Generalized Type-I Hybrid Censoring Scheme in Estimation Competing Risks Chen Lifetime Populations

Different types of censoring scheme are investigated; however, statistical inference on censoring scheme which can save the ideal test time and the minimum number of failures is needed. The generalized type-I hybrid censoring scheme (GHCS) solves this problem. Competing the risk models under the GHCS when time to failure has Chen lifetime distribution (CD) is adopted in this research with consideration of only two cases of failure. Partially step-stress accelerated life tests (ALTs) are applied to obtain enough failure times in a small period to achieve a highly reliable product. The problem of parameter estimation under maximum likelihood (ML) and Bayes methods is discussed. The asymptotic confidence interval as well as the Bayes credible interval is constructed. The validity of theoretical results is assessed and compared through simulation study. Finally, brief comments are reported to describe the behaviour of the estimation results.


Introduction
Information about the lifetime products is presented in complete or censored data with respect to time or cost considerations. e complete failure time data is used when all the units under the test fail through the determined period of time. However, the censoring failure time data is used when some units under the test fail through a determined period of time. Various types of censoring are available and the common types are called type-I and type-II censoring schemes.
e first scheme has a prefixed test time and a random number of failures but the second scheme has a prefixed number of failures and a random test time. In serval cases of censoring, the test is required to run joint case of type-I and type-II censoring schemes described by the hybrid censoring scheme (HCS). e HCS can be described statistically as follows: suppose (τ, T m ) denote the ideal test time and the time of m failure which is used for statistical inference, respectively, and the test is removed at the only one time of them. en, HCS is defined under type-I and type-II censoring schemes and is called type-I HCS and type-II HCS. e test is removed at min (τ, T m ) in the type-I HCS, but at max (τ, T m ) in the type-II HCS, there is more information about the type of censoring presented by [1][2][3]. Furthermore, type-I censoring scheme or type-I HCS may satisfy the properties that the test has the smallest number of failures or maybe zero. However, type-II censoring scheme or type-II HCS satisfies the properties that the test has the largest number of failures; see [4]. e problem that appeared in these censoring schemes can be overcome in the generalized form of HSC; see [5] as type-I GHCS and type-II GHCS.
(1) In type-I GHCS, suppose n independent units are put under test and the prior integers r and m satisfy that 1 ≤ r < m ≤n and prior time τ. If the smallest number r is satisfied before τ (T r < τ), then the test is terminated at min (T m , τ) and the observed test times data are given by t � t 1;n < t 1;n < · · · < t r;n < · · · < t s;n , r ≤ s ≤ m. (1) However, if the smallest number r does not satisfy before τ (τ < T r ), then the test is terminated at T r and the observed test times data are given by t � t 1;n < t 1;n < · · · < t r;n . (2) Finally, if the largest number m is satisfied before τ (T m < τ),, then the test is terminated at T m and the observed test times are given by t � t 1;n < t 1;n < · · · < t m;n .
erefore, the type-I GHCS saves the minimum number which is necessary for statistical inferences.
(2) In type-II GHCS, let n independent units be put under test and the two prior times τ 1 and τ 2 such that τ 1 < τ 2 and integer m satisfies 1 ≤ m ≤ n. If the required number of failures is observed before τ 1 (T m < τ 1 ), then the test is terminated at τ 1 and the observed test times data are given by t � t 1;n < t 1;n < · · · < t m;n < · · · < t s;n , m ≤ s ≤ n. (4) On the other hand, if the required number of failures observed satisfies τ 1 < T m < τ 2 , then the test is terminated at T m and the observed test times data are given by t � t 1;n < t 1;n < · · · < t m;n .
Finally, if the required number of failures is observed to satisfy τ 1 < τ 2 < T m , then the test is terminated at τ 2 and the observed test times are given by t � t 1;n < t 1;n < · · · < t s;n , 1 ≤ s ≤ m. (6) In life testing experiments, the common problem is that units fail due to several fetal risks which are known as competing risks problem. e effect of any risk factor in the presence of other risk factors need to be assessed. is problem has been discussed early in [6][7][8][9][10] and recently in [11]. Under the consideration of two causes of failure, the competing risks model in the presence of type-I GHCS is presented as follows.
For a randomly selected n independent unit, a life testing experiment with priors integers r and m, 1 ≤ r < m ≤ n, is considered. At each step of the experiment, time T i;n and the cause of failure ρ i are recorded for i � 1, 2, . . . , d, where d satisfies r < d < m and ρ i ∈ 1, 2 { }. en, the joint likelihood function of type-I GHCS where t � {(t 1;n , ρ 1 ), (t 2;n , ρ 2 ), . . ., (t d;n , ρ d )} under the competing risks model is reported as To obtain more information about the lifetime of products industrial process, accelerated life tests (ALTs) present a suitable manner for reducing test time rather than using conditions. As we see in [12], ALTs are presented in different types; one of them is constant-stress ALTs, in which the test is kept with a constant level of stress; see [13][14][15]. e second type is called progressive-stress ALTs, in which the stress is kept with a continuously increasing level; see [16][17][18]. e third type is called step-stress ALTs, in which the stress level is changed through a prior time or the number of failures; see [19,20]. Furthermore, the ALTs can be done under the accelerated condition which is known by partial ALTs; see [21][22][23][24][25][26].
is paper aims to build and analyze type-I GHC competing risks sample under the model of partially stepstress ALTs from Chen lifetime products. e results of statistical analysis are built under maximum likelihood and Bayes method for point and interval estimation. e performances of the developed results are assessed and compared with mean squared error (MSE), average interval length (AL), and probability coverage (PC) through the Monte Carlo study.
is paper is structured as follows: the model formulation and abbreviation are presented in Section 2. e MLEs of model parameters as well as the asymptotic confidence intervals are investigated in Section 3. Bayes estimation with credible intervals is discussed in Section 4. e quality points and interval estimators are assessed via the Monte Carlo study in Section 5. Finally, the discussion and conclusion are presented in Section 6 ( Table 1). However, if r is satisfied after the time τ, then the test is terminated at T r . e test is running under conditions until a fixed time η; then, the test is ruining under accelerated conditions. Considering that, the failure time has an independent CD and two independent causes of failure to satisfy the following assumptions:
where λ is the accelerated factor. e random variable W is distributed with Chen lifetime distribution with PDF and is given by and f j1 (z) is given by (8). e CDF, S j2 (z), and hazard rate function h j2 (z) are given by (4) Under competing risks type-I GHC sample and partially step-stress ALTs model, the test is terminated at T r at τ < T r and min(τ, T m ) at τ > T r . en, the random sample of the total lifetime W is described by where d denotes the number of fail units, where r < d < m and k and m − k are the numbers of fail units under using and accelerated conditions, respectively. For this model, we can consider three different cases, τ < η, τ � η, or τ < η. Hence, the joint likelihood function of the observed values

Maximum Likelihood Estimation
When only two independent causes of failure and the test are running under the model of partially step-stress ALTs with type-I GHCS, the test information sample is used to obtain the point and interval MLEs which is reported in this section as follows.

3.1.
MLEs. e joint likelihood function (18) under CDF (9) and (14) for the observed type- Hazard failure rate function of t ji ρ i e cause of failure form i-th unit failure time CD(α, β) where integers k and (d − k) are denoted to failure units under using and stress conditions, respectively, and integers m 1 and m 2 denoted failure units under causes (ρ 1 , ρ 2 ). en, the log-form from (19) is reduced to e partial derivatives of log-likelihood function (20) are reduced to the likelihood equations solved with some numerical methods to obtain the estimates as follows: is reduced to Also, is reduced to which is reduced to e likelihood equations are reduced to two nonlinear equations which are solved numerically with any iteration method such as Newton Raphson to obtain α and λ which are used in (22) and (23) to present β 1 and β 2 .

Interval Estimation. For the parameters vectors
and the Fisher information matrix Σ is given by which is computed as the negative expectation of second partial derivatives (27). e approximate information matrix is used as the approximate form of the Fisher information matrix Σ specially in a large sample. e approximate information matrix Σ 0 at the maximum likelihood estimates Θ �(α, β 1 , β 2 , λ) is given by e asymptotic normality distribution of estimating α, β 1 , β 2 , and λ with mean (α, β 1 , β 2 , λ) and a variance co- erefore, 100(1 − ξ) intervals estimation of parameters vector Θ �{α, β 1 , β 2 , λ} are computed by where Θ i denotes the parameters estimate and value Σ ii denotes the diagonal of variance covariance matrix Σ − 1 0 with standard normal probability ξ/2.

Bayesian Approach with MCMC
Information about the model parameters and the information which is obtained from the life sample is used in this section to build the Bayes approach with the MCMC method. Besides, the estimators of parameters of CD and noninformative about accelerated factor are computed under squared error loss (SEL) function and independent prior distributions. erefore, independent gamma prior is adapted as follows: where Θ �(α, β 1 , β 2 , λ). en, the posterior distribution of Θ is defined by en, the Bayes estimate for any function ϕ(Θ) under SEL function is given by Generally, the ratio in (35) needs numerical approximation to compute, such as numerical integration and Lindley approximation. However, MCMC methods are the important tools that were applied recently with high accuracy and are obtained as follows.

Gibbs with MH Method.
e posterior distribution in (34) with prior distribution (33) and likelihood function (19) is calculated as; Mathematical Problems in Engineering en, the conditional PDFs of the posterior distribution is given by From equations (37) and (38), the conditional posterior PDFs are reduced to two conditional gamma density equations (38) and (39). Two functions are plotted similar to the normal distribution in (37) and (40). en, the process of generation from posterior distribution under the conditional posterior distribution by using Gibbs with the MH algorithms with normal proposal distribution [27] is given as follows: (1) Begin with initial vectors Θ (0) �(α (0) , β (0) 1 , β (0) 2 , λ (0) ) and indicator κ � 1. (3) e two values α (κ) and λ (κ) are generated from conditional densities (37) and (40) by MH algorithms with normal proposal distributions. e symmetric normal distributions are applied with mean α (κ− 1) or λ (κ− ) and variance obtained from the diagonal of the approximate information matrix, respectively. Also, the generated values are accepted with acceptance probability min [1, (π 4 , respectively, with respect to uniform (0, 1).
) is a built vector with Gibbs manner. (5) Put κ � 1 + 1 and then repeat steps 2-4 N times. (6) e Bayes estimates and the corresponding variance are given by where N is the number of iteration used to get stationary distribution.     measure point estimate and average lengths (AL) and the probability coverage (CP) are used to measure interval estimate. For Bayes estimation with MCMC methods, prior parameters are selected to satisfy the expectation of gamma prior as E(Θ l ) � (a l /b l ) � Θ l , l � 1, 2, 3. en, informative prior information (Prior 1) and noninformative prior are obtained when the posterior distribution is proportional with the likelihood function (Prior 0). Also, Chan is built for 11,000 iterations with the first 1000 as bur-in. Average Bayes estimates, mean squared errors (MSEs), coverage      (1) From all tables, the proposed methods serve well for all choices.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors have no conflicts of interest regarding the publication of the paper.