On Estimating the Parameters of the Beta Inverted Exponential Distribution under Type-II Censored Samples

: This article aims to consider estimating the unknown parameters, survival, and hazard functions of the beta inverted exponential distribution. Two methods of estimation were used based on type-II censored samples: maximum likelihood and Bayes estimators. The Bayes estimators were derived using an informative gamma prior distribution under three loss functions: squared error, linear exponential, and general entropy. Furthermore, a Monte Carlo simulation study was carried out to compare the performance of different methods. The potentiality of this distribution is illustrated using two real-life datasets from difference ﬁelds. Further, a comparison between this model and some other models was conducted via information criteria. Our model performs the best ﬁt for the real data.


Introduction
In many life-testing and reliability studies for engineering or medical sciences, the information on failure times for all experimental units may not be obtained ultimately by the experimenter. Due to this, there are many situations in which it is pre-planned to remove units before failure, and these obtained data are called censored data. The most common censoring schemes in life-testing experiments are type-I and type-II censoring schemes. The type-II censoring scheme is used often in toxicology experiments and life-testing applications, where it has proven to save time and money. Many authors have addressed Bayesian and non-Bayesian estimations based on type-II censored samples or different types of samples, including [1], who derived maximum likelihood estimation (MLE) and Bayes estimation under different types of loss functions for exponentiated Weibull distribution based on type-II censored samples. Prakash [2] discussed the properties of the Bayes estimator and the minimax estimator of the parameter of the inverted exponential distribution. The moments of the lower record value and the estimation of the parameter were presented, based on a series of observed record values by the maximum likelihood (ML). Furthermore, Dey and Dey [3] derived the MLE of the generalized inverted exponential distribution parameters in the case of the progressive type-II censoring scheme with binomial removals. Singh and Goel [4] studied a three-parameter beta inverted exponential distribution (BIED). They derived the non-central moments, inverse moments, moment-generating function, inverse-moment-generating function, and mode. Furthermore, they examined the distributional properties of order statistics. Moreover, a statistical inference about the distribution parameters based on a complete sample was investigated. Garg et al. [5] studied the MLE of the parameters and the expected Fisher information under a random censoring model of the generalized inverted exponential distribution. Bakoban and Abu-Zinadah [6] considered the four-parameter beta generalized inverted exponential distribution for complete samples. In their research, the MLE, the Fisher information matrix, and the confidence interval were found. Besides that, the Monte Carlo simulation was discussed to illustrate the theoretical results of the estimation. Finally, applications on real datasets were provided. Aldahlan [7] applied the ML method to estimate the inverse Weibull inverse exponential distribution parameters. One real dataset about time between failures for repairable items was applied. This article focuses on estimation methods based on the type-II censoring scheme. Two estimation methods were used to estimate the unknown parameters for the beta inverted exponential distribution (BIED): MLE and Bayes estimation. The proposed distribution has three parameters (scale parameter λ and shape parameters α and β). The cumulative distribution function (CDF) and the probability density function (PDF) of BIED, respectively, are: and: where B(α, β) = 1 0 w α−1 (1 − w) β−1 dw is the beta function. Equation (1) could also be written as a regularized incomplete beta function: where B(y, α, β) is the incomplete beta function, such that: The inverse of CDF is called the quantile function and is given by: , 0 < u < 1.
The survival and hazard function of the BIED, respectively, are given by: and: The layout of this article is as follows: In Section 2, the estimation of the unknown parameters for the BIED under type-II censored samples is introduced. A simulation study is discussed in Section 3. In Section 4, an application with real data is provided. Finally, the conclusion is given in Section 5.

Method of Estimation
In this section, we derive the ML and Bayesian estimators for the unknown parameters of the BIED based on type-II censored samples.

Bayes Estimation
Bayes estimators for the BIED are obtained based on type-II censored samples in this subsection. Singh and Goel [4] derived the Bayes estimators for the BIED based on complete samples under the SE loss function. They considered the gamma prior distribution for the unknown BIED parameters. The prior distribution is denoted by π(θ), which tells us what is known about θ without observing the data. Bayes theorem is based on the posterior distribution, which is defined as π * (θ|x) and given by (see [11]): where θ is continuous and L(x |θ) is the likelihood function. Furthermore, Equation (19) could be written as: where k is called the normalizing constant, necessary to ensure that the posterior distribution π * (θ|x) integrates or sums to one.
Here, we derive the Bayes estimates for α, β, and λ under three types of loss functions: squared error (SE), linear exponential (LINEX), and general entropy (GE). Moreover, four cases are considered first when α is unknown, while β and λ are known. A second case is when β is unknown, while α, λ are known. A third case is when the scale parameter λ is unknown. Finally, a fourth case is when both β and λ are unknown. Two techniques are used to compute the estimates: the standard Bayes and importance sampling techniques for the first three cases. The last case is computed via the importance sampling technique.

Case 1: Bayes Estimators When α Is Unknown
Assume α is unknown and has the following prior distribution α ∼ Gamma(a, b); thus, the prior for α is given by: By combining (9) and (21), the posterior distribution of the unknown parameter α is given by: where k 1 is the normalizing constant, defined as: Therefore, the Bayes estimator of α, denoted by ϕ(α), is obtained under three types of loss functions and two techniques as follows.

i. SE Loss Function
The symmetric loss function SE is defined as: Then, the Bayes estimator of ϕ(α) under the SE loss function, denoted byφ SSE C (α), and can be found using Equations (24) and (22).

iii. GE Loss Function
According to Calabria and Pulcini [13], the GE loss function of ϕ(θ) can be defined as: The Bayes estimator of ϕ(α) under the GE loss function, denoted byφ SGE C (α), can be found by using Equations (22) and (28).
where k −1 1 is defined in (23). The Bayes estimator of α using the importance sampling technique can be derived by rewriting the posterior density function in Equation (22); thus: .
The posterior density function of α can be considered as: where: The Bayes estimators of ϕ(α) under the SE, LINEX, and GE loss functions based on the importance sampling technique, denoted byφ ISE C (α),φ ILE C (α), andφ IGE C (α), respectively, could be found using the following Algorithm 1.
Algorithm 1 Importance sampling technique when α is unknown based on type-II censored samples. where The Bayes estimators of α are found numerically under these three loss functions by the NIntegrate function via Mathematica 11.

Case 2: Bayes Estimators When β Is Unknown
Suppose β is unknown and has the following prior distribution β ∼ Gamma(c, d), given by: By combining (9) and (35), the posterior distribution of the unknown parameter β is given by: where k 2 is the normalizing constant and can be written as: Therefore, the Bayes estimator of β, denoted by ϕ(β), is obtained under three types of loss function and two techniques as follows.

i. SE Loss Function
The Bayes estimator of ϕ(β) under the SE loss function, denoted byφ SSE C (β), can be found by using Equations (24) and (36).

iii. GE Loss Function
The Bayes estimator of ϕ(β) under the GE loss function, denoted byφ SGE C (β), can be found by using Equations (28) and (36).
The Bayes estimator of β using the importance sampling technique can be derived by rewriting the posterior density function in Equation (36); thus: .
The posterior density function of β can be considered as: where: The Bayes estimators of ϕ(β) under the SE, LINEX, and GE loss functions based on the importance sampling technique, denoted byφ ISE C (β),φ ILE C (β), andφ IGE C (β), respectively, could be found using the following Algorithm 2.
Algorithm 2 Importance sampling technique when β is unknown based on type-II censored samples. where The Bayes estimators of β under the three loss functions cannot be computed analytically through the two techniques used. They can be found numerically using the NIntegrate function via Mathematica 11.

Case 3: Bayes Estimators When λ Is Unknown
Suppose that the scale parameter λ is unknown and has the following prior distribution λ ∼ Gamma( f , ν), given by: By combining (9) and (46), the posterior distribution of the unknown parameter λ is given by: where k 3 is the normalizing constant and can be written as: The Bayes estimator of λ is indicated by ϕ(λ), and this estimator is obtained under three types of loss function and two techniques as follows.

i. SE Loss Function
The Bayes estimator of ϕ(λ) under the SE loss function, denoted byφ SSE C (λ), can be found by using Equations (24) and (47).

iii. GE Loss Function
The Bayes estimator of ϕ(λ) under the GE loss function, denoted byφ SGE C (λ), can be found by using Equations (28) and (47).
where k −1 3 is defined in (48). The Bayes estimator of λ using the importance sampling technique can be obtained by rewriting the posterior density function in Equation (47); thus: .
The posterior density function of λ can be considered as: where: The Bayes estimators of ϕ(λ) under the SE, LINEX, and GE loss functions based on the importance sampling technique, denoted byφ ISE C (λ),φ ILE C (λ), andφ IGE C (λ), respectively, could be found using the following Algorithm 3.

Algorithm 3
Importance sampling technique when λ is unknown based on type-II censored samples.
The Bayes estimators of λ under the three loss functions cannot be computed analytically through the two techniques used. It can be found numerically using the NIntegrate function via Mathematica 11.

Case 4: Bayes Estimators When λ and β Are Unknown
Consider that both parameters λ and β are unknown. Suppose β ∼ Gamma(c, d) and λ ∼ Gamma( f , ν). Therefore, the joint prior distribution is given by: By combining (9) and (57), the joint posterior distribution of the unknown parameters λ and β is given by: where k 4 is the normalizing constant and can be written as: The Bayes estimators of λ and β are derived under three types of loss function as follows.

Algorithm 4
Importance sampling technique when λ and β are unknown based on type-II censored samples.
The Bayes estimators of λ and β under the three loss functions can be found numerically using the NIntegrate function via Mathematica 11.

Simulation Study
Simulation studies were conducted using Mathematica 11 to clarify the performance of the proposed estimators. Simulation results are given for the ML and Bayesian methods based on type-II censored samples. Furthermore, biases and mean-squared errors (MSE) were considered to illustrate the performance of the different estimators, defined as: The ML estimates of the parameters α, β, λ, S(x 0 ) and h(x 0 ) could be found using the following Algorithm 5.
Algorithm 5 ML method of the parameters α, β, λ, S(x 0 ) and h(x 0 ) based on type-II censored samples.
It is clear from Tables 1-3 that the MSEs of the estimates decrease as the sample size increases. Based on biases, the parameter λ was underestimated for all sets of true values, except in Table 2, the parameter λ was overestimated. Likewise, the parameter α was overestimated. Besides, we noticed that when the percentage of censoring was 90%, the MSE of the estimates in most cases was better.  Bayes estimates of the parameters α, β, λ, S(x 0 ), and h(x 0 ) could be found using the following Algorithm 6.
1. For given true parameter values selected as (α, β, λ), generate a random sample of size n from Equation (5).

Arrange
Step 1 in ascending order to obtain X 1:n < X 2:n < X 3:n · · · < X n:n . 3. Obtain the censored sample according to the censoring percentage. 4. For given values of the hyperparameters parameters a, b, c, d, f , ν, compute the Bayes estimates of the parameters α, β, λ via the standard Bayes technique for the first three cases using the NIntegrate function under the SE, LINEX, and GE loss functions as shown in Table 4. 5. For given values of the hyperparameters parameters a, b, c, d, f , ν, compute the Bayes estimates of the parameters α, β, λ via the importance sampling technique for all cases under the SE, LINEX, and GE loss functions, as shown in Table 4. 6. Compute the Bayes estimates of S(x 0 ) and h(x 0 ) for the four cases from (6) and (7), respectively, using the estimates in the previous steps. 7. Repeat Steps 1-6 1000 times. 8. Calculate the mean, bias, and MSE for each estimate.
The simulation study was carried out with the true value of parameter (α = 0.8, β = 4, λ = 3) and different sample sizes n = 30, 50, and 100 for all four cases. Furthermore, we considered three different values of the LINEX shape parameter (τ = 0.001, τ = 2, τ = 5) and three values of the GE shape parameter (q = −1, q = 3, q = −3). The Bayes estimates for first three cases were derived based on the standard Bayes and importance sampling techniques. For the last case, this was obtained via the importance sampling technique. The values of the hyperparameters for the standard Bayes technique are (a = 2, b = 4, c = 5, d = 2, f = 2, ν = 5). Further, the values of the hyperparameters for the importance sampling technique are (a = 80, b = 0.1, c = 14, d = 2, f = 30, ν = 0.001) with a sample size of N = 1000. For given S(x 0 ) = 0.72904 and h(x 0 ) = 0.81786 at x 0 = 1, the ML and Bayes estimates of the S(x 0 ) and h(x 0 ) were computed. The four cases were calculated as follows.   Table 5. According to Table 6, we note that the ML estimates give better values than the Bayes estimates via the importance sampling technique; 4. From Table 5, the Bayes estimates under the GE loss function (q = −3) are considered the best estimates of S(x 0 ); 5. From Table 7, the Bayes estimates of β, S(x 0 ), and h(x 0 ) via the standard Bayes technique perform the best based on MSEs and biases at n = 30. Furthermore, when n = 50, 100, the ML estimates of β, S(x 0 ), and h(x 0 ) perform the estimates better than the Bayes estimates; 6. When β is unknown, the Bayes estimates of β via the importance sampling technique perform the best at n = 30, under the LINEX loss function (τ = 5). For n = 50, 100, the ML estimates of β give the best estimates. Furthermore, based on the MSEs and biases, the ML estimates of S(x 0 ) and h(x 0 ) give the best estimates (see Table 8); 7. Based on the MSEs, the ML estimates of λ, S(x 0 ), and h(x 0 ) perform the estimates better than the Bayes estimates based on the two techniques (see Tables 9 and 10); 8. From Table 11, the ML estimates of λ, S(x 0 ), and h(x 0 ) perform the best based on the smallest MSEs. Besides, the Bayes estimates of β via the importance sampling technique perform the best under the LINEX loss function (τ = 5).

Application
The performance of the BIED based on type-II censored samples is illustrated through two real datasets. The BIED model was compared with other lifetime models, such as the inverse exponential distribution (IED) as a special case of the BIED, introduced by Lin et al. [14], the inverse Weibull distribution (IWD) introduced by Keller and Kamath [15], the Weibull inverted exponential distribution (WIED) defined by [16], the Weibull exponential distribution (WED) (see [17]), and the odd Fréchet inverse exponential distribution (OFIED) introduced by Alrajhi [18]. Moreover, the model selection criteria were considered, which included the Akaike information criterion (AIC), log-likelihood ( ), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), and Hannan-Quinn information criterion (HQIC). The smallest values of the AIC, BIC, CAIC, and HQIC, and the highest value determine the best-fit model for the data. For more details about these criteria and their uses, see [19,20].
where (Θ) denotes the log-likelihood function, p is the number of parameters, and n is the sample size: • Aluminum coupons' cut: The following data consisting of 102 observations were used by Birnbaum and Saunders [21] and correspond to the fatigue life of 6061-T6 aluminum coupons in cycles (×10 −3 ) with the maximum pressure of 26,000 psi. These coupons were cut parallel to the direction of rolling and oscillated at 18 cycles per second. Descriptive statistics of these data are presented in Table 12. Based on the descriptive statistics in Table 12, we observed that the skewness value was close to zero; thus, the distribution of the fatigue life of the 6061-T6 aluminum coupons' cut dataset was approximately normal, while the variance was 3884.30, which indicate high variability in the dataset.
According to Figure 1, we can note that there were no outliers in the fatigue life of the 6061-T6 aluminum coupons' cut data. The ML estimates and Bayesian estimates via the standard Bayes technique of the BIED parameters for the fatigue life of the 6061-T6 aluminum coupons' cut data are presented in Table 13 at two censoring percentages of 80% and 90%.  The ML estimates of the model parameters and the performance of the BIED against other models for the fatigue life of the 6061-T6 aluminum coupons' cut data are presented in Table 14 at two censoring percentages of 80% and 90%.
In Table 14, the values of the AIC, BIC, CAIC, and HQIC show that the BIED was the best model for analyzing the fatigue life of the 6061-T6 aluminum coupons' cut data. Furthermore, we can consider that the WED is a good alternative model for these data. The estimated PDF and estimated CDF of the models for the fatigue life of the 6061-T6 aluminum coupons' cut data at two censoring percentages of 80% and 90% are shown in Figures 2 and 3. (a) Estimated PDFs.
(b) Estimated CDFs.   • Patients suffering from acute myelogenous leukemia: The following data consisting of 33 observations were studied by Feigl and Zelen [22] and represent the survival times (in weeks) of patients suffering from acute myelogenous leukemia. Descriptive statistics of these data are presented in Table 15. According to the descriptive statistics in Table 15, we observed that the distribution of the survival times (in weeks) of patients suffering from acute myelogenous leukemia data was positively skewed, while the variance was 2181.17, which indicates high variability in the dataset.
According to Figure 4, we can note that there were no outliers in the survival times (in weeks) of patients suffering from acute myelogenous leukemia data. Additionally, the ML estimates and Bayesian estimates via the standard Bayes technique of the BIED for this dataset are presented in Table 16 at two censoring percentages of 80% and 90%.
The ML estimates of the model parameters and the performance of the BIED against other models for the survival times (in weeks) of patients suffering from acute myelogenous leukemia data are shown in Table 14 at two censoring percentages of 80% and 90%.    Table 17, based on the values of the AIC, BIC, CAIC, and HQIC, shows that the BIED was the best model for fitting the survival times (in weeks) of patients suffering from acute myelogenous leukemia data. Further, the estimated PDF and estimated CDF of the models for this dataset at two censoring percentages of 80% and 90% are shown in Figures 5 and 6.

Conclusions
In this article, the maximum likelihood and Bayes estimators of the BIED were derived based on type-II censored samples. The invariance property was used to estimate the survival and hazard functions. Furthermore, in the Bayesian estimation, three loss functions were used with two techniques. The gamma distribution was assumed as a prior distribution for the shape and scale parameters. Besides, it can be concluded that the MSEs of the ML estimates and Bayes estimates for the unknown parameters decreased as the sample size increased. Furthermore, when α was unknown, the Bayes estimates gave better estimates via the standard Bayes technique. The ML estimates gave better estimates than the Bayes estimates using the two techniques when λ was unknown. For the third case, when β was unknown, the ML estimates gave better results than the Bayes estimates as the sample size increased. Likewise, when β and λ were unknown, the Bayes estimates via the importance sampling technique under the LINEX loss function (τ = 5) gave better results than the ML estimates as the sample size increased. Two real datasets were applied, and the BIED provided a better fit than the other compared distributions.