Several types of monitoring data are available. One is the censoring scheme, which is a popular problem in life testing experiments. The oldest censoring projects are the so-called “type-I”, and the other is “type-II”. In practice, there are usually two random variables, i.e., time and the number of failures of items. This strategy of censoring projects shows how the examiner imagines the experiment based on a predetermined time. A random number of units is accounted for the first type-I of a censoring scheme, which means it may be assumed the exact time of stopping experiment. While the predetermined number of failure units and a random time in the type-II censoring scheme. In these two types of censoring schemes, companies cannot be removed from an experiment until the final stage or the number of units fail. This process allows the detection of some units that are defective after running the experiment. The mixture of these types of censoring schemes is the so-called hybrid censoring system [1]. To remove elements from the test at any stage of the trial is known as a progressive censoring scheme [2]. The topic of progressive censoring has developed in different scientific fields, and has attracted much attention in recent years. Several authors have studied this type of data [3,4]. There are different types of progressive censoring schemes. The idea of the progressive type-I censoring scheme is to test time τ and determine the number m of failure units, and suppose n independent elements are tested under the censoring scheme r={r1,r2,…,rm}. The failure unit is removed at min(τ,Tm), where T_m is the stopping time of the number of failure units m. After each failure time (T_i, r_i), survival units are removed from the trial, where i=1,2,…,J and J≤m. In a progressive type-II censoring project, the number m of failure units and r={r1,r2,…,rm} are determined, and we suppose n independent units are examined and the experiment is stopped at T_m. After each failure time (T_i, r_i), survival units are removed from the test, where i=1,2,…,m. The lifetime products come from different production lines [5,6]. The exact likelihood inference using bootstrap algorithms was studied [7], as was the type-II progressive censoring scheme [8,9] and two censoring schemes [10]. Consider manufactured products that come from two production lines η1 and η2 under the same conditions. Assume two independent samples S₁ and S₂ are chosen from these lines for experimental testing. The experiment runs under some consideration of time and cost, and the experimenter reports that it terminates after a predetermined time or number of failures. This is called a joint censoring scheme [11]. The procedure of joint progressive type-II censoring was described previously, where the sample size S₁ +S₂ is taken as S₁ from line η1 and S₂ from line η2. The integers m and r={r1,r2,…,rm} are determined to satisfy the form S1+S2+∑ i=1mri. The element r₁ is removed immediately from the experiment. We observe the first failure unit, say T₁ and has line W₁ from line η1 or η2, say (t1,ω1). Also, the number r₂ is removed from the test after we examine the second failure unit, say T₂ and has line W₂, say (t2,ω2). The experiment continues until (tm,ωm) is observed, where w_i takes the value 1 or 0, depending on lines η1 or η2. The result of the previous examination t={(t1,ω1),(t2,ω2),…,(tm,ωm)} is called the joint progressive type-II censoring procedure. The concept of a balanced joint progressive type-II censoring scheme was considered by [12] for analytically more straightforward estimators than the other type of progressive censoring procedure. Several authors have discussed statistical inference using different distributions, such as two exponential distributions [12]. The procedure of lifetime using Weibull distributions was investigated [13]. The interpretation of the balanced joint progressive type-II censoring procedure starts with samples of size S₁ +S₂, taken from production lines η1 and η2, respectively. Integers m and the integers r={r1,r2,…,rm} are determined to satisfy m+∑ i=1m-1ri< min(S1,S2). The failure times and types are observed, say (ti,ωi), i=1,2,…,m. Fig. 1 shows the main idea of a joint progressive type-II censoring scheme. This study discusses the properties of Chen lifetime estimation procedures under a joint progressive type-II censoring scheme. The Chen lifetime distribution with two parameters was introduced by [14]. This study’s objective is to build a balanced joint progressive type-II censoring procedure for the Chen lifetime distribution and parameter estimation with the maximum likelihood estimator (MLE) and Bayes methods. The developed methods are also used to measure the same Chen lifetime products’ relative merits under the same conditions. Estimators are evaluated through numerical data analysis and assessed through a simulation study. The remainder of this article is organized as follows. The main principle and model formulation are given in Section 2. Point MLE and interval estimators are introduced in Section 3. Section 4 discusses Bayes point and credible intervals. Estimators under numerical examples and simulation studies are discussed in Section 5. We summarize some comments which are extracted from numerical methods in Section 6.

Assume two production lines, and a random sample of size S₁ +S₂, where S₁ comes from line η1 and S₂ from line η2. The integers m and r={r1,r2,…,rm} are determined to satisfy m+∑ i=1m-1ri< min(S1,S2). Suppose t₁ is observed from some units that are taken from line η1, then, r₁ survival component is removed from S₁ and r₁ +1 survival component is removed from S₂ when the second failure t₂ is observed if t₂ is chosen from the line η2. In that case, r₂ +1 survival component is removed from S₁ − r₁ −1, and r₂ survival component is removed from the sample S₂ − r₂ −1. The test continues in this manner until the mth failure t_m is observed. If the final failure is from line η1, then the survival components S1-m-∑ i=1m-1ri are removed from η1, and S2-(m-1)-∑ i=1m-1ri are removed from η2. If the final failure belongs to line η2, then the survival units S1-(m-1)-∑ i=1m-1ri are removed from η1, and S2-m-∑ i=1m-1ri are removed from η2. Fig. 1 shows the scheme of joint balanced progressive type-II censoring. The observed data t={(t1,ω1),(t2,ω2),…,(tm,ωm)} are called balanced joint progressive type-II censoring samples. Under consideration that S₁ comes from the line η1, and it has independent and identically distribution of lifetimes {X1,X2,…,Xs1} and S₂ comes from the line η2, and ithas independent and identically distribution of lifetimes {X1*,X2*,…,Xs2*}. These samples distributed with populations have probability density (PDFs) and cumulative distribution (CDFs) functions are given, respectively, by the functions f_j(.) and Fj(.), j = 1, 2. Then the balanced joint progressive type-II sample t={t1,t2,…,tm} is taken from {X1,X2,…,Xm1,X1*,X2*,…,Xm2*}, where m = m₁ +m₂, m₁ is the number of failed units from line η1, and m₂ is the number of failed units from line η2. The observed balanced joint progressive type-II censoring sample is t={(t1,ω1),(t2,ω2),…,(tm,ωm)}, where ωi takes the value 1 or 0, depends on line η1 or η2, m1=∑ i=1mωi and m2=∑ i=1m(1-ωi).

The joint likelihood rule under two progressive type-II censoring samples t={(t1,ω1),(t2,ω2),…,(tm,ωm)} is

(1) L(t)∝(R1(tm))I1(R2(tm))I2(h1(tm))m1(h2(tm))1-m1×∏ i=1m-1(h1(ti))mi(h2(ti))1-mi(R1(ti)R2(ti))ri+1,

where

(2) Ij=Sj-(m-1)-∑ i=1m-1ri,j=1,2,

and R_j(.) and h_j(.) are reliability and hazard rate functions, respectively. Under the described model, the probability density functions (PDFs) and cumulative distribution functions (CDFs) of the tested unit and chosen from two lines η1 and η2 have Chen lifetime distributions with PDFs given by

(3) 0,~ \left(\alpha _{j}\lambda _{j}\right)> 0. \label{eqn-3} \end{equation}$$]]> fj(t)=αjλjtαj-1 exp{tαj} exp{λj(1- exp{tαj})},t>0,(αjλj)>0.

Reliability and hazard rate functions, respectively, are given by

(4) Fj(t)=1- exp{λj(1- exp{tαj})},

(5) Sj(t)= exp{λj(1- exp{tαj})},

and

(6) hj(t)=αjλjtαj-1 exp{tαj},

where αj and λj are the respective shape and scale parameters of the Chen distribution. Hence, a bathtub-shaped failure rate is noticed when αj≥1, and an exponential form can be obtained when αj=1 [15]. Fig. 2d plots the properties of the Chen distribution. It is clearly seen that h(t) provides a bathtub-shaped curve when α=1.

The joint likelihood function in Eq. (1) without a normalized constant under a Chen lifetime distribution is defined as

(7) L(α1,α2,λ1,λ2|t)      ∝(α1,λ1)m1(α2,λ2)m2exp{(α1−1)∑​mi=1ωilogti+∑​mi=1ωitiα1+λ1∑​m−1i=1(ri+1)(1−exp{ tiα1 })     +λ1I1(1−exp{ tiα1 })+(α2−1)∑​m−1i=1(1−ωi)logti+∑​mi=1(1−ωi)tiα2+λ2∑​m−1i=1(ri+1)     ×(1−exp{ tiα2 })+λ2I2(1−exp{ tiα2 })}.

After taking the logarithms of both sides, the joint likelihood function in Eq. (7) becomes

(8) ℓ(α1,α2,λ1,λ2|t)∝m1(α1,λ1)+m2(α2,λ2)+(α1-1)∑ i=1mωi logti+∑ i=1mωitiα1+λ1∑i=1m-1(ri+1)(1- exp{tiα1})+λ1I1(1- exp{tiα1})+(α2-1)∑i=1m-1(1-ωi)logti+∑i=1m(1-ωi)tiα2+λ2∑ i=1m-1(ri+1)(1- exp{tiα2})+λ2I2(1- exp{tiα2}),

which is used to represent the point and interval estimators of underlying parameters.

We need to use Bayes approaches with the MCMC method because of the dimensionality of the model. Bayes estimation requires prior information about the model parameters, which are considered in this study to be independent gamma priors. Then, the available prior information is modeled as

(24) 0\right),\quad i=1,2,3,4, \label{eqn-24} \end{equation}$$]]> θi→distributedasgamma(ai,bi),(ai,bi>0),i=1,2,3,4,

where θ=(α1,α2,λ1,λ2). The joint distribution of prior densities is formed by

(25) P*(θi)=∏ i=14biaiΓ(ai)θiai-1 exp{-biθi}.

Following this, the information about the model parameters is obtained from the prior information and the data, which provides the posterior distribution as

(26) P(α1,α2,λ1,λ2|t)=P*(α1,α2,λ1,λ2|t)L(α1,α2,λ1,λ2|t)∫ α1 ∫ α2 ∫ λ1 ∫ λ2 P*(α1,α2,λ1,λ2|t)L(α1,α2,λ1,λ2|t)dλ2dλ1dα2dα1,

where the denominator of the fraction can be removed since it contains no information about θ. The proportional form from posterior distribution (26) with prior distribution (25) and likelihood rule (7) is defined as

(27) P(α1,α2,λ1,λ2|t)∝α1m1+a1-1α2m2+a2-1λ1m1+a3-1λ2m2+a4-1× exp{-b1a1-b3λ1+(α1-1)∑ i=1mωi logti+∑ i=1mωitiα1+λ1∑ i=1m-1(ri+1)(1- exp{tiα1})+λ1I1(1- exp{tiα1})-b2a2-b4λ2+(α2-1)∑ i=1m(1-ωi) logti+∑ i=1m(1-ωi)tiα2+λ2∑ i=1m-1(ri+1)(1- exp{tiα2})+λ2I2(1- exp{tiα2})}.

The Bayes estimators are computed with respect to the loss rule; then the Bayes method of any function π(α1,α2,λ1,λ2) under the rule of the squared-error loss (SEL) function is presented by

(28) π^(α1,α2,λ1,λ2)=EP(π(α1,α2,λ1,λ2))=∫ α1∫ α2∫ λ1∫ λ2P(α1,α2,λ1,λ2|t)dλ2dλ1dα2dα1.

The integrals in Eqs. (26) and (28) generally cannot be obtained in explicit form, but can be solved by approximation, such as numerical integration or Lindley approximation. One of the most frequently applied methods is the MCMC method, which is used to compute point and interval estimates as follows. The full conditional distributions can be described as

(29) P1(α1|t,λ1)∝α1m1+a1-1 exp{-b1a1+α1∑i=1mωi logti+∑i=1mωitiα1-λ1∑ i=1m-1(ri+1)exp{tiα1}-λ1I1 exp{tiα1}},

(30) P2(α2|t,λ2)∝α2m2+a2-1 exp{-b2a2+α2∑i=1mωi logti+∑i=1m(1-ωi)tiα2-λ2∑ i=1m-1(ri+1)exp{tiα2}-λ2I2 exp{tiα2}},