Adaptive Type-II Progressive Censoring Schemes based on Maximum Product Spacing with Application of Generalized Rayleigh Distribution

In this paper, parameters estimation for the generalized Rayleigh (GR) distribution are discussed under the adaptive type-II progressive censoring schemes based on maximum product spacing. A comparison studies with another methods as maximum likelihood, and Bayesian estimation by use Markov chain Monte Carlo (MCMC) are discussed. Also, reliability estimation and hazard function are obtained. A numerical study using real data and Monte Carlo Simulation are performed to compare between different methods.


1.Introduction
introduced the GR distribution. A random variable has the GR distribution with parameters and , say if its cumulative distribution function (cdf) and probability density function (pdf) are given by respectively, where , > 0 and > 0.
The maximum likelihood estimators, the modified moment estimators, estimators based on percentiles, least squares estimators, weighted least squares estimators and the modified L-moment estimators by using extensive simulation techniques based on complete data are obtained to estimate parameters for GR distribution by Kundu and Raqab (2005). The Maximum likelihood and Bayesian approaches based on progressive Type-II censoring are obtained for GR distribution by Raqab and Madi (2011). The maximum likelihood and Bayesian estimation of the parameters for GR distribution and reliability function under type-I progressive hybrid censoring scheme had discussed by Tomer et al. (2014). The maximum likelihood estimation of the shape and scale parameters concerning GR distribution based on ranked set sampling and its some modifications had discussed by Esemen and Gürler (2018). In censoring scheme, the most important used censored schemes are Type-I and Type-II censoring. In general, censoring scheme called Progressive censoring scheme are placed on a life testing experiment and is a predetermined number of units to be failed, for more information see Balakrishnan and Sandhu (1995) and Balakrishnan (2007)  The maximum product spacing estimation (MPSE) method was introduced by Cheng and Amin (1983) and independently discussed by Ranneby (1984) as an alternative method to maximum likelihood estimation (MLE) method for continuous distributions, for example see Singh et al. (2014) and to more information see Ekström (2006). Ng et al. (2012) introduced estimation of the parameters for a threeparameter Weibull distribution based on progressively Type-II right censored samples using the MPSE method. See for example Singh et al. (2016) and Almetwally and Almongy (2019). Basu et al. (2018) introduced product of spacing estimator for a Progressive hybrid Type-I censoring scheme with binomial removals.
The goal of this paper is parameters estimation for the GR distribution under adaptive type-II progressive censoring scheme based on MPSE. The comparison is done by using the methods (MLE, MPSE and Byesian) which are proposed in this study. The optimal censoring scheme depend on using three different methods for optimality criteria (mean squared error (MSE), Bias and relative efficiency (RE)). To evaluate the performance of the estimators, extensive simulation techniques is carried out. Application of real data set is introduced to confirm the validity of the model and the scheme.
The paper is organized as follows: In section 2, we introduce adaptive Type-II progressive censoring scheme based on MPSE. Parameters estimation of the GR Experiment terminates before time T Experiment terminates after time T . . . distribution under adaptive Type-II progressive censoring scheme are derived in section 3, while in section 4 the Reliability Function estimation is discussed. In section 5, Monte Carlo simulation study to compare the performance of the parameters estimation for different methods is presented. In section 6, two applications of real data sets are studied. Finally, the results and conclusion of the current study are discussed.

2.Adaptive Type-II Progressive Censoring Schemes of MPSE
Based on the observed sample 1: : < ⋯ < : : from an adaptive Type-II progressive censoring scheme, the maximum product spacing under adaptive Type-II progressive censoring scheme can be written as:

3.Parameters Estimation of the GR Distribution
In this section, MLE, MPSE and Bayesian estimation by use MCMC of the GR distribution under adaptive type-II progressive censored data have been discussed.

3.1.MLE Method
The likelihood function of GR distribution under adaptive type-II progressive censored data can be written as to obtain the normal equations for the unknown parameters, we differentiate equation (3.2) partially with respect to the parameters and and equate them to zero. The estimators for ̂ and ̂ can be obtained as in the solution of the following equations: But the equations (3.3) and (3.4) have to be performed numerically using a nonlinear optimization algorithm.

3.2.MPSE Method
The maximum product spacing under adaptive Type-II progressive censoring scheme for GR distribution can be written as:

3.3.Bayesian Estimation
We consider the Bayesian estimation of the unknown parameters and . The Bayesian estimates is considered under assumption that the random variables and have an independent gamma distribution is a conjugate prior to the GR distributions under adaptive type-II progressive censored scheme. Assumed that ∼ ( , ) and ∼ ( . ) according to Kundu and Howlader (2010) and Madi and Raqab (2009) then, the joint prior density of and can be written as where all the hyper parameters , , are known and non-negative. According to Kundu and Howlader (2010), the hyper-parameters can be chosen to suit the prior belief of the experimenter in terms of location and variability of the gamma prior distribution.
Based on the likelihood function in Equation (3.1) and the joint prior density in Equation (3.9), the joint posterior of adaptive type-II progressive censored of GR distribution with parameters and is The conditional posterior densities of Θ = ( , ) are as follows:

4.Reliability Function
In this section, we propose the estimation of reliability and hazard function using MLE and MPSE methods of time, say ( = 0.25) and ( = 0.5). So on, on this basis and by using the invariance property we estimate the reliability and hazard function. The estimates of the reliability and hazard function is given as:

5.Simulation Study
In this section; Monte Carlo simulation is done to estimate the parameters based on adaptive Type-II progressive censoring scheme for MLE, MPSE and Bayesian methods. Using R language and using the following steps.
Simulation Algorithm Scheme: Monte Carlo experiments were carried out based on the following data generated form GR Distribution by using the quantile function We could define the best scheme as the scheme which minimizes the Bias, mean squared error (MSE) and the relative efficiency (RE) of the estimator. That is, the objective function (to be minimized in this case) would be The simulation methods are compared using Bias, MSE and RE for the parameters and their performance was assessed through a Monte Carlo simulation study, we defined the number of repeated-samples to 1000.
The convergence of MCMC estimation of and can be showed in Figure           From simulation results of parameter estimation for model (GR distribution under adaptive censoring scheme), we note that: the Bias, MSE and RE for all the estimators for parameters decrease when the sample size n and effective sample size m increase in all cases. The efficiency of the model (Bias, MSE and RE for all the estimators when parameters decrease) increases when the ratio of the effective sample size (r) increases. Three different samples schemes were applied on adaptive type-II progressive censoring scheme and that to get to the most effective scheme, the efficiency is the best for scheme 2 followed by scheme 3 then scheme 1. The schemes 1 and 2 not affected by the changes in time t, whatever the changes in time t, there is a stability in the numerical and practical results of schemes 1 and 2. See table (1), where scheme 1 is 1 = 2 = ⋯ = m−1 = 0, m = − , it is type-II scheme and scheme 2 is 1 = − 2 = 3 = ⋯ = m−1 = 0 so not affected by the time, while scheme 3 is 1 = 2 = ⋯ = m−1 = 1, m = − 2 − 1 affected by the time. The parameters of the model were estimated with various values of parameters, we note that when increases, the estimated efficiency decreases with constancy of the . The results prove that, the efficiency of the MPSE over MLE, and MCMC over MLE, then MPSE and MCMC are good alternative estimator of MLE, where alternative estimator has lesser in the Bias and MSE for MLE estimators for parameters. The MPSE has more relative efficiency than MLE for all parameters, but Bayesian estimates is the best method in this study.

6.Application
We present the numerical results of the parameter estimation of GR distribution under adaptive type-II progressive censoring scheme of a two real data.

6.1.Real Data I
Badar and Priest (1982) discussed the real data set of sample size 69 observed failure times, the data set is represented the data measured in GPA, for single carbon fibers and impregnated 1000 carbon fiber tows. Mahmoud et al (2016) discussed the real data to inferences for new Weibull-Pareto distribution (NWPD) based on progressively type-II censored Data. Almetwally and Almongy (2018) used this data to fit of the Marshall-Olkin Extended Weibull (MOEW) distribution under adaptive type-II progressive censoring scheme.
We computed the Kolmogorov-Smirnov (KS) distance between the empirical and the fitted distributions.    The convergence of MCMC estimation of and can be showed in Figure       The convergence of MCMC estimation of and can be showed in Figure

7.Conclusion
In this article, parameters estimation for the GR distribution are discussed under adaptive type-II progressively censoring sample with different schemes based on the MPSE, MLE and the Bayesian methods. In this model, the estimators based on MPSE method behave quite better than the estimators based on the MLE but the Bayesian estimation is the best one, where the bias, MSE and RE are less than from the other methods. In generally, The MPSE and Bayesian methods can be used as alternative methods for the MLE method. The parameters of the model were estimated by using various values of parameters, we note that when θ increases, the model efficiency decreases with constancy of α. In case of reliability estimation, we note that when θ increases, the estimated model efficiency decreases (reliability decrease and hazard increase) with constancy of α. By checking the previous results, we note that MPSE is better than MLE. We can conclude that the MPSE method is a good alternative method to the usual MLE method in many situation. We hope that the finding in this paper will be useful for researchers and statistician, where such types of things were required and also in cases where we have small sample size for analysis of lifetime data.