Burr-X Model Estimate using Bayesian and non-Bayesian Approaches

Email: aalhossain@jazanu.edu.sa Abstract: The present paper is aimed at developing Bayesian and Maximum Likelihood estimations (ML) of the Burr type-X model of distribution when data are gathered from Type-II cumulative censoring with binomial eliminations. The procedures for getting the (ML) evaluations of the parameters are examined. The Bayes technique to get both point and interval estimators of the parameters are illustrated. The expected termination time for Type-II cumulative censoring with binomial eliminations is analyzed after carrying out the computation. Classical and Bayes procedures are improved in the case of parameter estimation and evaluated the expected test time for Burr-X model under cumulative censoring wit binomial sweep. A simulation study is performed to compare the implementation of the various procedures and for the expected termination time of the test. Finally, illustrative examples are given and the results from emulation studies determining the achievement of the suggested techniques are presented.


Introduction
Censoring is normal in life-distribution research due to limits and other constraints on data gathering. Censoring happens when exact lifetimes are recognized only for a part of the individuals or units under investigating while for the rest of the lifetimes data on them is partial. It is known that the most common censored tests are type-II censoring for reserving money and time. For more details about the progressive censoring schemes and applications, we refer to the monograph Balakrishnan and Aggarwala (2000). The assessment of parameters from various lifetime models in the context of cumulative Type-II censored samples have been considered by many authors including Balakrishnan et al. (2003;Fernandez, 2004;Ng et al., 2004;Soliman, 2005;Wu et al., 2007;Ku and Kaya, 2007;Banerjee and Kundu (2008). However, there is some work presented in the Bayesian model. Amin (2008) expanded Bayesian techniques in the case of parameter appreciation and assessment of future surveillances from the classical Pareto model. In Bayesian framework, many authors, see among others, Zellner (1995;Torney, 2005;Yan et al., 2009;Naranjo et al., 2014) had done some work. For some related classical estimation on cumulative Type-II censoring with random cancellations or binomial eliminations, one may refer to Tse et al. (2000;Wu, 2003). Also, we may mention the following related researchers for different hypotheses: Cohen (1963) illustrated the gradually censored specimens in life testing. Efron (1982) studied the bootstrap and other resembling plans. Balakrishnan and Sandu (1995) gave a simple computation algorithm for generating gradually Type-II censored specimens. Soliman et al. (2011;Mahmoud et al., 2013) discussed Bayesian inference and expectation of Burr types model for cumulative first failure censored pattern. Al-Hossain (2015a) considered generalized and inverse Gaussian distributions for the power inverse Gaussian. Al-Hossain (2016) studied the inference from the Exponentiated Weibull distribution given adaptive cumulative censored information. Sutikno and Ratih (2014) presented the Gaussian Copula Marginal degradation for design maximum information with implementation.
Recently, many appreciation issues for the Burr-X model Bayesian and non-Bayesian approaches, when the duration of life is gathered in the case of Type-II cumulative censoring, where the number of units taken away at each failure time obeys a binomial model of distribution is illustrated. For example, Clifford et al. (2015) proposed a Bayesian approach to person perception which outlining the theoretical position and a methodological framework for testing the predictions experimentally. Ahmad et al. (2015) obtained a Bayes estimates of the parameters and MLE for Burr-X model in the case of double type-II censored specimen of dual generalized order statistics. AL-Hussaini et al. (2014) studied the Bayesian expectation bounds of order statistics established on gradually type-II censored competing risks information from a common class of distributions. Ali (2014) presented a mixture of the inverse Rayleigh distribution with properties and estimation in a Bayesian framework. Okasha and Matter (2015) discussed more parameters burr-XII model and its application to heavy tailed lifetime information. Silva and Cordeiro (2015) introduced a new family of distributions by compounding the Burr-XII and power series distributions. Wang and Lee (2014) displayed the M-estimator procedure founded on the Bisquare objective function to evaluate the Burr-III parameters. Okasha (2014) considered the E-Bayesian method for computing estimates of the unknown parameter and some survival time parameters e.g., hazard and reliability functions of Lomax model in the case of type-II censored information. Pathak and Chaturvedi (2014) derived an Estimation of the reliability function for two-parameter exponentiated Rayleigh or Burr type X distribution. Karam and Jbur (2014) offered Bayesian analyzes of the Burr type distribution under double type censored samples using different priors and loss functions. Furthermore, a short version of this paper has been previously published in conference proceedings (Al-Hoaasin, 2015b).
In this study, classical and Bayesian techniques are expanded in the case of parameter assessment and evaluated the expected test time for Burr-X distribution under cumulative censoring with binomial eliminations. Finally, this paper is arranged as the model formulation of the problem is given in section 2. In section 3, the techniques for finding the (MLE) of the parameters θ and p are presented. Interval estimations of the parameters and both point are obtained. Bayesian techniques are using to illustrate point and interval estimations in sections 4. In section 5, the prospective duration in the case of Type-II cumulative censoring with the influence of different p is examined. Illustrative examples and the outcomes from emulation studies imposing the implementation of our suggested technique have been shown in section 6. In the end, we conclude the work in section 7.

Model Description
Assume the lifetime of a specific unit has a Burr-X model with probability density function (pdf), the corresponding cumulative distribution function (cdf) and the conditional likelihood function may be expressed as in Al-Hossain (2015b): where, x,θ>0. For more details see Al-Hossain (2015b).
From (1) and (2) Assume that a single unit being taken away from the test at the ith failure, i = 1, 2,⋅⋅⋅, m-1 is independent of the others but with same probability p. After that, the number R i of units extracted at the ith failure, i = 1, 2, ⋅⋅⋅, m-1, follows a binomial distribution with parameters Employing (4), (6) and (7), the full likelihood function is given by: And: With: It is easy to see that both c* and T(x) is independent of θ and p. For more details see Al-Hossain (2015b).

Maximum Likelihood Estimation
The methods for getting the MLE's of the parameters θ and p in the context of gradually Type-II censoring information with binomial eliminations are considered. Interval and both point estimations of the parameters are obtained.

Point Estimation
The functions L 1 and L 2 are estimators of p and θ respectively. Consequently, the (MLE) of θ may be written as maximizing Equation 9 directly due to L 1 in Equation 9 does not depend on p. Similarly, L 2 in Equation 10 independent of θ so the MLE of p may be gotten directly by maximizing Equation 10. Therefore, by considering the logarithms of L 1 (θ) and L 2 (p), the MLE's of θ and p may be obtained as in Al-Hossain (2015b).

Interval Estimation Bootstrap Confidence Intervals
Here, the parametric bootstrap percentile technique assumed by Efron (1982) to build confidence intervals for the parameters is used. For more details see (Al-Hossain, 2015b).

Approximate Interval Estimation
In this section, the MLEs of θ and p based on the approximate confidence intervals under asymptotic distributions are presented. Also, the MLEs of the parameters θ and p, in the case of the asymptotic variances and covariances are obtained considering the elements of the inverse of Fisher data matrix: The approximate (observed) asymptotic variance covariance matrix for the MLEs can be derived by neglecting the anticipation factor E. Then we have: With: To compute the approximate confidence intervals for θ and p, so the asymptotic normality of the MLE may be used. Then, (1-δ) 100% approximate confidence intervals for θ and p yield: where, Z δ/2 is the percentile of the standard normal distribution with right-tail probability δ/2.

Bayes Estimation
The choice of the Loss Function (LF) is determining by an integral part of Bayesian estimation methods. In literature, many various kinds of loss functions have been considered to describe different types of loss structures. The symmetric Square Error Loss (SEL) is one of the most common loss functions. A loss function should display the consequences of various errors. Many asymmetric (LFs) may exist in the literature, however, LINEX Loss Function (LLF) is excessively utilized due to a natural expansion of SEL. Mathematically representation of LLF can be written as: One more advantageous of asymmetric loss function is the General Entropy (GE) loss: where, the lower value happens at u u = ɶ and the Bayes expectation value BG u ɶ of u is: Under GE loss (Soliman, 2005).

Point Estimation
In this section, due to θ and p considered as independent random variables, we use the gamma prior distribution with the parameters α, β for θ, in some equations which are found in (Al-Hossain, 2015b).

Symmetric Bayes Estimation
Square-error loss function is so-called briefly as SEL function. The estimator of this function is the posterior mean. So, applying of the posterior densities and the Bayes estimators of the variables θ and p are given in (Al-Hossain, 2015b).

Asymmetric Bayes Estimation
LINEX loss function if replacing u by θ and then θ by u, respectively, the Bayes estimate BL θ ɶ of the variables θ and p comparative to LINEX loss function may be expressed as: where, the integration of Equation 21 is obtained by the numerical integration methods.
General Entropy loss function replacing u by θ in And using Equation 25 in (Al-Hossain, 2015b), one can get: Replacing also u by p in (19), the Bayes estimate BG p ɶ of parameter p relative to the general entropy loss function becomes:

Interval Estimation
Highest Posterior Density Interval (HPDI). In general, the Bayesian procedure to interval estimation is highly more direct than the ML method, for more details (Al-Hossain, 2015b).

Expected Duration
In many industrial processes, life test is conducted in order to assess the quality of a product. Typically, n products are placed under test and their times to failure are observed. These observed lifetimes are then used to estimate the life distribution of the product. However, in many applications, life tests are usually terminated before the complete lifetimes of n products are observed. Data from this censored test consist of times to failure of failed units and running times on unfailed units. Following Balakrishnan and Aggrawala (2000), under progressive interval type II censored with random or binomial eliminations conditioning on R, the expected the value of X m is presented by (Al-Hossain, 2015b): By the same way, setting m = n and r i = 0 for all i = 1,..., m in Equation 25, the anticipated time of an entire sampling case with n test units may be written as: Finally, carrying out the expectation on the life hand side and the right side of Equation 25 with respect to the R, the anticipated ending point for gradually type-II censoring with binomial cancellations is estimated from the following equation: where, g(r 1 ) = n-m, g(r i ) = n-m-r 1 -⋅⋅⋅-r i-1 , i = 2,⋅⋅⋅, m-1 and P (R, p) is given in Equation 7. The ratio of the expected time under various schemes to the expected time under entire sampling, i.e., Ratio of Expected Experiment Times (REET) is defined in (Al-Hossain, 2015b).       for the different parameters n, m, p and θ numerically based on Type-II progressive censoring with binomial eliminations. We examined some values like: n = 6, 10, 15, θ = 1, 2, 3 and p = 0.1,⋅⋅⋅, 0.9. The results are presented in Table 3a-4b with remarking that when m = n be in agreement with the complete sample plan.  Table 3a-5b, it is noticed that the anticipated of ending time for Type-II cumulative censoring sample is getting close to that of the entire sample when m is getting bigger. For a constant value of m, the expected experiment time of type-II cumulative with binomial eliminations goes down while the sample size n getting bigger. Also, with respect to binomial eliminations, it is obvious that for constant values of n and m, the REET becoming close to one faster for going up of p. All results are because of the fact that a high cancellation of probability shows a large number of dropouts. Therefore, the cancellation of the probability p plays a considerable factor in the time needed to accomplish the experiment. In all cases, a large number of test units n would abbreviate the experiment period of the test while the underlying schedule is Type-II with binomial eliminations.

Conclusion
The problem of estimating the parameters of Burr distribution type-X based on a mean square error while data are gathered in the context of Type-II censoring with binomial eliminations are considered. Interval and both point assessments of the variables and the anticipated time to accomplish the test has been computed and interpreted for various censoring schedules. The outcomes illustrate that the maximum likelihood estimation for various estimators of the variable is going down while the cancellation probability p becoming large, furthermore, the corresponding time needed to accomplish the test going up significantly. A numerical example is given to illustrate the procedures and the accuracy of prediction intervals is investigated via Monte Carlo simulation. Furthermore, this study shows how simulation can be a helpful and illuminating way to approach problems in Bayesian analysis. Moreover, the simulation has presented some interesting properties of the Bayes estimates. After an extensive study of results. Bayesian problems of updating estimates can be handled easily and straight forwardly with simulation. Since we can express the distribution function of the Burr Type-X as well as its inverse in closed form, the inversion method of simulation is straightforward to implement. The study can further be extended by considering generalized versions of the distribution under the variety of circumstances.