INTRODUCTION
The Weibull distribution has the widest variety of applications in many areas,
including life testing and reliability theory. The most used methods which are
considered to be the traditional methods, are maximum likelihood and the moment
estimation (Cohen and Whitten, 1982). Sinha
(1986) used Bayes and the maximum likelihood estimators of reliability function
and hazard rate for Weibull distribution using Lindley’s approximation
method. Singh et al. (2002) estimated exponentiated
Weibull shape parameters by using Bayes and maximum likelihood estimators. Hossain
and Zimmer (2003) obtained maximum likelihood to estimate Weibull parameters
for complete and censored samples. Hahn (2004) showed
that Jeffreys’ prior applied to panel models with fixed effects yields
posterior inference which is not always free from the incidental parameter problem.
Sinha and Sloan (1988) obtained the Bayes estimator
of three parameters of the Weibull distribution and compared the posterior standard
deviation estimate counterparts with numerical examples given. Assoudou
and Essebbar (2003) used the independent MetropolisHasting algorithm to
estimate Bayesian using Jeffreys’ noninformative prior. Singh
et al. (2005) estimated Bayes and maximum likelihood for twoparameters
exponentiated Weibull distribution when the sample was available from typeII
censoring scheme. Soliman et al. (2006) estimated
the Weibull distribution by using the maximum likelihood estimator and Bayesian
estimator under squared error loss function and Linex loss function for a given
shape parameter and several unknown parameters. Singh et
al. (2008) estimated generalizedexponential by maximum likelihood and
obtained Bayes estimator using Lindley’s expansion. Preda
et al. (2010) used maximum likelihood and Bayesian methods to estimate
the modified Weibull by Lindley’s expansion under various loss functions.
The objective of this study is to estimate the survival function and hazard rate of the Weibull distribution for right censoring data by using Bayesian estimator with Jeffreys prior and the extension of Jeffreys prior and maximum likelihood estimator. We compare the performance of these estimators through simulate study under several conditions and used the mean square error to determine the best estimator.
MATERIALS AND METHODS
Maximum likelihood estimation: Let (t_{1},...,t_{n}) be the set of n random lifetime from Weibull distribution with parameters θ and p.
The probability density function of Weibull distribution is given by:
The likelihood function is:
where, δ_{i} = 1 for failure and δ_{i} = 0 for censored observation and S(.) is the survival function.
The logarithm of the likelihood function can be expressed as follows (Klein
and Moeschberger, 2003):
To obtain the equations for the unknown parameters, we differentiate Eq. 1 partially with respect to the parameters θ and p and equal it to zero. The resulting equations are given below, respectively:
Following Hossain and Zimmer (2003), let U(θ)
equals to zero, then the maximum likelihood estimator is:
The shape parametric p cannot be solved analytically and for that we use the Newton Raphson method to find the numerical solution.
Following Soliman et al. (2006), the estimate
of the survival function and hazard rate of Weibull are:
Bayesian using Jeffreys prior information: The Jeffreys prior is the square root of the determinant of the Fisher information matrix parameters per observation as:
The Fisher information matrix of parameters per observation is:
Where:
Then:
The posterior probability density function of θ and p given the data (t_{i},...,t_{n}) is obtained by dividing the joint probability density function with the marginal density function as follows:
Following Sinha (1986), the survival function for the
Weibull distribution is:
Where:
Following Sinha (1986) hazard rate of Weibull distribution
is:
Bayesian using extension of Jeffreys prior information: The extension of Jeffreys prior is by taking g_{2}(θ, p)∝ [I(θ, p)]^{c}, cεR^{+}
Then:
The posterior probability density function of θ and p is obtained by dividing the joint probability density function with a marginal density function as follows:
The estimated survival function for the Weibull distribution is:
Where:
The estimated hazard rate of Weibull distribution is:
The integrals in Eq. 610 cannot be solved
analytically and for that we used Lindley’s Expansion to solve the parameters
approximation.
Lindley’s expansion: Sinha (1986) considered
the Lindley’s Expansion for the survival of Bayes estimator by using the
following:
Where:
Substituting u = p/θt^{p1} the Bayes hazard estimator can be obtained in a similar manner.
For extension of Jeffreys prior estimator, we substitute:
Simulation study: In this simulation study, we have chosen n = 25, 50
and 100 to represent small moderate and large sample size and the following
steps are employed (Ahmed and Ibrahim, 2011).
• 
Generate lifetime X with different sample sizes n = 25, 50
and 100 from Weibull distribution 
• 
Generate censored time C with different sample sizes n = 25, 50 and 100
from Uniform distribution (0, b) and the value of b depends on the proportion
of censored observation, where, we consider 20% of censored data 
• 
The observed time T is the minimum of the failure and censored times,
T_{i} = min (X_{i}, C_{i}) and we defined delta
as follows: 
δ_{i} = 1if X≤C and δ_{i}
= 0 if X>C 
• 
The value of parameters chosen were θ = 0.5 and 1.5,
p = 0.8 and 1.2. The four values of extension Jeffreys prior were c = 1,
5, 10 and 15, the considered values of θ, p and c are meant
for illustration only and other values can also be taken for generating
the samples from Weibull distribution 
• 
The maximum likelihood from Eq. 3 and 4
were used to estimate the survival function and hazard rate, respectively
for Weibull distribution. Bayesian using Lindley’s approximation from
(11) calculate the survival function and subsequently the hazard rate following
Sinha (1986) 
• 
Steps 15 are repeated 10,000 times and the Mean Square Error (MSE) for
each method was calculated. The results are displayed in Tables
16 for the different choices of the parameters and
extension of Jeffreys prior 
In Table 13, when we compare the Mean
Square Error (MSE) of estimated survival function of Weibull distribution for
censored data by Maximum likelihood (MLE) and Bayesian using Jeffreys prior
and extension of Jeffreys prior, we found the Maximum likelihood (MLE) give
smallest value compared to the others. However, it is clear from the Table
13 for the survival function, Bayesian using the extension
of Jeffreys prior is better than the other estimators when θ = 0.8, p =
0.5 with c = 5, 10 and 15. Additionally, Bayesian using the extension of Jeffreys
is better than the other estimators when θ =1.2 with c = 10 and 15. When
the number of sample size increases the Mean Square Error (MSE) decreases in
all cases (Ahmed et al., 2010).
Table 1: 
MSE estimated survival function of Weibull distribution for
n = 25 

Table 2: 
MSE estimated survival function of Weibull distribution for
n = 50 

Table 3: 
MSE estimated survival function of Weibull distribution for
n = 100 

Table 4: 
MSE estimated hazard rate of Weibull distribution for n =
25 

In Table 46, when we compared the hazard
estimators of Weibull distribution with censored data by Maximum likelihood
(MLE) and Bayesian using Jeffreys prior and the extension of Jeffreys prior
by Mean Square Error (MSE) we found that the Maximum likelihood (MLE) give smallest
vaule compared to the others. However, Bayesian using the extension of Jeffreys’s
is better than the other estimators when θ = 1.2, p = 0.5. Bayesian using
the extension of Jeffreys is better than the other estimators when θ =
1.2 , p = 1.5, with c = 5, 10 and 15. When the number of sample size increases
the Mean Square Error (MSE) decreases in all cases. Following (Ahmed
and Ibrahim, 2011).
Table 5: 
MSE estimated hazard rate of Weibull distribution for n =
50 

Table 6: 
MSE estimated hazard rate of Weibull distribution for n =
100 

CONCLUSION
The simulation results show that if MSE is accepted as an index of precision, the Maximum Likelihood estimates of survival function and hazard rate are more efficient than their Bayesian counterparts. However, the extension of Jeffreys is better than MLE for certain conditions. All methods produced a decrease of MSE as the sample size increases.