# Hybrid Censoring Schemes with Mixture Gamma Failure Distribution

Authors: Hassan M. Aljohani; Nada M. Alfaer
Journal: Journal of Mathematics
 (2021)

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## Abstract

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 type-II 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.


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## Body

## 1. 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. These kinds of data have received huge attention in recent literature. The main reason for studying this type of data is to understand the underlying mechanisms and then provide useful information for prediction. There are two types of censoring schemes. The 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, sayn. 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. The 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. The 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. This 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]. The T-2HCS was also studied under accelerated life tests from Gompertz distribution by Almarashi and Abd-Elmougod [7]. The 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. This 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. The 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∗=maxxr:n,T, where 0≤r≤n and T are chosen numbers and xr: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 collectingr-th failure of items before the moment of T1 is reached, and then the experiment can be terminated at T1(2)
Case II is to collectr-th failure of products, which occurs between two prefixed times, and then the experiment terminated at the last failure item xr:n(3)
Case III is to collect ther-th failure occurring before time T2 is reachedThis 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 T1. If the r-th failure products occur early, then the examiner can continue until the moment of unit time T1 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. The 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 [15–17] 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. The huge problem of applying the Bayes theorem is detecting the posterior distribution, which is more complicated for an analytical solution. The 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]. The use of these tools dates back to the early 70 years [21] and was generalised and extended by Hastings [22]. The 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. The 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.


## 2. Gamma Distribution

Suppose that the lifetimeX has a gamma distribution with two parameters α and β that are the shape and the scale, respectively. Thus, X∼Gα,1/β, with density(1)fx;α,β=1Γαβαxα−1exp−xβ,x>0;α,β>0,where the exception αβ, the variance αβ2, and Xt,t>0. The cumulative distribution function CDF, survival function Rt, and failure rate function Ht are given by(2)Fx;α,β=∫0x1Γαβαuα−1exp−uβdu=1−sx;α,βexp−xβ,x>0;β>0,α≥1,where the part sx;α,β=∑j=0α−11/j!x/βj and(3)Rt|α,β=sx;α,βexp−tβ,t>0;β>0,α≥1,(4)Ht|α,β=tα−1Γαβαsx;α,β,t>0;β>0,α≥1,where α≥1,β>0, and 0<ω≤1. The coefficient of variation CV of the distribution Gα,1/β is defined as(5)C.V=1α.The random variableX of lifetime is said to have a mixture of gamma distribution, if it has the following CDF for each element in the vector Ωn=ω1,ω2,…,ωn,α1,α2,…,αn,β1,β2,…,βn greater than zero and x>0 as follows:(6)fx;Ωn=∑i=1nωifx;αi,βi,x>0;αi,βi>0,0<ω≤1.If the vectorΩ2=ω1,ω2,α1,α2,β1,β2, then the following CDF is given by(7)fx;Ω2=ω1Γα1β1α1xα1−1exp−xβ1+1−ω1Γα2β2α2xα2−1exp−xβ2,x>0;α1,β1,α2,β2>0,where ω2=1−ω1. The corresponding cumulative function CDF, survival function Rt, and failure-rate function Ht are given by(8)Fx;Ω2=2−ω1sx;α1,β1exp−xβ1−1−ω1sx;α2,β2exp−xβ2,(9)Rt|ω1,α1,β1,α2,β2=ω1sx;α1,β1exp−tβ1+1−ω1sx;α2,β2exp−tβ2−1,(10)Ht|ω1,α1,β1,α2,β2=ω1/Γα1β1α1xα1−1exp−x/β1+1−ω1/Γα2β2α2xα2−1exp−x/β2ω1sx;α1,β1exp−t/β1+1−ω1sx;α2,β2exp−t/β2−1,where α1,α2≥1,β1,β2>0, and 0<ω1≤1. The coefficient of variation CV is given by(11)C.V=ω1β1α1+1−ω1β2α2αβ.Figure1 displays an example single gamma distribution with parameters α=2 and β=2 (a), and mixed gamma distribution with α=20,13, β=10,15 with ω=4 (b).Figure 1
Examples of the scaled gamma distribution: (a) single distribution and (b) mixture of two distributions.
(a)(b)The previous forms will be used for the estimation in Section3, where hybrid censoring schemes requires fx;Ω2 and Fx;Ω2 in (7) and (8). These forms are used usually to build the likelihood.


## 3. Likelihood Function

At the start of this section, the likelihood function and the maximum likelihood of parametersα and β are showed. The likelihood function uses the sample of T-2HCS x is(12)Lα,β|x=n!n−s0!∏i=1s0fxi1−Ft0n−s0,where(13)T−2HCSs0,t0=R,xR:n,for the case1,wherexR:n>T1,n1,T1,for the case2,wheren1>R,n,xn:n,for the case3,whereT2>xn:n.Hence,R refers to the number of failures in sample and ∑i=1rRi<n. Using (1) and (2) in (12) and if the vector Ω1=α1,α2,β1,β2, then we get the likelihood function of α,β under the generalized T-2HCS is(14)LΩ1|x=n!n−s0!1Γαβαs0∏i=1s0xiα−1exp−xiβ∑j=0α−11j!t0βjexp−t0βn−s0.The log-likelihood function is given by(15)ℓΩ1x=logn!n−s0!−s0αlogβ−s0logΓα+α−1∑i=1s0logxi−∑i=1s0xiβ+n−s0log∑j=0α−11j!t0βj−t0β.The ML estimate ofα and β can be obtained by solving the following equations:(16)s0logβ−s0∂/∂αΓαΓα+∑i=1s0logxi+n−s01α−1+lnt0/βt0/βα−1t0/βα−1=0,(17)s0α1β+∑i=1s0xiβ2+n−s0t0β2−∑j=1α−11/j−1!t0/β2j−2∑j=0α−11/j!t0/βj=0.It can be seen that it is not easy to solve the above equations and it can be solved using the iterative method. There are many ways to computeα^ 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 ofα and β, let α0 and β0, then compute ℓx|Ω10 in (14)(ii)
Generateϵ1 from a normal distribution, and generate ϵ2 from a normal distribution as well(iii)
Letα∗=αi−1+ε1 and β∗=βi−1+ε2, then compute ℓx|Ω1∗(iv)
Ifℓx|Ω1∗−ℓx|Ω1i−1≥logU0,1, let αi=α∗ and βi=β∗; otherwise, αi=αi−1 and βi=βi−1(v)
Updateε1 and ε2Using simulated ML samples with the following equation:Letζ=α,β,Rt,Ht,CV, then(18)ζ^j=1N−M−1∑i=M+1Nζii,Varζ^j=1N−M−2∑i=M+1Nζ^ji−ζ^j2.The 95% credible interval,ζ^i±Zγ/2Varζ^i, where Z is the standard normal variate and γ is the acceptance ratio. Hence, the main reason for updating ϵ1 and ϵ2 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(19)IXx|θ,β=IWx|θ,β−IW|xx|θ,β,where IXx|Ωi and i=1,2 indicate the observation, IWx|Ωi is the complete information, and IW|xx|Ωi is the missing information. The Fisher information IXx|Ωi is given by(20)IXx|Ωi=E−∂2ℓx|Ωi∂Ωi2|x∈R−Var∂ℓx|Ωi∂Ωi2|x∈R,and to derive the risk of IXΩ1|x, the form of excepted information and variance was used:(21)RIXx|Ωi=E−∂2ℓx|Ωi∂Ωi2|x∈R−IWx|Ωi+Var∂ℓx|Ωi∂Ωi2|x∈R,where RIXx|Ωi indicate the risk and the estimator E−∂2ℓx|Ωi/∂Ωi2|x∈R is biased with bias E−∂2ℓx|Ωi/∂Ωi2|x∈R−IWx|Ωi, see [27, 28] for more details.If the vectorΩ2, then we get the likelihood function of α,β under the generalized T-2HCS x:(22)LΩ2|x=n!n−s0!∏i=1s0ω1Γα1β1α1xiα1−1exp−xiβ1+1−ω1Γα2β2α2xiα2−1exp−xiβ2ω1∑j=0α1−11j!t0β1jexp−t0β1+1−ω1∑j=0α2−11j!t0β2jexp−t0β2−1n−s0,where α1,α2≥1,β1,β2>0, and 0<ω1≤1. The log-likelihood equation is(23)ℓΩ2|x=logn!n−s0!+log∑i=1s0ω1Γα1β1α1xiα1−1exp−xiβ1+1−ω1Γα2β2α2xiα2−1exp−xiβ2+n−s0logω1∑j=0α1−11j!t0β1jexp−t0β1+1−ω1∑j=0α2−11j!t0β2jexp−t0β2−1.ML estimation will be used to computeα^1,α^2, β^1, and β^2:(i)
Starting with an initial guess ofω^1,α^1,α^2, β^1 and β^2, say ω^10,α^10,α^20, β^10 and β^20, by using (3) compute then compute ℓΩ2|x0 in (23)(ii)
Generateε1 and ε2, ε3, ε4, and ε5 from a normal distribution(iii)
Letα1∗=α1i−1+ε1, α2∗=α2i−1+ε2, β1∗=β1i−1+ε3, β2∗=β2i−1+ε4, and ω1∗=ω1i−1+ε5, then compute ℓΩ2|x∗(iv)
IfℓΩ2|x∗−ℓΩ2|xi−1≥logU0,1, ω1i=ω1∗, αi=α∗, and βi=β∗ otherwise ω1i=ω1i−1, αi=αi−1, and βi=βi−1(v)
Updateε1, ε2, ε2, and ε4Use simulated ML samples with the following equations.Letζ=ω1,ω2,α1,α2,β1,β2,Rt,Ht,CV, then(24)ζ^j=1N−M−1∑i=M+1Nζii,and Varζ^j=1N−M−2∑i=M+1Nζ^ji−ζ^j2.The 95% credible interval,ζ^i±Zγ/2Varζ^i, where Z is the standard normal distribution and γ is the acceptance ratio. The ML function and estimate point and interval will be used in Section 5. Also, the tools of MCMC algorithms are introduced. However, it is easy to provide that ML often does poorly in estimation since the level of noise usually corrupts the data. Posterior techniques have been recommended to improve ML.


## 4. Bayes Estimation

Bayes estimators are proposed to estimate unknown parameters of gamma distribution based on T-2HCS. Bayesian tools require prior distributions to describe the unknown parameters. The gamma distribution is chosen as prior for many reasons. Gamma prior has a fixable form. The posterior distribution ofα and β given the data is(25)Pα,β|x=Lx|α,βP1αP2β∫0∞∫0∞Lx|α,βP1αP2βdαdβ.Thus, the Bayes estimate ofgα,β is(26)Egα,β=∫0∞∫0∞gα,βLx|α,βP1αP2βdαdβ∫0∞∫0∞Lx|α,βP1αP2βdαdβ.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 gα,β. We assume that the parameters α and β are independent, and the prior distributions of α and β are(27)π1α|a1,b1=b1a1Γa1αa1−1exp−b1α,α>0;a1,b1>0,π2β|a2,b2=b2a2Γa2β−a2−1exp−b2β,β>0;a2,b2>0,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. The joint posterior function of α and β can be obtained as follows:(28)πα,β|x=n!n−s0!b1a1b2a2αa1−1β−a2−α−1ΓαΓa1Γa2s0×∏i=1s0xiα−1exp−xiβ−b2β−b1α∑i=0α−11i!t0βiexp−t0βn−s0.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(29)πα|β,x∝αa1−1β−αΓαs0expαlog∑i=1s0xi−b1+n−s0log∑i=0α−11i!t0βi,and the posterior function of β given α is(30)πβ|α,x∝β−a2−α−1s0exp−∑i=1s0xiβ−b2β+n−s0log∑i=0α−11i!t0βi−t0βn−s0.The Metropolis-Hastings tool using the normal distribution to generate random samples from (11) is applied. These 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 Rt, Ht, and the corresponding confidence interval can be investigated. The proposed algorithm of the MCMC method is considered as follows:(1)
Start with guess valueβ^=β0, α^=α0, a1=a10, a2=a20, b2=b20, N= number of iteration and M= burn-in(2)
Compute the posterior distribution in (11)(3)
Generateα∗=αi−1+ε where ε∼N0,τ1(4)
Generateβ∗=βi−1+ε where ε∼N0,τ2(5)
Generateu from a uniform distribution, U0,1, αi=α∗,u≤πα∗,βi−1|xqαi−1|α∗/παi−1,βi−1|xqα∗|αi−1αi−1,O.w(6)
Also, generateu from a uniform distribution, U0,1, βi=β∗,u≤παi−1,β∗|xqβi−1|β∗/παi−1,βi−1|xqβ∗|βi−1βi−1,O.w(7)
Updatea1, a2, b2, τ1, and τ2(8)
Using simulated samples, estimate the parameters as follows:Letζ=α,β,Rt,Ht,CV, then ζ^j=1/N−M−1∑i=M+1Nζii and Var (ζ^j)=1/N−M−2∑i=M+1Nζ^ji−ζ¯^j2, where j=1,2,3,4,5. The main reason of updating the variance of the proposal is to converge. This procedure depends on the acceptance rate. Hence, to update the hyperparameters a1, a2, b1, and b2 can be computed as(31)a1=α2i−1v1,andb1=αi−1v1,where αi−1 obtained from the previous acceptance and v1=1 is used. Similarly, for the hyperparameters, a2 and b2 can be computed as follows:(32)a2=αi−1v2+2,andb2=αi−1a2−1.For maximum a posteriori (MAP) estimates, it is obtained by computing(33)ζ^MAP=argmaxζζ^K.There is a small change to posterior mean using temperature,Tk, with Tk=2/log1+k; this is the one choice of annealing schedule, see [23] for more detail. The reason for the use of temperature is that it gives a quick good estimation of the parameter. If the vector Ω2 in (22), then the parameter ω1 can be modeled using bate distribution, ω∼Betaa5,b5, with density given by(34)πω1|a5,b5=Γa5+b5Γa5+Γb5ω1a5−11−ω1b5−1,0≤ω≤1,a5,b5>0.Similarly, stepi−1 might provide a mean ω1i−1 and variance V5 for which correspond to(35)a5=b5ω1i−11−ω1i−1,b5=1−ω1i−12−v51−ω1i−1v5,where v5=0.001 is used. Sometimes, the posterior mean is complex and more complicated. So, the MCMC algorithms are implemented to estimate the parameters using hybrid censoring schemes. In Section 5, the proposed methods will be investigated and compared.


## 5. 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:Case 1: if the number of failures isr=81, the first time is T1=3.5 and the second time of finishing experiments is T2=4. Since the last failure is x81:100<T1, the experiment would have terminated in this case at T1=3.5, and the sample 0.39,0.81,…,3.31 and 3.33.Case 2: if the number of item failures isr=91, the first time is T1=3.5, and the second time is T2=4. Since T1<x91:100<T2, the latest failure will be collected is x91:100=3.70 and the sample is 0.39,0.81,…,3.68 and 3.70.Case 3: if the number of failures isr=93, the first time to stop is T1=3.5, and the second time to stop is T2=4. Since T2<x93:100, the experiment will be finished by T2=4 and the sample is 0.39,0.81,…,3.70 and 3.75, see [30] for explicit motivation.The MLEs and Bayes were applied based on the three schemes to estimateβ and α. The 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 a1=a2=b1=b2=0.001 and updated using the forms in (16) and (22). The number of iterations is 10,000. They were removing the first 1000 values, which is described as burn-in. The Bayesian Information Criterion (BIC) is applied, where it is defined as plogn−2ℓΘ^, p is the number of parameters.Also, consider Table1. 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.Table 1
The results of the proposed methods for the parameters using a single gamma distribution.
NoMethodEstimation of parametersa^1b^1a^2b^2α^β^ℓΘ^BIC1MLE————13.10786.6432−26147.2652303.31————(10.0244, 16.1912)(5.9873, 7.2991)(−26149.36, −26145.16)—MCMC PM0.0010.0010.0010.00113.13346.6563−26132.8152274.42————(10.2845, 15.9823)(6.0517, 7.2608)(−26134.75, −26130.87)—MCMC PM1.66620.12912.39328.736612.98966.5946−26145.0552298.90(1.6547, 1.6777)(0.1290, 0.1291)(2.3815, 2.4049)(8.5070, 8.9661)(10.0904, 15.8888)(5.9768, 7.2124)(−26147.18, −26142.92)—MCMC MAP0.0010.0010.0010.00112.83266.5239−26133.7852276.36MCMC MAP1.66210.12892.44099.567312.78886.5041−26146.1352301.062MLE————11.25005.2846−31790.9963591.00————(8.7029, 13.7971)(4.7678, 5.8014)(−31793.0526, −31788.93)—MCMC PM0.0010.0010.0010.00111.17975.2587−31776.405563561.83————(8.6348, 13.7245)(4.7423, 5.7750)(−31778.4529, −31774.35)—MCMC PM0.98760.09942.21485.630610.97395.1753−31788.2763585.57(0.9842, 0.9911)(0.0994, 0.0994)(2.2050, 2.2247)(5.5498, 5.7114)(8.5376, 13.4101)(4.6819, 5.6687)(−31790.17, −31786.38)—MCMC MAP0.0010.0010.0010.00110.87625.1323−31777.4263563.86MCMC MAP0.78730.08872.20895.524810.78345.0936−31789.0963587.203MLE————10.44464.7340−32351.6464712.34————(8.1702, 12.7190)(4.2796, 5.1884)(−32353.57, −32349.72)—MCMC PM0.0010.0010.0010.00110.55474.7808−32336.9864683.01————(8.2484, 12.8609)(4.3202, 5.2414)(−32339.02, −32334.95)—MCMC PM1.00820.10042.15014.455210.29664.6778−32348.7164706.46(1.0044, 1.0120)(0.1004, 0.1004)(2.1407, 2.1594)(4.3717, 4.5387)(7.9718, 12.6213)(4.2127, 5.1429)(−32350.72, −32346.69)—MCMC MAP0.0010.0010.0010.00110.23424.6512−32337.9864685.01MCMC MAP1.08260.10402.21505.633210.12364.6057−32349.9864709.01Figure2 shows an example of the path of the progress of the MCMC algorithms (PM): (a) and (d) show the values of the proposal for α and β; (b) and (e) show the boxplot to show the median; (c) and (f) show the density of the estimated parameters.Figure 2
Plots of the monitoring MCMC algorithm whereα and β are estimated. Scheme 1 at R=59 points is used and t0=2.87 and n=60. (a, d) Trace plots of α^ and β^ at the transient period of 10000 iterations. (b, c) Summary of the MCMC analysis for α^ and β^. (e, f) Histograms of α^ and β^ for scheme 1.
(a)(b)(c)(d)(e)(f)The contour of ML function is shown in Figure3, and the summary of the numerical results is shown in Table 1; hence, there is symmetry about the 45o line. The red dots show the location of the ML, and the dark areas show the 95% confidence fields for α and β. The results of estimated parameters α1, α2, β1, β2, and ω1 for the mixed gamma distribution are shown in Table 2. Hence, ω2=1−ω1, and the hyperparameters are chosen to be a1=a2=a3=a4=b1=b2=b3=b4=0.001 and a5=b5=2, where a1 and b1 are the hyperparameters for α1; a2 and b2 are the prior parameters for α2; a3 and b3 are the parameters for the distribution of β1; and a4 and b4 are the parameters for the distribution of β2. 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.Figure 3
Plots of ML results for two-component models showing the ML estimates (red points) with approximate95% confidence fields (gray areas), and the border interval is marked (green lines) for different samples: (a) case 1, (b) case 2, and (c) case 3.
(a)(b)(c)Table 2
The results of the proposed methods for the parameters using the mixed distribution.
NoMethodEstimation of parametersα^1α^2β^1β^2ω^ℓΘ^BIC1MLE10.44657.260410.89017.80490.3713−26182.4552382.48(5.6236, 15.2695)(3.8872, 10.6336)(8.6102, 13.1700)(6.7141, 8.8957)(0.2016, 0.5411)(−26187.07, −26177.83)—MCMC PM10.39727.124512.80936.52140.3073−26107.1352231.85(4.5699, 16.3643)(6.4403, 7.8087)(9.8544, 15.7642)(5.2619, 7.7810)(0.1106, 0.5039)(−26175.48, −26038.79)—MCMC MAP10,01917.909710.05806.44430.4203−26111.2852240.142MLE10.92516.66489.70855.79240.456831195.3762408.78(5.1289, 16.7213)(4.1246, 9.205)(6.1803, 13.2367)(3.6051, 7.9796)(0.3845, 0.5292)(−46360.62, −16030.12)—MCMC PM10.39755.151710.94315.16280.4903−31696.6563411.35(5.0573, 15.7377)(0.2605, 10.0430)(8.4918, 13.3944)(4.1696, 6.1560)(0.4536, 0.5269)(−31851.80, −31541.50)—MCMC MAP10.28574.161310.90835.14540.4919−31697.8263413.683MLE9.14315.63118.43984.18670.4101−32373.9164765.91(5.0695, 13.2167)(2.8756, 8.3867)(7.0168, 9.2768)(2.5939, 5.7795)(0.2930, 0.5271)(−32378.93, −32368.88)—MCMC PM10.34424.278410.42414.72950.4303−32136.0664290.20(8.5418, 12.1466)(4.1081, 4.4486)(7.9943, 12.8540)(3.7599, 5.6990)(0.2333, 0.6273)(−32258.16, −32013.95)—MCMC MAP9.20573.981010.20484.63940.4592−32298.0764614.23


## 6. Conclusion

This 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. The 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 Tables1 and 2 as follows:(1)
The results of proposal methods in Table2 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)
The thin number helps to provide the independent sample(5)
The Bayesian information criterion shows that the results improved in case I, whenr-th failures are collected before reaching the moment of Ti(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



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*Source: 6641078-2021-04-05.xml*