and Bayesian approaches, using statistics, such as Markov chain Monte Carlo methods. Bayes estimators and corresponding highest posterior density confidence intervals will be tabled. Also, Markov chain Monte Carlo simulation is implemented to compare the performances of the different methods and the real dataset is analyzed for illustrative purposes.


Introduction
Over the last century, monitoring items and products are incredibly important tasks for managers in companies and different institutions. Censoring scheme data can be a result of monitoring. ese kinds of data have received huge attention in recent literature. e main reason for studying this type of data is to understand the underlying mechanisms and then provide useful information for prediction. ere are two types of censoring schemes. e first is called T-1, while the other is T-2, which are applied in many scientific applications. More precisely, suppose that a number of units are examined, say n. In the case of T-1 censoring schemes, the experiment will be stopped when a predetermined time of the experiment has been reached, say T, and all items are failed after that time is not spotted. While T-2 censoring schemes, the experiment is stopped when a predetermined number is chosen, say r, from a number of n items, that has failed. Hence, the assumption is not to replace the number of failed items. So, there are two random variables r and T in these types of censoring schemes, see Leese and Patel and Gajjar [1,2] for an explicit motivation. In contrast, the T-2HCS can be explained as a mixture of two kinds of censoring schemes. e first is T-1 censoring scheme, and the second is T-2 censoring scheme. Suppose that a number of identical units, say n, are tested. e test will be finished when a preselected number, say r, from a number of items that are failed, or when a predetermined time, say, T, on the test has been reached. is type of technique was first introduced by Epstein [3]. Several authors have studied T-2HCS. For example, Bartholomew [4] made a statistical theory about the mean of an exponential distribution, where the sample is chosen from T-2HCS, while Barlow and Madansky [5] solved estimation using a truncated or censored sample. Exact point and interval estimation for an exponential parameter using T-2HCS was studied by Chen and Bhattacharyya [6]. e T-2HCS was also studied under accelerated life tests from Gompertz distribution by Almarashi and Abd-Elmougod [7]. e gamma distribution was studied to derive integral representations for the first and the second type of Chebyshev polynomials by Cesarano [8], who used the fundamentals of Hermite polynomials to find the integration of gamma distribution. Double gamma probability density function was studied [9], which can be used with wavelet basis, such Harr wavelet, to reduce spark from recoding of magnetic readings in single-fiber measurement procedure, see [10] for more details. is process is to understand the relationship between spectrum and the results of magnetic readings of features of nanofibers. It is easy to find the explanation of the mixing of censoring schemes. For example, Childs et al. [11] suggested a new approach to the schemes and is called T-2HCS. e definition of data is to suppose n identical products or items on the experiment, and then the experiment will be stopped at the random time, say T * � max x r: n , T , where 0 ≤ r ≤ n and T are chosen numbers and x r: n indicates the time of r-th failure item in a sample of size n. Under this definition, it is clear to extricate three cases: (1) Case I can be explained as collecting r-th failure of items before the moment of T 1 is reached, and then the experiment can be terminated at T 1 (2) Case II is to collect r-th failure of products, which occurs between two prefixed times, and then the experiment terminated at the last failure item x r: n (3) Case III is to collect the r-th failure occurring before time T 2 is reached is type of investigation might arise in a case when the examiner decides to add a specific condition Childs et al. [11], for example, at least number of failures r must be collected and then this might cost for the use of the testing facility until the moment of units time T 1 . If the r-th failure products occur early, then the examiner can continue until the moment of unit time T 1 happened to adjust the use of the testing facility. On the other hand, the examiner will surely choose to continue until r-th failure of items is collected, see Childs et al. [11] for more details. Several results for the T-2HCS can be found by Kundu and Pradhan [12] who studied the estimation of the parameters of the generalized exponential distribution, and Dube et al. [13] who studied the log-normal distribution using T-2HCS. e main ideas behind the theory of gamma distribution can be found in the early eighteenth century. It was first introduced by Leonhard Euler and Christian Goldbach in 1729 [14]. However, the distribution of gamma has two parameters, which appeared to be more popular in econometrics and certain other scientific areas of study. For example, it is often used to model waiting times, such as life testing. Also, the gamma distribution is more common in the Bayesian approach, which is applied as a conjugate prior distribution for various types of inverse scale parameters such as the variance of noise, see [15][16][17] for more details. More recently, gamma distribution was generalized by Stacy [18]. Also, Suksaengrakcharoen and Bodhisuwan [19] presented a new distribution, which is called the mixed generalized gamma distribution. e huge problem of applying the Bayes theorem is detecting the posterior distribution, which is more complicated for an analytical solution. e MCMC tools have been involved in many published articles in the past century. Scientists have increasingly applied the approach to drawing a sample from complex and complicated posterior Chib and Greenberg [20]. e use of these tools dates back to the early 70 years [21] and was generalised and extended by Hastings [22]. e designs of the MCMC approaches were improved by Geman and Geman [23], who added a new tool, simulated annealing steps, which is a stochastic optimization rule. Recently, Julian et al [24] gave a good review of the use of MCMC algorithms. Also, the convergence monitoring process is more effectively improved by Gelman and Rubin [25] in different ways. More information on this process can be found by Aykroyd [26]. Within this article, the investigation of T-2HCS is considered. e project will be based on the background of the gamma distribution. In Section 3, the likelihood function is considered, and also survival function, failure-rate function, and coefficient of variation are investigated. In Section 4, Bayesian approaches are used to introduce a model. Section 5 gives the application of real data which has been analyzed, and the MCMC algorithms for estimating the underlying parameters and its interval estimation is obtained. Section 6 displays conclusions and discussion.
e previous forms will be used for the estimation in Section 3, where hybrid censoring schemes requires f(x; Ω 2 ) and F(x; Ω 2 ) in (7) and (8). ese forms are used usually to build the likelihood.

Likelihood Function
At the start of this section, the likelihood function and the maximum likelihood of parameters α and β are showed. e likelihood function uses the sample of T-2HCS x is where R, x R:n , for the case 1, where x R:n > T 1 , n, x n:n , for the case 3, where T 2 > x n:n .
Hence, R refers to the number of failures in sample and r i�1 R i < n. Using (1) and (2) in (12) and if the vector Ω 1 � α 1 , α 2 , β 1 , β 2 , then we get the likelihood function of (α, β) under the generalized T-2HCS is e log-likelihood function is given by Journal of Mathematics e ML estimate of α and β can be obtained by solving the following equations: It can be seen that it is not easy to solve the above equations and it can be solved using the iterative method.
ere are many ways to compute α and β, one of these is ML estimation to compute α and β, which can be described as follows: (i) Starting with an initial value of α and β, let α (0) and β (0) , then compute ℓ(x|Ω 1 ) (0) in (14) (ii) Generate ϵ 1 from a normal distribution, and generate ϵ 2 from a normal distribution as well (v) Update ε 1 and ε 2 Using simulated ML samples with the following equation: Let ζ � (α, β, R(t), H(t), CV), then where Z is the standard normal variate and c is the acceptance ratio. Hence, the main reason for updating ϵ 1 and ϵ 2 is to obtain the answers more quickly. In this topic, the missing value method of Louis [27] can be mentioned to obtain the asymptotic confidence intervals. Hence, the complete information contains missing information, and this can be written as where I X (x|Ω i ) and i � 1, 2 indicate the observation, I W (x|Ω i ) is the complete information, and I W|x (x|Ω i ) is the missing information. e Fisher information I X (x|Ω i ) is given by and to derive the risk of I X (Ω 1 |x), the form of excepted information and variance was used: where R(I X (x|Ω i )) indicate the risk and the estimator E − ((z 2 ℓ(x|Ω i ))/zΩ 2 i )|x ∈ R is biased with bias E − ((z 2 ℓ(x|Ω i ))/zΩ 2 i )|x ∈ R − I W (x|Ω i ), see [27,28] for more details.

Bayes Estimation
Bayes estimators are proposed to estimate unknown parameters of gamma distribution based on T-2HCS. Bayesian Journal of Mathematics tools require prior distributions to describe the unknown parameters. e gamma distribution is chosen as prior for many reasons. Gamma prior has a fixable form. e posterior distribution of α and β given the data is us, the Bayes estimate of g(α, β) is It is not easy to evaluate the proportion of the two integrals in Equation (26). Because the integral is too complex and complicated, the MCMC method is proposed to gather samples from the complex form in (25) and then compute an approximation to the exact Bayes estimate of g(α, β). We assume that the parameters α and β are independent, and the prior distributions of α and β are where the parameters of the prior distributions a, b, c, and d are suggested to be known for the first initial step of the run and chosen to describe the prior behavior about the unknown parameters using the mean and chosen variance. e joint posterior function of α and β can be obtained as follows: It is not easy to solve (11). So, the numerical method such as MCMC methods can be used. From (11), the posterior function of α given β is and the posterior function of β given α is e Metropolis-Hastings tool using the normal distribution to generate random samples from (11) is applied. ese algorithms are written to estimate parameter in (11), and each step of iteration provides values of β and α and then can be used to give R(t), H(t), and the corresponding confidence interval can be investigated. e proposed algorithm of the MCMC method is considered as follows: (1) Start with guess value β � β 0 , α � α 0 , a 1 � a 0 1 , a 2 � a 0 2 , b 2 � b 0 2 , N � number of iteration and M � burn-in (2) Compute the posterior distribution in (11) (3) Generate α ( * ) � α (i− 1) + ε where ε ∼ N(0, τ 1 ) e main reason of updating the variance of the proposal is to converge. is procedure depends on the acceptance rate. Hence, to update the hyperparameters a 1 , a 2 , b 1 , and b 2 can be computed as where (α) (i− 1) obtained from the previous acceptance and v 1 � 1 is used. Similarly, for the hyperparameters, a 2 and b 2 can be computed as follows: For maximum a posteriori (MAP) estimates, it is obtained by computing (33) ere is a small change to posterior mean using temperature, T k , with T k � (2/(log(1 + k))); this is the one choice of annealing schedule, see [23] for more detail. e reason for the use of temperature is that it gives a quick good estimation of the parameter. If the vector Ω 2 in (22), then the parameter ω 1 can be modeled using bate distribution, ω ∼ Beta(a 5 , b 5 ), with density given by Similarly, step (i − 1) might provide a mean ω (i− 1) 1 and variance V 5 for which correspond to where v 5 � 0.001 is used. Sometimes, the posterior mean is complex and more complicated. So, the MCMC algorithms   Journal of Mathematics are implemented to estimate the parameters using hybrid censoring schemes. In Section 5, the proposed methods will be investigated and compared.

Application to Real-Life Data
A real application set of carbon fibers is collected by Nichols and Padgett [29]; these data give 100 observations. According to generalized T-2HCS, three different schemes can be obtained from data: e MLEs and Bayes were applied based on the three schemes to estimate β and α. e confidence intervals are also calculated -the parameter estimation of β and α -using the MLEs and Bayes methods. In the single distribution of gamma, the ML and Bayes estimates are applied when the prior parameters are a 1 � a 2 � b 1 � b 2 � 0.001 and updated using the forms in (16) and (22). e number of iterations is 10,000. ey were removing the first 1000 values, which is described as burn-in. e Bayesian Information Criterion (BIC) is applied, where it is defined as p log(n) − 2ℓ(Θ), p is the number of parameters.
Also, consider Table 1. Firstly, two approaches are applied: the first is to fix the hyperparameters and the other is to update the hyperparameters. It can be seen that there is no significant difference between these two methods. In general, α ∼ (10.1236, 13.1078) and β ∼ (4.6057, 6.6432). According to the BIC, it is clear to identify the posterior mean estimates as the best model. e contour of ML function is shown in Figure 3, and the summary of the numerical results is shown in Table 1; hence, there is symmetry about the 45 o line. e red dots show the location of the ML, and the dark areas show the 95% confidence fields for α and β. e results of estimated parameters α 1 , α 2 , β 1 , β 2 , and ω 1 for the mixed gamma distribution are shown in Table 2. Hence, ω 2 � 1 − ω 1 , and the hyperparameters are chosen to be a 1 � a 2 � a 3 � a 4 � b 1 � b 2 � b 3 � b 4 � 0.001 and a 5 � b 5 � 2, where a 1 and b 1 are the hyperparameters for α 1 ; a 2 and b 2 are the prior parameters for α 2 ; a 3 and b 3 are the parameters for the distribution of β 1 ; and a 4 and b 4 are the parameters for the distribution of β 2 . In general, it can be seen that posterior mean gives slightly better results than our Bayesian methods, but the MLEs are simpler to use and much faster and are implemented easily. Finally, we believe mixed models can improve the result of estimation.

Conclusion
is article has introduced two parameter estimation models that are proposed using MCMC algorithms. Extensive simulation studies have shown that our models work well across a signal and mixture of gamma distribution T-2HCS. e MCMC algorithms were applied to calculate the approximate proposal methods and the corresponding credible intervals. Finally, we summarize some comments about the results shown in Tables 1 and 2 as follows: (1) e results of proposal methods in Table 2 show that the mixed model is good to use and estimate the parameters, and it is more acceptable (2) Bayesian approaches give slightly better results than MLEs (3) Updating the variances of the proposal gives the answer more quickly (4) e thin number helps to provide the independent sample (5) e Bayesian information criterion shows that the results improved in case I, when r-th failures are collected before reaching the moment of T i (6) Some parameters proposed each iteration, while some of them stop for some steps; this technique helps to converge (7) Tables show that proposed methods work well for estimating parameters

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.