Trimmed samples are widely employed in several areas of statistical practice, especially when some sample values at either or both extremes might have been contaminated. The problem of estimating the parameters of Burr distribution type III based on a trimmed samples and prior information will be considered. In this paper, we study the estimation of unknown parameters based on doubly censored type II. The problem discussed using maximum likelihood method and Bayesian approach to estimate the shape parameters of Burr type III distribution. The numerical illustration requires solving nonlinear equations, therefore, MathCAD (2001) statistical package used to asses these effects numerically.
Burr [
However, outliers may occur in the data set. Trimmed samples are widely employed in several areas of statistical practice, especially when some sample values at either or both extremes might have been contaminated. The problem of estimating the parameters of Burr distribution type III based on a trimmed sample and prior information is considered.
Many authors discuss different methods of estimation for Burr type XII distribution. AlHussaini and Jaheen [
The objective of this paper is to obtain the estimators of the unknown shape parameters of Burr type III based on doubly censored type II. The problem discussed using maximum likelihood method and Bayesian approach to estimate the shape parameters of Burr type III distribution requires solving nonlinear equations, therefore, numerical study is carried out to asses these effects using MathCAD (2001) statistical package.
The Burr family of distributions has, in recent years, assumed an important position in the field of life testing because of its uses to fit business failure data, also it includes the wellknown exponential and Weibull distributions as special cases. Burr [
Burr family of distributions includes twelve types of cumulative distribution functions, which yield a variety of density shapes. Many standard theoretical distributions, including the Weibull, exponential logistic, generalized logistic, Gompertz, normal, extreme value, and uniform distributions are special cases or limiting cases of Burr family of distributions. The simple closed form of these distributions has been applied to studies in simulation. Burr [
The twelve distributions of Burr are listed in Burr [
The distribution of Burr
The expectation of the distribution is obtained as follows.
If
The variance
The mode
The median
In this section we will obtain the estimation of Burr parameters based on trimmed samples using Bayesian and non Bayesian methods of estimation.
In this section, the estimation problem of Burr type III with two parameters (
The asymptotic variances covariance matrix of the parameters (
The elements of matrix
In estimation problem, it is required to study the properties of the derived expressions for the estimators theoretically or analytically. Sometimes it seems very difficult to study the properties of the estimators theoretically because of the complicated formula of the estimators. Consequently, a simulation study will be set up, treating separately the sampling distribution of the estimators. Simulation studies have been performed using MathCAD (2001) for illustrating the theoretical results of the estimation problem. The simulation procedures will described below.
Generate a random sample of size 10, 20, 40, 60, 80, and 100 from Burr type III distribution. The generation of Burr type III distribution is very simple if
Choose censored failure
For each sample and for the four sets of parameters distribution were estimated under doubly censored type II.
NewtenRaphson method was used for solving the nonlinear equations for
Equations (
Results are tabulated in Table
The maximum likelihood estimator, the standard deviation, and covariance of the Burr distribution with two parameter under doubly censored sample when (a) (








MLE  Standard deviation  MLE  Standard deviation  
5  4.619  3.689  7.841  4.341 
58.814  
10  7  2  4.534  3.524  4.672  4.172  50.042 
9  4.338  3.308  4.722  3.202  22.784  
 
10  6.874  3.874  6.654  4.154  66.589  
20  15  2  6.483  2.483  6.439  3.939  64.752 
18  6.147  3.147  6.307  3.807  41.058  
 
20  6.498  3.498  6.802  4.302  74.941  
40  30  3  5.384  3.384  6.471  3.971  62.603 
38  5.287  2.287  5.629  3.129  39.913  
 
30  5.692  2.692  6.794  4.294  62.054  
60  45  3  5.542  2.542  6.678  4.178  47.829 
58  5.201  2.201  5.489  2.989  16.475  
 
40  5.448  2.448  6.824  4.324  78.475  
80  65  5  5.707  2.707  6.671  4.171  66.942 
78  5.334  2.334  5.391  2.891  11.313  
 
50  6.689  3.689  6.841  4.341  58.814  
100  75  5  6.524  3.524  6.672  4.172  50.042 
98  6.308  3.308  5.702  3.202  30.784 








MLE  Standard deviation  MLE  Standard deviation  
5  6.689  3.689  6.841  4.341  58.814  
10  7  2  6.524  3.524  6.672  4.172  50.042 
9  6.308  3.308  5.702  3.202  30.784  
 
10  6.874  3.874  6.654  4.154  66.589  
20  15  2  6.483  3.483  6.439  3.939  64.752 
18  6.147  3.147  6.307  3.807  41.058  
 
20  6.498  3.498  6.802  4.302  74.941  
40  30  3  6.384  3.384  6.471  3.971  62.603 
38  5.287  2.287  5.629  3.129  39.913  
 
30  5.692  2.692  6.794  4.294  62.054  
60  45  3  5.542  2.542  6.678  4.178  47.829 
58  5.201  2.201  5.489  2.989  16.475  
 
40  5.448  2.448  6.824  4.324  78.475  
80  65  5  5.707  2.707  6.671  4.171  66.942 
78  5.334  2.334  5.391  2.891  11.313  
 
50  6.689  3.689  6.841  4.341  58.814  
100  75  5  6.524  3.524  6.672  4.172  50.042 
98  6.308  3.308  5.702  3.202  30.784 








MLE  Standard deviation  MLE  Standard deviation  
5  5.952  2.952  5.159  3.159  69.043  
10  7  2  5.712  2.712  4.914  2.914  55.876 
9  5.819  2.819  4.249  2.249  38.874  
 
10  5.984  2.984  5.147  3.147  68.497  
20  15  2  5.147  2.147  4.271  2.271  24.984 
18  5.078  2.078  4.014  2.014  14.872  
 
20  5.618  2.618  5.081  3.081  69.187  
40  30  3  5.491  2.491  4.941  2.941  57.078 
38  5.107  2.107  4.028  2.028  23.174  
 
30  5.347  2.347  4.627  2.627  70.842  
60  45  3  5.148  2.148  4.489  2.489  58.247 
58  5.004  2.004  4.309  2.309  41.872  
 
40  5.334  2.334  4.658  2.658  74.194  
80  65  5  5.238  2.238  4.748  2.748  60.827 
78  4.997  1.997  4.102  2.102  47.145  
 
50  0.447  0.553  1.39  0.19  1.987  
100  75  5  0.474  0.526  1.317  0.117  1.854 
98  1.218  0.218  1.234  0.034  1.823 








MLE  Standard deviation  MLE  Standard deviation  
5  1.395  0.395  2.717  0.717  42.175  
10  7  2  1.501  0.501  2.585  0.585  35.214 
9  1.014  0.014  2.461  0.461  18.258  
 
10  1.284  0.284  2.648  0.648  40.954  
20  15  2  1.481  0.481  2.441  0.441  33.254 
18  1.008  0.008  2.314  0.314  16.418  
 
20  1.658  0.658  2.995  0.995  68.284  
40  30  3  1.018  0.018  2.147  0.147  50.258 
38  1.001  0.001  1.654  0.346  52.547  
 
30  1.228  0.228  2.954  0.954  68.952  
60  45  3  1.109  0.109  2.756  0.756  53.218 
58  1.019  0.019  2.341  0.341  40.021  
 
40  1.721  0.721  2.481  0.481  77.248  
80  65  5  1.856  0.856  2.441  0.441  80.514 
78  1.007  0.007  1.761  0.239  54.951  
 
50  1.395  0.395  2.717  0.717  42.175  
100  75  5  1.501  0.501  2.585  0.585  35.214 
98  1.014  0.014  2.461  0.461  18.258 
From Table
The Bayesian approach allows both sample and prior information to be incorporated into analysis, which will improve the quality of the inferences. In this section, Bayesian estimators and posterior variance of the shapes parameters are obtained in the case of double censored type II. The prior distribution could be “none informative” such as a flat, uniform distribution, assuming equal probability for the parameter value within a realistic range, and when the prior could be more informative such as a normal distribution or other possible distribution that represents an initial assessment of what is known about the parameter before the collection of data and the analysis. Moreover, a numerical examples are given for illustration study.
The likelihood function takes the form in (
It is well known that under a squared error loss function, the Bayes estimator of the parameter will be its posterior expectation. To obtain the posterior mean and posterior variance, a numerical integration is required. Then, the posterior mean and posterior variance of the shape parameters
The posterior mean and the posterior variance of
Repeat Steps
Solve the nonlinear equations to obtain posterior variance of the shape parameter
The posterior mean and the posterior variance of the estimators for the shape parameters
Numerical results are summarized in Table
The posterior mean and posterior variance of the Burr distribution with two parameter distribution under doubly censored sample when (




 

Posterior mean  Posterior variance  Posterior mean  Posterior variance  
5  1.017  0.685  0.671  0.765  
10  2  7  0.701  0.514  0.569  0.648 
9  0.608  0.338  0.408  0.491  
 
10  1.010  0.742  0.948  0.785  
20  2  15  0.871  0.537  0.653  0.676 
18  0.691  0.338  0.527  0.527  
 
20  0.437  0.674  0.741  0.576  
40  3  30  0.288  0.582  0.687  0.477 
38  0.119  0.449  0.354  0.345  
 
30  0.609  0.762  0.671  0.758  
60  3  45  0.547  0.684  0.507  0.862 
58  0.489  0.449  0.407  0.647  
 
40  0.827  0.761  0.718  0.901  
80  5  65  0.667  0.895  0.651  0.763 
78  0.524  0.612  0.581  0.522  
 
50  0.718  0.761  0.718  0.901  
100  5  75  0.651  0.895  0.651  0.763 
98  0.581  0.612  0.581  0.522 
Under the assumption that minimal loss function occurs
Now, in (
Repeat Steps
Solve the nonlinear equations to obtain the posterior variance of the shape parameter
The posterior mean and the posterior variance of the estimators for the shape parameter
Numerical results are summarized in Table
The expectation mean and variance of the Burr distribution with two parameter distribution under doubly censored sample when (




 

Expectation mean  Variance  Expectation mean  Variance  
5  1.698  0.396  2.596  1.631  
10  2  7  1.353  0.205  1.700  0.810 
9  1.189  0.292  1.663  0.726  
 
10  1.468  0.267  2.552  1.819  
20  2  15  1.238  0.166  2.032  1.619 
18  1.237  0.178  1.773  1.152  
 
20  1.382  0.294  2.343  1.198  
40  3  30  1.092  0.200  1.411  0.848 
38  1.022  0.196  0.936  0.847  
 
30  1.075  0.197  1.543  1.237  
60  3  45  1.013  0.174  1.018  1.062 
58  0.988  0.170  0.935  1.138  
 
40  1.275  0.162  1.007  1.188  
80  5  65  0.986  0.182  0.340  1.193 
78  0.906  0.151  1.762  2.269  
 
50  0.992  0.761  0.718  0.901  
100  5  75  0.745  0.895  0.651  0.763 
98  0.604  0.612  0.581  0.522 
The Bayes estimator
The equations (
Repeat Steps
Solving the nonlinear Bayesian for posterior variance of the shape parameter
The posterior mean and the posterior variance of the estimators for the shape parameter
Numerical results are summarized in Table
The expectation mean and variance of the Burr distribution with two parameter distribution under doubly censored sample when (




 

Expectation mean  Variance  Expectation mean  Variance  
5  1.641  0.685  3.825  0.765  
10  2  7  1.375  0.514  2.614  0.648 
9  1.255  0.338  1.661  0.491  
 
10  1.474  0.742  3.423  0.785  
20  2  15  1.299  0.537  2.490  0.676 
18  1.147  0.338  1.550  0.527  
 
20  1.294  0.674  3.076  0.576  
40  3  30  1.087  0.582  1.378  0.477 
38  1.016  0.449  1.263  0.345  
 
30  1.032  0.762  1.672  0.758  
60  3  45  0.977  0.684  1.173  0.862 
58  0.924  0.449  1.038  0.647  
 
40  1.162  0.761  0.860  0.901  
80  5  65  1.034  0.895  0.396  0.763 
78  0.888  0.612  0.185  0.522  
 
50  0.992  0.761  0.718  0.901  
100  5  75  0.745  0.895  0.651  0.763 
98  0.604  0.612  0.581  0.522 
Assume that the parameters
Hence, under a squared error loss function, the Bayes estimator of the parameter will be its posterior expectation. To obtain the posterior mean and posterior variance a numerical integration is required. Then, the posterior mean and posterior variance of the shape parameters
The posterior mean and the posterior variance of
Repeat Steps
Solve the nonlinear Bayesian for posterior variance of the shape parameter
The posterior mean and the posterior variance of the estimators for the shape parameter
Numerical results are summarized in Table
The posterior mean and posterior variance of the Burr distribution with two parameter under doubly censored sample when (




 

Posterior mean  Posterior variance  Posteriormean  Posterior variance  
5  0.495  0.204  1.017  0.719  
10  2  7  0.384  0.214  0.701  0.645 
9  0.258  0.217  0.608  0.201  
 
10  0.234  0.391  1.010  0.604  
20  2  15  0.128  8.215  0.871  0.556 
18  0.087  0.319  0.691  0.493  
 
20  0.482  0.339  0.437  0.911  
40  3  30  0.352  0.337  0.288  0.425 
38  0.241  0.190  0.119  0.307  
 
30  0.852  0.199  0.609  0.827  
60  3  45  0.729  0.160  0.547  0.647 
58  0.635  0.139  0.489  0.558  
 
40  0.957  1.171  0.827  0.728  
80  5  65  0.721  0.206  0.667  0.661 
78  0.408  0.078  0.524  0.337  
 
50  0.992  0.761  0.718  0.901  
100  5  75  0.745  0.895  0.651  0.763 
98  0.604  0.612  0.581  0.522 
It is noted that the posterior mean decreases when
This project was founded by the deanship of scientific research (DSR), King Abdulaziz University, Jeddah under Grant no. (124/130/1432). The authors, therefore, acknowledge with thanks DSR technical and financial support.