For many years, the Weibull distribution as a statistical model has a broad
range of applications in life testing and reliability theory with the major
advantage of providing reasonably accurate failure analysis and failure forecasts
with infinitesimally small samples. Some comparisons of estimation methods for
Weibull parameters using complete and censored samples have been discussed (Hossain
and Zimmer, 2003). Also, an array of methods has been proposed for the estimation
of parameters of the Weibull distribution. Carroll (2003)
examined the use and applicability of Weibull model in the analysis of survival
data from clinical trials and illustrated the practical benefits of a Weibull
based analysis. The application of the Weibull distribution in the modeling
and analysis of survival data has also been described extensively by Mudholkar
et al. (1996). Kantar and Senoglu (2008)
reported their findings on the comparative study for the location and scale
parameters of the Weibull distribution with a given shape parameter. A new class
of shrinkage estimators for the shape parameter in an independently identically
distributed two-parameter Weibull model under censored sampling was introduced
by Singh et al. (2008).
Ibrahim and Laud (1991) have shown that Generalized
linear models are suitable for modeling various kinds of data consisting of
exponential family response variables with covariates. They provide two theorems
that support the use of Jeffreys's priors for Generalized linear models with
intrinsically fixed or known scale parameters. Singh et
al. (2005) compared Bayes and classical estimators for two-parameter
Exponentiated Weibull distribution when sample was available from type-II censoring
scheme. The estimates were obtained under squared error loss function as well
as under LINEX loss function using non-informative type of priors for the parameters.
Hahn (2004) showed that the Jeffreys prior applied
to panel models with fixed effects yields posterior inference which is not always
free from the incidental parameter problem.
The objective of this study is to estimate the survival function of the Weibull distribution with right censoring data by using Bayesian estimator and Maximum likelihood estimator. We also compare these estimators by using mean square error and mean percentage error to get the best estimator under several conditions.
MAXIMUM LIKELIHOOD ESTIMATION
, tn) be the set of random lifetime from Weibull distribution with parameters θ and p.
The probability density function of Weibull distribution is given by:
The likelihood function for right censoring as in Klein and
Moeschberger (2003) is
Following Soliman et al. (2006) the estimate
of the survival function of Weibull is
where, θML is the solution from partial derivative of In (L)
Eq. 2, with respect to θ equal to zero (Hossain
and Zimmer, 2003)
The shape parameter p is assumed to be known.
, tn) be a random sample of size n with distribution function F (t, θ, p ) and probability density function f (t, θ, p).
In the Weibull case, we assumed that the probability density function of the lifetime is given by Eq. 1.
Jeffrey prior information: Following Al-Kutubi and
Ibrahim, (2009a) the Jeffery prior equation method is,
where, k is a constant
The joint probability density function is obtained by multiplying the likelihood
function and Jeffrey prior as follows (Soliman et al.,
The marginal probability density function is obtained by integrating the joint
probability density function with respect to the scale parameter given as follows
(Singh et al., 2005):
The posterior probability density function of θ given the data (t1,
, tn) is obtained by divide the joint probability density function
with the marginal density function, following (Singh et
The estimator for survival function obtain as in (Al-Kutubi
and Ibrahim, 2009b) is
Extension of jeffrey prior information: Based on Al-Kutubi
and Ibrahim (2009a), the extension of Jeffrey prior is by taking g2
(θ) ∝ [(I (θ))]c, c ε R+, giving
where, k is a constant
The joint probability density function is obtained by multiplying the likelihood function and extension of Jeffrey prior information as in Eq. 6,
The Marginal probability density function is obtained by integrating the joint probability density function with respect to the scale parameter as given in Eq. 7,
Then the posterior probability density function of θ given the data (t1,
, tn) is obtained by dividing the joint probability density function with the marginal density function as in Eq. 8
The estimator for survival function is then
RESULTS AND DISCUSSION
In this simulation study, we have chosen n = 25, 50, 100 to represent small
moderate and large sample size, where the percentage of censoring is 30% for
each sample (Carroll, 2003). The values of parameter
chosen were θ = 0.5 and 1.5 and p = 0.8 and 1.2. The two values of Jeffery
extension were c = 0.4 and 1.4 (Al-Omari et al.,
2010). The number of replication used was R = 1000 (Sinha,
1986). The Mean Square Error (MSE) and Mean Percentage Error (MPE) calculated
to compare the methods of estimation are as follows:
(Al-Kutubi and Ibrahim, 2009a).
The results obtained from the simulation study are presented in Table
1 and 2 for the MSE and the MPE respectively of the three
estimators for all sample size and p-values (Al-Kutubi and
Three values of estimators which are MLE estimator, Jeffrey prior and extension
of Jeffrey prior are shown in each row of both Table 1 and
2. The best method is the one that gives the smallest value
of (MSE) and (MPE), (Al-Omari et al., 2010).
As shown Table 1, comparison can be made on survival function
estimators of Weibull distribution in Maximum likelihood, Bayesian estimator
by Mean Square Error (MSE) (Soliman et al., 2006).
Results show that when c = 0.4, the maximum likelihood is the best compared
|| MSE estimated survival function of Weibull distribution
|| MPE estimated survival function of Weibull distribution
However, when the c = 1.4, extension of Jeffrey is the best compared to others.
Following (Kantar and Senoglu, 2008) for the effect
of the shape parameter, when the shape parameter p increase for = 0.5 the Mean
Square Error (MSE) decreases, on the other hand, the mean square error increases
when the shape parameter increases for = 1.5. When the sample size n increase
the mean square error decrease for all cases (Al-Omari et
From Table 2, when we compared survival function estimators
of Weibull distribution in Maximum likelihood, Bayes using Jeffery prior and
extension of Jeffery prior by Mean Percentage Error (MPE), we found out that
the maximum likelihood is the best when c = 0.4 and the extension of Jeffrey
is better than others when the c = 1.4. The shape parameter p increases when
= 0.5 the mean percentage error (MPE) decreases. On the other hand, the mean
percentage error increases when the shape parameter increases that is when =
1.5. When the sample size n increases the Mean Percentage Error (MPE) decreases
for all cases (Al-Omari et al., 2010).
The study concludes that the extension of Jeffrey prior is the best estimator when the value of extension of Jeffrey is 1.4. On the other hand, the maximum likelihood method is still better than others when the value of extension of Jeffrey is 0.4.
When the number of sample size increases the Mean Square Error (MSE) and Mean Percentage Error (MPE) decrease in all cases.