Abstract
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.
1. 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, Note that Burr type XII distribution can be derived from Burr type III distribution by replacing with . 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.
Let denote an absolutely continuous distribution function with the corresponding density function and let , , be the corresponding dual GOS. Then, the joint probability density function of the first dual GOS is the following: for , , with parameters , , , and such that for all . For simplicity, we shall assume . If and , then it gives the joint probability density function of reversed order statistics from the independent and identically distributed random sample coming from . If , then reduces to the th lower -record value of the iid random variables. Various distributional properties and some applications of the related topics are studied by Burkschat et al. [9], Ahsanullah [10], Jaheen [11], Mbah and Ahsanullah [12], Barakat and El-Adll [13], and W. Kim and C. Kim [14].
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.
2. Maximum Likelihood Estimation
For , suppose that , , and (, is a real number) are dual generalized order statistics drawn from BurrIII . Using (1) and (2), we can get the following likelihood function: From (3), the log-likelihood function is proportional to To derive the maximum likelihood estimators (MLE’s) and of and , where and for . From (5), MLE of is expressed by Substituting (7) in (6), MLE of can be written as MLE of the parameter is obtained by solving the nonlinear equation (8). Substituting MLE of the parameter in (7), we can obtain MLE of the parameter .
The asymptotic variance-covariance matrix of MLE for the parameters and is given by the elements of the Fisher information matrix:
From (4), the asymptotic variance-covariance matrix for MLE is obtained by the following: with where and .
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: where Assume that . For each , we can have the expectation of , which is If we use the transformation , then the expectation of is given by To get the expectation of , we need to compute By the same transformation method , the expectation of , for , is given by For , we can compute the expectation of , which is Using the same transformation method , the expectation of is the following: To get the expectation of , we should compute With the transformation and , the expectation of , when , is given by Now, we can get each entry of the Fisher information matrix as follows: Using (15), (17), (19), and (21), all entries can be explicitly expressed, depending on .
3. Bayes Estimation
In this section, we want to estimate the parameters and under squared error loss (SEL) function, which is defined as 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 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 is 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 , based on Lindley’s approximation, the approximate Bayes estimator of a function , under the SEL function, leads to where and for , Note that is the th element of the inverse of the matrix , , where . Moreover, (27) is to be evaluated at the MLE’s of and .
Now, we apply Lindley’s approximation (27) to our case, where and . The elements can be obtained as Let , , and . Then, . Then, we can rewrite this as where Also, the values of can be obtained as follows for ; Note . Then, we get Substituting all the above components to (27), the Bayes estimate of the function given in (27), under the SEL function, becomes where
From (35), we can deduce the values of the Bayes estimates of the parameters and as follows.
If , then , , and . Hence, If , then , , and . Hence, Note that (35), (37), and (38) are to be evaluated at MLE’s .
4. 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 , , and be the lower record values of size which can be obtained from the dual GOS scheme as a special case by taking and . 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.(1)For and , samples of lower record values of size were generated from Burr type III distribution. Burr type III lower record values are generated using the inverse cdf, , where is the uniformly distributed random variate.(2)MLE’s, and of the parameters and , are calculated by iteratively solving (7) and (8) with and .(3)For given vlaues of prior parameters , the Bayes estimates of and are computed from (37) and (38) with and .(4)The above steps are repeated 1,000 times to evaluate the root mean squared error (RMSE) of MLE and Bayes estimates for the different sample sizes . Note that where is the true value and is the th estimate of evaluated at .
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was funded by the Korea Meteorological Administration Research and Development Program under Grant CATER 2012-3081.