American Open Journal of Statistics, 2011, 1, 33-45
doi:10.4236/ojs.2011.12005 Published Online July 2011 (http://www.SciRP.org/journal/ojs)
Copyright © 2011 SciRes. OJS
Estimation Using Censored Data from Exponentiated
Burr Type XII Population
Essam K. AL-Hussaini, Mohamed Hussein
Department of Mathematics, Alexandria University, Egypt
E-mail: dr_essak@hotmail.com
Received May 3, 2011; revised May 25, 2011; accepted June 2, 2011
Abstract
Maximum likelihood and Bayes estimators of the parameters, survival function (SF) and hazard rate function
(HRF) are obtained for the three-parameter exponentiated Burr type XII distribution when sample is avail-
able from type II censored scheme. Bayes estimators have been developed using the standard Bayes and
MCMC methods under square error and LINEX loss functions, using informative type of priors for the pa-
rameters. Simulation comparison of various estimation methods is made when n = 20, 40, 60 and censored
data. The Bayes estimates are found to be, generally, better than the maximum likelihood estimates against
the proposed prior, in the sense of having smaller mean square errors. This is found to be true whether the
data are complete or censored. Estimates improve by increasing sample size. Analysis is also carried out for
real life data.
Keywords: Exponentiated Distribution, Proportional Reversed Hazard Rate Model, Lehmann Alternatives,
Maximum Likelihood and Bayes Estimation, Burr Type XII Distribution, Subjective Prior, SE
and LINEX Loss Functions, MCMC
1. Introduction
Analogous to the Pearson system of distributions, Burr [1]
introduced a system that includes twelve types of cumu-
lative distribution functions (CDF) which yield a variety
of density shapes. This system is obtained by considering
CDF’s satisfying a differential equation which has a so-
lution, given by:
1
1 exp d , F x x x
where x is chosen such that F x
is a CDF on the
real line. Twelve choices for x , made by Burr, re-
sulted in twelve distributions from which types III, X and
XII have been frequently used. The flexibilities of Burr
XII distribution were investigated by Hatke [2], Burr [3],
Rodrigues [4] and Tadikamalla [5].
In a different direction, it was Takahasi[6] who first
noticed that the 3-parameter Burr XII probability density
function (PDF) can be obtained by compounding a
Weibull PDF with a gamma PDF. That is, if X|θ~ Weibull
(θ, β) and θ~ gamma (γ, δ) then the compound PDF, say
g(x|β, γ, δ), is given by
1 1
0
1
1
1
, , e e d
1 ,
x
gx x
x x
which is the 3-parameter Burr XII (β, γ, δ) PDF.
If δ = 1, this PDF reduces to the 2-parameter Burr XII
(β, ), whose PDF, CDF, SF, and HRF are given, for x > 0,
(β, > 0), by:
1
1
, 1 gx x x
, (1.1)
, 1 1 Gx x ,
(1.2)
, 1 , 1
G
R x Gx x ,
(1.3)
1
,
,
1 ,
G
G
gx
x
x
x R x
. (1.4)
The Burr XII and its reciprocal Burr III distributions
have been used in many applications such as actuarial
science, as in Embrechts et al. [7] and Klugman[8],
quantal bioassay as in Drane et al. [9], economics, as in
McDonald and Richards [10], Morrison and Schmittlein