Classical and Bayesian Inference of Unit Gompertz Distribution Based on Progressively Type II Censored Data

Abstract In this article, two frequentist approaches and a Bayesian approach employing progressive Type II censored data are used to estimate parameters of a unit Gompertz distribution. In frequentist approach, besides conventional maximum likelihood estimation, maximum product of spacing method is proposed for parameter estimation as an alternative approach to common maximum likelihood method. Both Newton-Raphson and stochastic expectation minimization algorithms are used for computing the MLEs, while Bayes estimates are obtained using both the product of spacing function and the likelihood function. Additionally, the highest posterior density (HPD) credible intervals are compared with the approximate confidence intervals (CIs) for the parameters of the model that were derived using both traditional approaches. Moreover, percentile bootstrap technique is utilized to compute confidence intervals. Numerical comparisons are presented of the proposed estimators with respect to various criteria quantities using Monte Carlo simulations. Further, using different optimality criteria, an optimal censoring scheme has been suggested. Besides, one-sample and two-sample prediction problems based on observed sample and predictive intervals under Bayesian framework are discussed. Finally, to demonstrate the proposed methodology in a real-life scenario, maximum flood level data is considered to show the applicability of the proposed methods.


Introduction
The Gompertz distribution was first introduced in the year 1825 by Benjamin Gompertz and it became very popular among demographers and actuaries.This distribution is a generalization of the exponential distribution and has wide applicability in different spheres, especially in medical and actuarial studies.It possesses some relation with some well-known distributions such as exponential, double exponential, Weibull, extreme value (Gumbel distribution) or generalized logistic distribution (see Willekens, 2001), which makes it useful for actuarial and the medical studies.Mazucheli et al. (2019) proposed a new transformed distribution by using the transformation X ¼ exp ð−YÞ, where random variable Y follows Gompertz distribution.The authors emphasized that the new transformed distribution gives a very satisfactory fit than Beta and Kumaraswamy distributions for some specific data sets.This new transformed distribution is called the Unit-Gompertz (UG) distribution.The UG distribution has the following probability density function (p.d.f.) f ðz; a, bÞ ¼ abz −ðbþ1Þ e −aðz −b −1Þ , 0 < z < 1, a > 0, b > 0: (1) The corresponding cumulative distribution function (c.d.f.) is given by Fðz; a, bÞ ¼ e −aðz −b −1Þ : (2) The p.d.f. of this distribution can have variety of shapes while the hazard rate function is constant, increasing and upside-down bathtub-shaped.However, UG distribution has an edge over the Gompertz distribution because of the fact that upside-down bathtub-shaped hazard function cannot be modeled using Gompertz distribution.For more properties of UG distribution, one can refer to the works of Mazucheli et al. (2019) and Anis and De (2020).The UG distribution has also been criticized for its support in the range of (0, 1) because traditional normal observed lifetime data may have values greater than zero.This is in addition to its flexible modeling capacity provided by the hazard rate curves.Thus, here two hypotheses are put out to explain why the UG distribution is required to suit the random phenomena in real-world contexts.The first justification for employing lifetime models with restricted support is because, from a practical standpoint, the lifetime of products cannot be limitless.For instance, physical attributes frequently produce data that may fall inside a certain finite range in many life test experiments.Data on fractions, percentages, per capita income growth, fuel efficiency of vehicles, individual height and weight, survival times from a fatal disease, etc. are likely to fall within some bounded positive intervals (see Kumaraswamy, 1980;G� omez-D� eniz et al., 2014;Mazucheli, Menezes, & Ghitany, 2018;Mazucheli et al., 2018aMazucheli et al., , 2018b)).A lifetime model with bounded support is a preferable choice when analyzing the samples, since it can give the collected data more weight and provide a better-fitting effect.On the other hand, there are also a lot of randomly generated data resulting from real-world circumstances, from which observations are in fact bounded with lower and upper ends.So, taking into account both fitting capability and lifetime data, distributions with finite support may occasionally be more appropriate in fitting practical phenomenon.Due to its adaptability, the UG distribution with unit bounded support could therefore be considered as a suitable model in reliability and lifetime studies as well as other field of applied disciplines.
After the introduction of UG distribution in literature, to the best of our knowledge, very little work has been conducted by researchers on this distribution.Notable among these works are: Jha et al. (2019) considered UG distribution to estimate multicomponent stress-strength reliability based on classical and Bayesian approaches using complete sample.Kumar et al. (2020) studied classical as well as Bayesian estimation of the parameters of the model based on lower record values and inter-record times.They also obtained prediction of future record values for the UG distribution.Furthermore, they derived single and product moments of lower record values.Jha et al. (2020) again considered this model using classical and Bayesian approaches to estimate the multicomponent stress strength reliability under progressive type II censoring samples.They also obtained bootstrap, asymptotic and highest posterior density confidence interval of reliability quantity.Recently, Arshad et al. (2023) studied this model under the framework of dual generalized order statistics.They obtained the parameters of the model using Markov Chain Monte Carlo and Lindley's approximation methods based on order statistics and lower record values.
In lifetime and reliability experiments, the necessity for censored data arises due to non-availability of complete information on failure times data for all experimental units.Although several censoring schemes have been developed over the years, yet researchers seem to prefer progressive type II censoring scheme over other conventional censoring schemes.Under this scheme, experimenter can reduce the total time on test and/or the number of failed items/survival test units can be withdrawn during the experiment at different stages which is not possible in case of conventional type II censoring scheme.The progressive type II censoring can be described as follows.Suppose that n identical and independent units are put on a life test.When the first failure occurs, X 1:m:n , R 1 of the n − 1 active units are randomly removed from the test.Similarly, when the second failure occurs, X 2:m:n , R 2 of the n − R 1 − 2 active units are randomly removed from the test, and the test continues.This is repeated m times, where 1 � m � n is prefixed, after which all R m remaining surviving items are removed.Here, is the progressive censoring scheme.The progressively type II order statistics generated through R are denoted by X ¼ ðX 1:m:n , X 2:m:n , :::, X m:m:n Þ: Independently, Cheng and Amin (1983) and Ranneby (1984) introduced the method of maximum product of spacing function as a competitive method to the method of maximum likelihood.The maximum product of spacing estimators (MPSEs) are obtained by maximizing the product of spacing (PS) function similar to maximum likelihood estimators (MLEs).Therefore, it is expected that the MPSEs retain most of the properties of the MLEs.Anatolyev and Kosenok (2005) stated that the MPSEs are more efficient than the MLEs for skewed distributions or in small sample cases for heavy tailed distributions.Nassar et al. (2018) compared the performance MPSE with MLE, moment estimation, L-moment estimation, least square and weighted least squares estimation, and Cram� er-von-Mises estimation to estimate the parameters of transmuted exponentiated Pareto (TEP) distribution.For some recent references, see for example, Almetwally and Almongy (2019), El-Sherpieny et al. (2020) Shakhatreh et al. (2022) and Yadav et al. (2021).The general condition to obtain the MPSEs is that f(z) > 0; 8z 2 ða, bÞ, and all the sample items are i.i.d.In our case, we assume that Z follows the unit Gompertz distribution with support (0, 1), which furnishes a ¼ 0 and b ¼ 1.
In the premise of the above, we have not come across any work related to estimation of the parameters of the UG distribution under progressive type II censoring using two frequentist and Bayesian methods along with optimal censoring scheme and one-sample and two-sample prediction problems based on observed sample under Bayesian framework.Our objectives in this study are: First, estimating the parameters of the UG distribution using two frequentist estimation approaches, namely; conventional maximum likelihood and MPS estimation methods.In addition, based on these two methods, we have obtained approximate confidence intervals (ACIs) for the parameters.Second objective is to obtain the MLEs by using both Newton-Raphson (NR) and Stochastic Expectation Maximization (SEM) algorithms.Third objective is to obtain the Bayes Estimates (BEs) of the UG distribution parameters based on likelihood (LK) and PS functions under squared-error loss function using independent gamma priors.Furthermore, Markov Chain Monte Carlo (MCMC) techniques are considered to compute the posterior functions and consequently, Bayes estimates and associated credible intervals are computed.Fourth objective is to obtain an optimal censoring scheme using different optimality criteria.Fifth objective is obtain Bayes predictive intervals based on one-sample prediction and twosample prediction methods.Using various choices of the effective sample size, the performance of the proposed methods is compared through a simulation study in terms of their bias, mean squared-error (MSE), sample standard error (SSE) and estimated standard error (ESE) and confidence intervals (CI) are compared in terms of their average confidence lengths (IL) and coverage probability (CP).A well-known data set on maximum flood level (in millions of cubic feet per second) for Susquehanna River at Harrisburg, Pennsylvania has been re-analyzed.
Rest of the paper is organized as follows: Classical point estimators based on progressive type II censored sample through LK and PS functions are investigated in Sections 2 and 3. Section 4 provides SEM algorithm to estimate MLEs.Section 5 provides asymptotic confidence intervals based on LK and PS functions.Section 6 provides Bayes estimates as well as associated credible intervals using each of the proposed frequentist functions.The simulated results are presented in Section 7. In Section 8, optimal censoring plans are presented.Prediction problem has been considered in Section 9.An application using real dataset is provided for illustrative purposes in Section 10.Finally, we conclude the paper in Section 11.

Maximum Likelihood Estimators
Based on the observed sample X ¼ ðX 1:m:n , X 2:m:n , :::, X m:m:n Þ from a type II progressive censoring scheme, R ¼ ðR 1 , :::, R m Þ, the LK function can be written as where The associated log-likelihood function (without constant term) can be expressed from (3) as First, we state the following theorem regarding the uniqueness and existence of the MLEs.
Theorem 2.1.The MLEs of a and b for a > 0 and b > 0 exist and unique.
Proof.The proof is deferred in the Appendix.
w Upon differentiating Eq. ( 4) with respect to a and b and equating to zero, the resulting equations must be satisfied to obtain the MLEs of a and b.The normal equations are where Very simple iterative procedure like bisection or Newton-Raphson method may be used to maximize Eqs. ( 5) and ( 6) to obtain the MLEs of a and b say â and b, respectively.

Maximum product of spacing estimation
Based on a progressive type II censored sample, we can write the PS function as follows with x 0:m:n ¼ v 0 ¼ 0 and x mþ1:m:n ¼ v mþ1 ¼ 1: The MPSEs of a and b can be obtain by maximizing (8) with respect to a and b or equivalently by maximizing the natural logarithm of the PS function in the following form The MPSEs of a and b denoted by e a and e b can be obtained by solving the following two normal equations @s n ða, bjXÞ @a ¼ and Since there are no closed from solutions for the MPSEs, therefore, one can adopt an iterative procedure to obtain MPSEs numerically from Eqs. ( 10) and (11).Cheng and Traylor (1995) stated that the MPSEs are consistent and exhibit similar asymptotic properties to the MLEs under more general conditions.Also, the MPSEs possess the invariance principle similar to the MLEs (see for more details Coolen & Newby, 1990).

Stochastic EM algorithm
It can be seen that in evaluating the MLEs using Newton-Raphson, one can face two difficulties.The first one is the sensitivity of the obtained estimators to the initial values of parameters, and the second is the calculation of the second-order derivatives of the loglikelihood based on progressive data sometimes can be tedious.Alternatively, to find the MLEs, we propose a version of expectation-maximization (EM) algorithm which is called stochastic EM (SEM) algorithm.First, we explain the idea of EM algorithm.The EM algorithm, proposed by Dempster et al. (1977), is an iterative technique and is widely used approach for computing the maximum LK estimates of incomplete data or missing information problems.Here, we treat the censored observations as data missing information and apply EM algorithm to find the MLEs.The EM algorithm includes two main steps; Expectation (E)-step and Maximization (M)-step.Assume the complete lifetimes are given by W ¼ ðX, ZÞ where X ¼ ðX 1:m:n , :::, X m:m:n Þ denotes the observed observations and Z ¼ ðZ 1 , :::, Z m Þ denotes the censored data where Z j ¼ ðZ j1 , :::, Z jR j Þ: Then the complete log-likelihood function based on the complete lifetimes, W, is proportional to To perform the E-step of the EM algorithm, we need to compute the conditional expectation of the complete log-likelihood conditionally on the observed data X, using the current value a ðkÞ and b ðkÞ of the parameters a and b as follows.Now In the M-step, we find a ðkþ1Þ and b ðkþ1Þ which maximize the conditional expectation given in Eq. ( 12).This is easily achieved by solving the following likelihood equations Observe that, the explicit expressions of the conditional expectations in Eqs. ( 13) and ( 14) can be written as follows.For any function gðz ij ; a, bÞ, i ¼ 1, 2, :::, m; j ¼ 1, :::, R i , we have x i:m:n gðy; a, bÞy −ðbþ1Þ e −aðy −b −1Þ dy Ð 1 x i:m:n y −ðbþ1Þ e −aðy −b −1Þ dy It is observed that the E-step of the EM algorithm involved complex integrals which cannot be solved in a closed form.The SEM algorithm is an alternative method of the EM algorithm where the expectation in the E-step is calculated using Monte Carlo simulations.It is useful for the cases when the E-step is hard to calculate exactly.The idea of approximating the E-step in EM algorithm by the Monte-Carlo technique, was first proposed by Wei and Tanner (1990).As mentioned by Wang and Cheng (2010), the approximation of Wei and Tanner (1990) have more time-consuming.Later, Diebolt and Celeux (1993) modified their idea by replacing the E-step with stochastic step through simulation technique.For more information about SEM, see for example, Tregouet et al. (2004), Zhang et al. (2014) and Arabi Belaghi et al. (2017).
The description of SEM method is as follows.Note that the conditional survival function of a random variable Z ij , given Z ij > x i:m:n , can be computed by We first generate independent R i number of samples z ij , i ¼ 1, 2, :::, m; j ¼ 1, :::, R i from the conditional survival function, Eq. ( 16), using the expression where u is a random variate from the uniform distribution, U ð0, 1Þ: Upon using this simulated sample, Eqs. ( 13) and ( 14) reduce to Therefore the SEM algorithm works as follows.Set initial values of a and b as a ð0Þ and b ð0Þ : Step(i) At k-th iteration, let ða ðkÞ , b ðkÞ Þ be the estimate of ða, bÞ: Step(ii) Using the expression (17), simulate z ij � z ij ða ðkÞ , b ðkÞ Þ, i ¼ 1, :::, m; j ¼ 1, :::, R i , where a and b are replaced by a ðkÞ and b ðkÞ , respectively.
Step(iv) If ja ðkþ1Þ − a ðkÞ j þ jb ðkþ1Þ − b ðkÞ j < �, for some pre-specified quantity �, then set a ðkþ1Þ and b ðkþ1Þ , as the MLEs of a and b, otherwise, set k ¼ k þ 1 and go to Step(ii).

Asymptotic confidence intervals (ACIs)
In this section, we construct three types of 100ð1 − cÞ% confidence intervals for the unknown parameters a and b: The first type of confidence interval is obtained by using the MLEs, the second by using MPSEs and the third by using parametric bootstrap method.

ACIs based on MLEs
From the log-likelihood function Eq. ( 4), the second order partial derivatives of l n ða, bjXÞ can be obtained directly with respect to a and b as follows It is observed that the asymptotic variance-covariance (VarCov) of the MLEs of a and b cannot be obtained in closed form because of the complicated nature of the expectations of the expressions .Therefore, we obtain the approximate asymptotic VarCov matrix for the MLEs by obtaining the inverse of the observed Fisher information matrix as follows Iðâ, bÞ ¼ − @ 2 l n ða, bjXÞ @a 2 − @ 2 l n ða, bjXÞ @a@b − @ 2 l n ða, bjXÞ @b@a − @ 2 l n ða, bjXÞ @b 2 Using the asymptotic properties of the MLEs, it is known that ðâ, bÞ � N 2 ðða, bÞ, Iðâ, bÞÞ, where Iðâ, bÞ is given in Eq. ( 23).Thus, the ð1 − cÞ% ACIs of the parameters a and b can be obtained as follows â6z c=2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi c varðaÞ q and b6z c=2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi c varðbÞ q , where z c=2 is the upper ðc=2Þ th percentile point of a standard normal distribution.

ACIs based on MPSEs
Here we construct the ð1 − cÞ% ACIs of the a and b based on the asymptotic properties of the MPSEs.First, we obtain the second order partial derivatives of the logarithm of PS function, s n ða, bjXÞ, given in Eq. ( 9) as follow and Due to the difficulties in obtaining the expectations of the expressions Eqs.(24-26), the asymptotic VarCov of the MPSEs of a and b cannot be obtained.Similar to the previous case in Subsection 5.1, we obtain the approximate asymptotic VarCov matrix for the MPSEs as Iðe a, e bÞ ¼ − @ 2 s n ða, bÞ @a 2 − @ 2 s n ða, bÞ @a@b − @ 2 s n ða, bÞ @b@a − @ 2 s n ða, bÞ @b 2 Based on the asymptotic properties of the MPSEs as in the case of the MLEs, (see, Cheng & Traylor, 1995) it follows that ðe a, e bÞ � N 2 ðða, bÞ, Iðe a, e bÞÞ, where Iðe a, e bÞ is given by Eq. ( 27).Therefore, the ð1 − cÞ% ACIs of a and b are e a6z c=2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi f varðaÞ q and e b6z c=2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi f varðbÞ q : (28)

Parametric bootstrap (Boot-p) CIs
The asymptotic normality of MLEs used to find the confidence interval is well-performed only when the sample size is large enough.Therefore, alternatively, in this subsection, the confidence intervals are computed using parametric percentile bootstrap (Boot-p) method even for small sample size.The bootsrap method, proposed by Efron and Tibshirani (1986), are widely used to estimate standard error or to build confidence intervals for the parameters as well as estimating the reliability and hazard functions.The generation of progressively type II Boot-p samples can be described as follows. Step respectively, where ½x� denotes the integral part of x.

Bayesian estimation
In this section, we consider Bayesian estimation of the unknown parameters of the UG distribution under progressively type II censored data.We mainly discuss the Bayes estimates and the associated credible intervals of the unknown parameter(s) based on LK and PS functions.In our Bayesian analysis, we have assumed only squared error loss function.However, other loss functions can be considered.Prior distributions play an important role for obtaining the Bayes estimates.However, there is no clear cut method to choose the best priors for the unknown model parameters for Bayesian estimation problem.In this regard, readers may refer to the works of Arnold and Press (1983).
In the premise of the above arguments, we consider the piecewise independent gamma priors for the parameters of the considered model.Thus the proposed priors for the parameters a and b may be taken as: and The hyperparameters a, b, c, d are known and non-negative.As the family of gamma distributions is highly flexible, and can provide different shapes based on parameter values and thus it can be considered as suitable priors of the model parameters.See for more details, Kundu and Pradhan (2009) and Dey, Singh, et al. (2016).Thus, the joint prior distribution for a and b is pða, bÞ / a a−1 b c−1 e −ðbaþdbÞ , a, b > 0: (31) Based on the observed sample x 1:m:n < x 2:m:n < � � � < x m:m:n from a type II progressive censoring scheme, the joint posterior density using joint prior distribution defined in Eq. ( 31), and the traditional LK function can be written as The marginal posterior probability density functions of a and b are given respectively as and Similarly, based on the PS function, Eq. ( 8) and the joint prior distribution defined in Eq. ( 31), the joint posterior density and the marginal posterior density functions of a and b can be, respectively, expressed as where and Suppose x is an estimator of parameter x, we propose to use squared-error (SE) loss function which is defined as From ( 38) the Bayes estimate is given by the posterior mean of x.For arbitrary function uða, bÞ, the Bayes estimator, namely ûB (or e u B ), is the expectation of the posterior distribution under squared error loss, which is given by using the LK function and using the PS function.
Next, we construct the credible intervals for a and b: Let h denote the parameter a or the parameter b: After obtaining the marginal posterior distribution of h, a symmetric 100ð1 − cÞ% Bayes credible interval of h, denoted by ½L h , U h �, can be obtained by solving the following equations and We need to apply suitable numerical method to compute the above intervals.

Bayesian Inference using MCMC approach
It is clear that the Bayes estimates of the unknown parameters a and b do not possess closed forms.Therefore, instead of applying numerical methods, we utilize Markov Chain Monte-Carlo (MCMC) approach to obtain the Bayes estimates and construct credible intervals of the unknown parameters.The advantage of using MCMC is its ability to construct Highest Posterior Density (HPD) credible intervals for a and b: We first draw random samples from the posterior density function, pða, bjxÞ and then, we use the simulated values to compute the Bayes estimates and HPDs of a and b: Here pða, bjxÞ denotes Xða, bjxÞ given in Eq. ( 32) or Wða, bjxÞ given in Eq. ( 35).Since the posterior density function pða, bjxÞ cannot be simulated easily, we utilize a random walk Metropolis-Hastings (MH) algorithm to generate samples from it.The purpose of using the random-walk MH algorithm is due to its flexibility in generating random samples from any proposal distribution, especially when the conditional posterior distributions of the parameters are unknown distributions.For our case, we propose normal densities Nðl i , ri Þ, i ¼ 1, 2, where l i ¼ a For constructing HPD credible intervals of a and b, we use the method proposed by Chen and Shao (1999) where ½x� denotes the integral part of x.Thus the HPD credible interval of a (or b) can be derived by choosing the credible interval which has the shortest length.
For the classical estimation, we estimate the unknown parameters using the MLE, MPS and SEM methods.For each of these methods, we have computed the absolute average bias (Bias), the sample standard deviation (SSE), the estimated standard deviation (ESE) using the observed information matrix based on the LK (or PS) function, the mean square error (MSE).Moreover, we have evaluated 95% Wald's confidence intervals using the observed information matrix based on the LK (or PS) function (CI), the Boot-p based on LK function (MBT) and Boot-p based of PS function (PBT).
For the Bayes estimations, we obtain the Bayes estimate by numerically solving Eq. (39) (BSL), the Bayes estimate by numerically solving (BSP) Eq. ( 40), MCMC based on LK fuction (MCL) and MCMC based on PS function (MCP).For each of these methods, we compute the absolute average bias (Bias), mean square error (MSE), credible intervals (CI) by numerically solving Eqs. ( 41) and ( 42) and HPD intervals using the generated MCMC samples based on LK (or PS) function.For a given sample, the random-walk MH algorithm is used to generate MCMC samples of size M ¼ 20,100 with burn-in period of N ¼ 100: With this respect for selecting the hyperparameters we follow the procedure given in Dey, Dey, et al. (2016) and Singh and Tripathi (2018).The process for the both types of estimation (classical and Bayesian) is replicated 1,000 times.The results of classical estimation are reported in Tables 1 and 2, and the results of Bayes estimation are reported in Tables 3 and 4.
From Table 1, it is observed that the Bias for all the estimators, in general, are reasonably small which indicates that the estimated values are close to the true parameter values.As expected, the Bias, MSE, SSE an ESE of all estimators are decreasing when sample sizes are increasing for all the cases.However, the MPS method presents less bias estimates than the MLE and SEM for the all cases.In addition, the SEM algorithm performs better than MLE based on this aspect.Clearly, the MSE of MPS and SEM is less than that of MLE.Moreover, MPS presents higher MSE as compared with SEM.The SSE for the SEM method are less than that of MLE and MPS methods while the ESE of the MLE is the while the ESE of the MLE is less that than of MPS and SEM.
With respect to 95% confidence interval, from Tables 2 and 3, the confidence intervals constructed by Boot-p methods based on LK function (MBT) have the smaller.In addition, the length of the confidence intervals is decreasing when the value of sample size is increasing.Moreover, as n increases the CPs of a and b become very close to the nominal level.Hence, the performance of the MLEs are satisfactory in terms of the biases and standard errors of the estimates.For the Bayesian estimates, the Bias of all proposed methods are also reasonably small.It is clear that the Bias, RMSE and the length of the credible/HPD intervals are decreasing and the HPDs of a and b become very close to the nominal level when the value of sample size is increasing.

Optimal censoring
In life-testing experiment, it is usually taken a fixed and pre-specified censoring scheme.However, choosing the optimum censoring scheme from a set of all possible schemes can be of important concerns to the estimation problem, as it may lead to efficient estimates for parameters.In this section, instead of considering pre-specified inspection censoring scheme, we investigate by using different techniques for selecting the optimum censoring scheme under the progressive type II censored data.The problem of identifying the optimal censoring scheme for different distributions has received considerable attention in the statistical literature.See for example, Abouammoh and Alshingiti (2009), Pradhan and Kundu (2013), Sultan et al. (2014), Dube et al. (2016), Sen et al. (2018) and Ashour et al. (2020).The first two criteria, proposed in this section, for selecting the optimal censoring scheme depend on the comparison of the Fisher information matrices (or equivalently of the variance-covraince (VarCov) matrix) of the MLEs of the unknown parameters as follow.

Criterion (I):
Minimizing the determinant of VarCov matrix of the MLEs.

Criterion (II):
Minimizing the trace of VarCov matrix of the MLEs.
In the case of one parameter distributions, criteria I and II are quite effective.However, if we have multi-parameter distributions, these criteria are not scale invariant (see, Gupta & Kundu, 2006).So we have to use other criteria which are scale invariant.The other two criteria, we adopted here, are based on the comparison of the precisions of the logarithm of MLE for p-th quantile of the UG distribution.For the UG distribution, the logarithm for p-th quantile is given by Then, the asymptotic variance of log ð Tp Þ, the MLE of log ðT p Þ based on the censoring scheme ðR 1 , :::, R m Þ, can be computed by using delta method as Varð log ð T p ÞÞ ¼ G T ðâ, bÞVarCovðâ, bÞGðâ, bÞ, where − log ðpÞ=âÞ b2

!
Table 5 presents the results of optimal censoring based on MLEs and MPSs with respect to Criteria I-VI for ða, bÞ ¼ ð1:25, 1:5Þ, ð0:5, 0:75Þ, m ¼ 5 and n ¼ 10, 15, 20: From Table 5, we observed that the censoring scheme ð0 � m, n − mÞ is the most preferred one among the other schemes based on all criteria.Moreover, the reported censoring schemes are almost the same under the criteria V and VI for the MLE with n ¼ 10, 15 and for the MPS with n ¼ 10, 20:

Bayesian prediction
The prediction of the censored or future observation(s) based on current sample (also known as informative sample) is important practical problem in applied statistics.For more information and recent references about this problem, see the articles Al-Hussaini (1999), Mousa and Jaheen (2002), Balakrishnan et al. (2010), Dey and Dey (2014), AL-Hussaini et al. (2015), Dey, Singh, et al. (2016) and Kotb and Raqab (2019).Our main objective in this section is to investigate the predictive estimate and the predicative interval of the j-th order statistic T ¼ X j:R k , j ¼ 1, 2, :::, R k ; k ¼ 1, 2, :::, m, based on progressively type II censored sample with respect one-sample and two-sample prediction problems.

One-sample prediction problem
Let X ¼ ðX 1:m:n , X 2:m:n , :::, X m:m:n Þ be a progressively type II censored sample with progressive censoring scheme R ¼ ðR 1 , :::, R m Þ: To simplified our notation, let X k:m:n ¼ X k and x k:m:n ¼ x k : Let T ¼ X j:R k denote the j-the order statistic out of R k removed unites at stage k, for j ¼ 1, :::, R k and k ¼ 1, 2 � � � , m: We assume that the predicted value of the j-th order statistic, X j:R k , should be take a value in the range ½x k , x kþ1 �: Then the conditional distribution of T given X ¼ x is given by where j ¼ 1, 2, :::, R k and k ¼ 1, 2, :::, m: Here, the p.d.f.f and the c.d.f.F are given in (1) and (2), respectively.Upon substituting Eqs.(1 and 2) and using binomial and negative binomial expansions in the above expression, we get, for 0 < x k < t < x kþ1 < 1, Let pða, bjXÞ be the joint posterior density Xða, bjXÞ given in Eq. ( 32) or the joint posterior density Wða, bjXÞ given in Eq. ( 35).Then, the Bayes prediction density function of T given X ¼ x, can be computed by taking the expectation of the conditional distribution of T given X ¼ x, f TjX , with respect to the joint posterior density, pða, bjXÞ, as follows.
From Eq. ( 43), the Bayes predication of T ¼ X j:R k , j ¼ 1, 2, :::, R k , given X ¼ x, can be obtained by Consequently, the predictive survival function of T ¼ X j:R k given X ¼ x is obtained as Therefore the 100ð1 − cÞ% Bayes predictive interval of T ¼ X j:R k can be computed by solving the following equations for the lower bound, L, and upper bound , U, as

Two-sample Bayesian predication
Let Y ¼ ðY 1 , Y 2 , :::, Y N Þ be an unobserved independent ordered sample of size N from the same population of the informative sample.This sample will be referred to as the future sample.Our aim here is to predict the j-th ordered statistic in the future sample, Y j , based on the progressively type II censored informative sample.The density function, Y j , is given by f Y j jX ðyjx, a, bÞ ¼ f Y j ðyja, bÞ ¼ j N j � � ðFðyja, bÞÞ j−1 ð1 − Fðyja, bÞÞ N−j f ðyja, bÞ: Upon substituting Eqs. ( 1) and (2) and using binomial expansion in the above expression, we get Then the Bayes predictive density function of Y j given x is given by Consequently, the predictive survival function of Y j given x, S � Y j jX , is obtained as Similarly, the 100ð1 − cÞ% Bayes predictive interval of Y j given x can be computed by solving the following equations for the lower bound, L and upper bound U The solutions of the above equations cannot be obtained analytically so we apply a numerical technique for solving them simultaneously.

Real data analysis
In this section, we analyze a data set as a real life application of the UG distribution.The data set represents 20 observations of the maximum flood level (in millions of cubic feet per second) for Susquehanna River at Harrisburg, Pennsylvania, and is reported in Dumonceaux and Antle (1973).
Dataset: 0:26, 0:27, 0:30, 0:32, 0:32, 0:34, 0:38, 0:38, 0:39, 0:40, 0:41, 0:42, 0:42, 0:42, 0:45, 0:48, 0:49, 0:61, 0:65, 0:74: By Mazucheli et al. (2019), the GU distribution fits the real data set in comparison to the beta, Kumaraswamy and McDonal distributions.The MLEs (standard errors) of the completed data set of a and b are 0:02ð0:02Þ and 4:14ð0:74Þ, respectively.For estimating the unknown parameters, we have considered 4 censoring schemes for LK function and 5 censoring schemes for MP function corresponding to the six optimal criteria presented in Section 8.Moreover, to compare the performance of the estimators based on these censoring schemes with that of the conventional Type-II (T 2 ) censoring scheme, this scheme is considered as well.The progressively censored samples are generated from the completed set by considering m ¼ 10 and using all the proposed censoring schemes and the generated samples are reported in Table 6.With respect to the hyperparameters, since we have no priori information is available, we consider a non-informative prior a ¼ b ¼ c ¼ d ¼ 0:001: The results of classical and Bayesian estimations are presented in Tables 7 and 8.In Table 7, we report the estimates (Est.), estimated standard error (ESE), 95% confidence interval (CI) and the length of these intervals (IL) of the parameters a and b using MLE, MPS and SEM methods.Moreover, 95% percentile bootstrap confidence interval (CIB) and their lengths (ILB) for the two parameters using MLE and MPS methods are also included.In Table 8, we report Bayes estimates (Est.), 95% credible/HPD intervals (CI/HPD) and their lengths (IL) of the two parameters using numerical calculations (DIR) and MCMC methods based on the LK and PS functions.From the reported values in Tables 7 and 8, it can be seen that, in terms of confidence/ceridable intervals, for the majority of the cases, the censoring scheme III has lower interval lengths for the classical estimation while the censoring scheme II has the lower interval lengths for Bayesian estimation.
With respect to the prediction problem, the predictive values, e x j:R k or Y j , and the 95% prediction intervals, (L,U), and their lengths (IL) using one-sample and two-sample techniques for the first three optimal censoring schemes of each of LK and PS functions and T 2 censoring scheme, with N ¼ 20 and m ¼ 10 are reported in Tables 9 and 10.From Table 10, it can be seen that, for a future sample of size 10, the length of the predication intervals become wider with the increase in j for j < 10 and for all the censoring schemes.
Table 9. One-sample predictive estimates of the real data set, e x j:R k , 95% prediction intervals, (L,U) and their lengths, IL, using LK and PS functions for the first three optimal censoring schemes, I, II and III and T 2 scheme.
LK function (MBT) have the smaller CI.Bayes estimates obtained by PS method out performs LK method in terms of both bias and MSE, while the length of the HPD intervals based on LK function are shorter than PS function.In regard to optimal censoring, the censoring scheme ð0 � m, n − mÞ is the most preferred one among the other schemes based on all criteria.However, the considered censoring schemes are almost the same under the criteria V and VI for the MLE with n ¼ 10, 15 and for the MPS with n ¼ 10, 20: In real data analysis, we observed that confidence/ceridable intervals based on censoring scheme III has the lower interval lengths for the classical estimation while the censoring scheme II has the lower interval lengths for Bayesian estimation.Finally, the Bayesian approach according to both the traditional LK and PS functions to estimate the parameters of the UG distribution under progressive type II censoring is recommended.We hope that the methodologies proposed in this work will be useful to applied statisticians.It will be interesting to study the methods of estimation under hybrid censored data.The work is in progress, and it will be reported later.

(
or b), ri is the estimated variance of a (or b).These samples are utilized to obtain the Bayes estimates and construct credible intervals for the parameters as given in Algorithm 1.The retained sample values, ða 1 , b 1 Þ, :::, ða M , b M Þ is a random sample from the posterior density pða, bjxÞ: To reduce the effects of the initial values on the simulated samples, we discard some of the initial N < M number of samples (burn-in).Now, using Monte-Carlo integration technique, the Bayes estimates of a and b under squared error loss function can be obtained as as follows.Let a ðNþ1Þ < a ðNþ2Þ < � � � < a ðMÞ and b ðNþ1Þ < b ðNþ2Þ < � � � < b ðMÞ be the ordered values of a j and b j for j ¼ N þ 1, :::, M: Consider the following 100(1-c)% credible intervals of a and b ða ðjÞ , a ðjþ ð1−cÞM ½ �Þ Þandðb ðjÞ , b ðjþ ð1−

2). Step (4): Repeat Step(2) and Step(3), for
(1): Compute the MLEs, â and b, based on the original progressively type II censored sample X ¼ ðX 1:m:n , :::, X m:m:n Þ: Step (2): Based on the computed MLEs in Step (1), â and b, generate a progressively type II censored sample with the same censoring scheme R and same values of ðn, mÞ, utilizing the algorithm given in Balakrishnan and Sandhu (1995).Step (3): Compute the MLEs, â� and b� , based on the generated bootstrap sample in Step(B times, where B is a pre-specified quantity.Then we have two series of estimators â� 1 , â� 2 , :::, â� ðBÞ : Then 100ð1 − cÞ% Boot-p confidence intervals of a and b are computed by â�

Table 1 .
Simulation results of classical estimation methods.
MLE: maximum likelihood estimator; MPS: maximum product of spacing estimator; SEM: stochastic EM; bias: absolute average bias; SSE: sample standard error; ESE: estimated standard error; MSE: mean square error.

Table 2 .
Simulation results of classical estimation methods.

Table 3 .
Simulation results of Bayesian methods.

Table 5 .
The optimal censoring scheme (optimum) with its value (value) based on MLEs and MPSs for the criteria (crit)I-VI when m ¼ 5 and n ¼ 10, 15, 20:

Table 6 .
The generated progressively type II censored sample from the real data set based on optimal censoring schemes with respect to criteria I-VI using the likelihood (LK) and product of spacing (PS) functions and type II (T 2 ) censoring scheme.