TheWeibull Generalized Exponential Distribution with Censored Sample: Estimation and Application on Real Data

Applied Statistics and Insurance Department, Faculty of Commerce, Mansoura University, Mansoura, Egypt Statistics Department, Faculty of Business Administration, Delta University of Science and Technology, Mansoura, Egypt Statistics Department, University of Tabuk, Tabuk, Saudi Arabia Mathematics Department, Faculty of Science, Helwan University, Cairo, Egypt Mathematical and Natural Sciences Department, Faculty of Engineering, Egyptian Russian University, Badr, Egypt


Introduction
e importance of any distribution comes from its flexibility, and one of the most important families is the Weibull family as it has many applications in industry, medicine, and many science fields. e first one that proposed the Weibull generalized family of distributions using the Weibull generator was done in [1]. e authors in [2] used the Weibull generalized family to generate the new distribution by assuming exponential distribution as a baseline distribution, which is denoted Weibull generalized exponential distribution (WGED). According to [2], the WGED is better than different distributions as generalized exponential distribution (see [3]), beta exponential distribution (see [4]), and the beta generalized exponential distribution (see [5]) in fitting many kinds of data. Almetwally et al. [6] discussed the estimation of the WGED parameters with progressive Type-II censoring (PTIIC) schemes by using the maximum likelihood and Bayesian estimation methods. e authors in [7] addressed the estimation of WGED parameters based on generalized order statistics, and they derived the submodels of generalized order statistics such as order statistics and record values. e authors in [8] introduced a heavy tailed exponential distribution by using the alpha power method for generalized continuous distribution. Teamah et al. [9] presented Fréchet-Weibull mixture exponential distribution along with a variety of statistical properties. e most commonly used censoring schemes are Type-I censored (or time censored) and Type-II censored (or failure-censored). ese two censoring schemes do not allow for units to be removed from the experiments while they are still alive. Progressive censoring is a more general censoring scheme that allows the units to be removed from the test (see [10]). Progressive censoring is useful in a life-testing experiment because it has the ability to remove life units from the experiment, so it saves time and money. Applications under progressive Type-II censoring (PTIIC) using different lifetime distributions have been discussed by many authors, for example, see [11][12][13][14]þ and [15].
Ng et al. [16] suggested an adaptive Type-II progressive censoring scheme (ATIIPCS) in which the effective number of failures m < n is fixed in advance and the progressive censoring scheme R 1 , . . . , R m is provided. Suppose that the experimenter fixes a time T, which represents the time of the experiment, but the test itself may be allowed to run overtime T. Let us denote that m is completely observed failure times by X i: m: n , i � 1, . . . , m. If the m th progressive censored failure time occurs before time T, the experiment will be terminated at the time X m: m: n . Otherwise, once the experiment time passes time T, but the number of observed failures has not reached m, we will terminate the experiment. Many authors had discussed applications under ATIIPCS using different lifetime distributions, for example, see [17][18][19][20][21] and [22].
Ng et al. [16] introduced PTIIC by using MLE. Almetwally et al. [20] introduced PTIIC by using MPS. Basu et al. [23] developed MPS estimator for a progressive hybrid Type-I censoring scheme with binomial removals. El-Sherpieny et al. [15] introduced progressive Type-II hybrid censoring based on the MPS method with application for power Lomax distribution. Almetwally et al. [21] discussed the Weibull parameter estimation under PTIIC by using MPS and MLE methods. Maximum product spacing for the stress-strength model based on progressive Type-II hybridcensored samples with different cases has been obtained by [24]. Parameters of the extended odd Weibull exponential distribution are estimated under the progressive Type-II censoring scheme with random removal using the maximum product spacing and maximum likelihood estimation methods by [25]. e MCMC algorithm for the Bayesian estimation, it was introduced by [26]. For more information, see [20,[27][28][29] and [30].
Because of the importance of the Weibull distribution and ATIIPCS in reliability studies, we had considered the ATIIPCS applied to items whose lifetimes under design conditions are assumed to follow WGED under the ATIIPCS with the random removal. e removals from the test are considered by using the binomial distribution. MLE, MPS, and approximate confidence intervals (CI) of the estimates are presented. Bayesian estimates, percentile bootstrap CI, and bootstrap-t CI are obtained. Monte Carlo simulation study, as well as application to real data, is performed to illustrate the theoretical results. e paper is organized as follows: Section 2 is devoted to model description and notations of the WGED parameters using the classical estimation method under APTIIC. In Section 3, we introduced the classical estimation method under APTIIC. In Section 4, we introduced the Bayesian estimation method under APTIIC. A simulation study is performed to illustrate the statistical properties of the parameters in Section 5. Two real data applications are analyzed in Section 6. Eventually, the concluded remarks are given in Section 7.

Model Description and Notations
Assume a random variable X > 0 has WGED with a vector of parameter Θ � (α, c, θ), and say that its cumulative distribution function (CDF) is given by e corresponding PDF is e quantile function of the WGE distribution is In the APTIIC, the scheme can be described as follows. Assume that we set n independent observations placed on a life testing and the progressive censoring scheme R i , i � 1, 2, . . . , m. At the time of the first failure, x 1 , R 1 ∼ binomal(n − m, p) units are randomly removed from the remaining (n − 1) surviving items. At the time of the second failure, x 2 , R 2 ∼ binomal(n − m − R 1 , p) units of the remaining n − 2 − R 1 , units are randomly removed, and so on, the test continues until the m th failure at which time, and all the remaining In APTIIC, the number of failures m, with removal probability p, and time T are fixed given by the experimenter. Suppose that an individual unit was being removed from the test is independent of the others but with the same removal probability p. en, the number of units removed at each failure time follows a binomial distribution which is, for If the m th progressively censored failure time occurs before T, the experiment will be terminated at the time X m: m: n . Otherwise, once the experimental time exceeds time T, but the number of observed failures has not reached m, we would terminate the experiment as soon as possible. e data form is as follows: X 1: m: n < X 2: m: n < . . . < X D: m: n < T < . . . < X m: m: n . e number of units removed at each failure time assumed to follow a binomial distribution with the following probability mass function: While for, i � 2, 3, ..., m-1, where 0 ≤ r i ≤ n − m − i− 1 j�1 r j . Furthermore, suppose that R i is independent of X i: m: n for all i. en, the joint likelihood function can be found.
where Pr(R � r) � Pr(R 1 � r 1 , R 2 � r 2 , . . . , R m− 1 � r m− 1 ), i.e., e MLE of p can be derived by maximizing equation (6) directly. Hence, the MLE of p is obtained by solving the following equation: Since, L 1 (x i: m: n , Θ) does not involve the binomial parameter p, then the likelihood function under ATIIPCS can be written as where A is a constant that does not depend on parameters and

The Classical Estimation Method under ATIIPCS
is section deals with MLE and MPS methods of the parameters WGED based on the ATIIPCS data with binomial removal.

MLE Method.
Using equation (10), the likelihood function for WGED based on ATIIPCS can be written as where A � n(n − R 1 − 1) · · · (n − m− 1 i�1 R i − (m − 1)) is a constant which does not depend on the parameters. e natural logarithm of the likelihood function equation can be obtained as follows: For convenience, let l(Θ) � ln L 1 (x i: m: n , Θ); hence, the partial derivatives of equation (14) are given as follows: and e MLE of Θ for the WGED parameters is the solution of equations (15), (16), and (17) by using the Newton-Raphson method. Furthermore, the asymptotic CI (ACI) can be approximated numerically by inverting Fisher's information matrix. us, the 95% ACI for α, c, and θ is easily obtained, respectively, as α ±

MPS Method.
With the use of equation (11), the product spacing function for WGED based on ATIIPCS can be written as where A is a constant which does not depend on the parameters. e natural logarithm of the product spacing function is 4 Complexity Let s(Θ) � ln S 1 (x i: m: n , Θ), then the partial derivatives by the MPS method of equation (19) are given as follows: and e MPS estimates for the WGED parameters can be obtained using the Newton-Raphson method. Moreover, we use the approximate 95% CI for α, c, and θ, respectively, as follows: Also, we propose different bootstrap CIs of population parameters under the MLE method based on ATIIPCS data with binomial removal for the WGED as a bootstrap percentile (BP) and bootstrap-t (BT). For more information about this algorithm, see [31,32] and [15].

Bayesian Estimation
In this section, we consider the Bayesian estimation for the parameters of WGED based on ATIIPCS under the assumption that the random variables Θ � (α, c, θ) have an independent gamma prior distribution. Assume that [6,33]); the prior joint density of α. c, and θ can be written as e posterior likelihood can be represented to be proportional to the product of the likelihood given in equation (13) and the joint prior's densities given by equation (22). at is, en, the posterior joint density of Θ is Using the squared error loss function (SE), the Bayesian estimators of the parameters Θ are obtained as follows: ese integrals are very hard to be solved analytically, so that the MCMC approach will be used. An important subclass of the MCMC techniques is Gibbs sampling and more general Metropolis-within-Gibbs samplers. Metropolis et al. [26] were the first to introduce this algorithm. e Metropolis-Hastings (MH) algorithm and the Gibbs sampling are the two most popular examples of an MCMC method. It is similar to acceptance-rejection sampling; the MH algorithm considers that, to each iteration of the algorithm, a candidate value can be generated from a proposal distribution. e MH algorithm generates a sequence of draws from WGED under ATIIPCS as follows: Algorithm 1 and 2 .
According to [34], we obtain Bayes credible intervals of the parameters Θ � (α, c, θ) as follows: Furthermore, for different bootstrap CIs of population parameters under the Bayesian estimation method based on ATIIPCS data with binomial removal for the WGED as BP and BT, see Tables 1 and 2.

Simulation Study
In this section, Monte Carlo simulation was done for comparison between maximum likelihood and Bayesian estimation methods under censoring scheme, for estimating parameters of WGED in a lifetime by R language. Monte Carlo experiments were carried out based on the following data generated from WGED, where X is distributed as WGED for different shape parameters: We made this simulation using different sample sizes n � 50 and 150, different censored sample sizes m, and set of different sample schemes, and p is 0.25, 0.5, and 0.75.
We could define the best scheme as the scheme, which minimizes the mean squared error (MSE(Θ)), bias of estimation, and length of CI (L.CI) of the estimator. For Bayes confidence credible intervals, denoted as CCI, the CI of MLE, MPS, and Bayesian estimation and associated CI are calculated.
We conclude remarks on the simulation as follows: (1) e simulation outcomes are recorded in Tables 1-4. e following concluding remakes are noticed based on these tables as follows.

Application of Real Data
In this section, we will apply the numerical results of the parameter estimation of WGED under ATIIPCS on two cases of real data, namely, electric data and carbon fibers data. distribution functions for the data to be 0.16538, and the corresponding p value is 0.6182. Figure 1 discusses the plot of the max distance between the two CDF curves, histogram, PP-plot, and QQ-plot for WGED. erefore, it indicates that WGED can be fitted to the electric data set. Table 5 shows the estimation of parameters and standard error (St.E) for complete electric data. Table 6 displays the sample of progressive Type-II censored data with R removal for electric data. Table 7 gives the estimation of parameters and standard error under the censored sample for electric data. Histogram plot, approximate marginal posterior density, and MCMC convergence of α, c, and θ are represented in Figure 2.
Histogram plot, approximate marginal posterior density, and MCMC convergence of α, c, and θ are represented in Figure 3. [36] discussed carbon fibers of 69 observed failure times. ese data sets represent the strength of items measured in GPA for single carbon fibers and impregnated 1000 carbon fiber tows. We computed the KS test distance is 0.07408, and the corresponding p value is 0.8605. erefore, it indicates that WGED can provide a good fit for the data set by using empirical cumulative distribution function (ECDF), histogram, PP-plot, and QQ-plot for carbon fiber data in Figure 4. Table 8 shows estimation of parameters and standard error for complete carbon fiber data.
(3) For t � 0 to N (a huge number 10,000, for example), given the candidate point (Θ * ), calculate the acceptance probability (5) Repeat steps 2-4, t + 1 times until we get N draws. (6) e Bayes estimate of Θ l , with respect to squared error loss function is N t�1 ((Θ t− 1 l ) t /N). (7) Repeat the above steps l times to get a Bayesian estimate of Θ l .

Conclusions
In this paper, we discussed MLE, MPS, and Bayesian estimation to estimate the parameter problem of the WGED based on ATIIPCS with random removal. We used Bayesian estimation under the square error loss function to calculate the unknown parameters for WGED under the assumption of independent gamma priors. e performance of the different estimator optimal censoring schemes is compared based on a simulation study to determine the optimal censoring schemes by using MSE, the Bias, and the L.CI. It is noticeable that the Bayesian estimation is better and more efficient than the MLE and MPS estimation according to the MSE. We applied two real data applications based on ATIIPCS of carbon fiber and electric data which are obtained, we deduce that these sets provide an excellent fit for the proposed distribution according to the p value, and also we plotted the PP-plots and other kinds of plots to make sure that the distribution is a good candidate to these real data sets.

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

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. 14 Complexity