Estimation of the Parameters of Burr Type III Distribution Based on Dual Generalized Order Statistics

The estimation of the parameters of Burr type III distribution based on dual generalized order statistics is considered by using the maximum likelihood (ML) approach as well as the Bayesian approach. The exact expression of the expected Fisher information matrix of the parameters in the distribution is obtained. Also, an approximation based on Lindley is used to obtain the Bayes estimator. To compare the maximum likelihood estimator and the Bayes estimator of the parameters, Monte Carlo simulation study is performed.


Introduction
Burr type III distribution with two parameters was first introduced in the literature of Burr [1] for modelling lifetime data or survival data. It is more flexible and includes a variety of distributions with varying degrees of skewness and kurtosis. Burr type III distribution with two parameters and , which is denoted by BurrIII ( , ), has also been applied in areas of statistical modelling such as forestry (Gove et al. [2]), meteorology (Mielke [3]), and reliability (Mokhlis [4]).
The probability density function and the cumulative distribution function of BurrIII ( , ) are given by, respectively, , > 0, > 0, > 0, Note that Burr type XII distribution can be derived from Burr type III distribution by replacing with 1/ . The usefulness and properties of Burr distribution are discussed by Burr and Cislak [5] and Johnson et al. [6]. Abd-Elfattah and Alharbey [7] considered a Bayesian estimation for Burr type III distribution based on double censoring.
Order statistics are widely used in statistical modelling and inference. As a unified approach to a variety of models of ordered random variables such as ordinary order statistics, upper record values, and sequential order statistics, the concept of generalized order statistics (GOS) was introduced by Kamps [8]. Based on GOS, Burkschat et al. [9] introduced the concept of dual generalized order statistics as a dual model of GOS and a unification of several models of decreasingly ordered random variables such as reversed order statistics, lower record values, and lower Pfeifer records.
In this paper, our main objective is to describe MLE, the exact expression of the expected Fisher information matrix, and Bayes estimation procedures for the parameters of Burr type III distribution based on dual GOS, assuming the conjugate priors. In Section 2, we consider MLE and obtain an exact expression of the expected Fisher information matrix of the parameters. In Section 3, Lindley's approximation is used to obtain Bayes estimates for the parameters. Finally, in order to compare MLE with Bayes estimators, Monte Carlo simulation is studied in Section 4.
Note that the Fisher information involves only a function of and so we need the marginal probability density function of th dual GOS based on the distribution function ( ) and the density function ( ). From Burkschat et al. [9], the marginal probability density function of th dual GOS is the following: The Scientific World Journal Assume that ̸ = −1. For each = 1, 2, . . . , , we can have the expectation of , which is If we use the transformation = 1/ ( ), then the expectation of is given by To get the expectation of ln , we need to compute By the same transformation method = 1/ ( ), the expectation of ln , for ̸ = −1, is given by For = −1, we can compute the expectation of , which is Using the same transformation method = 1/ ( ), the expectation of is the following: To get the expectation of ln , we should compute 4 The Scientific World Journal With the transformation = 1/ ( ) and (∑ ∞ =1 ( / )) 2 = ∑ ∞ =2 ∑ −1 =1 ( /( − ) ), the expectation of ln , when = −1, is given by Now, we can get each entry of the Fisher information matrix * as follows: Using (15)

Bayes Estimation
In this section, we want to estimate the parameters and under squared error loss (SEL) function, which is defined as 0 ( ,̂) = ( −̂) 2 for a parameter . Assuming that the parameters and are unknown, a natural choice for the prior distributions of and would be to assume that the two quantities are independent gamma distributions as in the following: where , , , and are chosen to reflect prior knowledge about and . By combining (3) and (24), the joint posterior density function of and can be put as follows: Under the SEL function, it is well known that the Bayes estimator of a function = ( , ) is the posterior mean of the function, which iŝ In general, the integral ratio in (26) cannot be expressed in a simple closed form. Hence, we use Lindley's approximation [15] to obtain a numerical approximation. In a two-parameter case, say ( 1 , 2 ) = ( , ), based on Lindley's approximation, the approximate Bayes estimator of a function = ( 1 , 2 ), under the SEL function, leads tô = ( 1 , 2 ) + 1 2 ( + * 30 12 + * 03 21 + * 21 12 + * 12 21 ) ) .

Simulation Study and Comparisons
In this section, we consider MLE and the approximate Bayes estimates for two parameters and of Burr type III distribution. To assess the performance of these estimates, we conducted a simulation study. Let (1) = 1 , (2) = 2 , . . ., and ( ) = be the lower record values of size which can be obtained from the dual GOS scheme as a special case by taking = −1 and = 1. MLE and Bayes estimates for the parameters of BurrIII ( , ) based on lower records are computed and compared through the Monte Carlo simulation study according to the following steps.

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The Scientific World Journal  Table 1 provides the averaged RMSE of MLE and Bayes estimates based on lower record values for two sets of prior parameters ( , , , ). To show the consistency of the result across varying data sets with large variability and differing sample sizes, we simulate data under two sets of parameters, each prior distribution with large variability. We see that the Bayes estimates are better than MLE in the sense of comparing RMSE of the estimates. As the sample size increases, RMSE of the estimates should decrease, which is the case in our computer simulation.