Bayesian Estimations from the Two-Parameter Bathtub- Shaped Lifetime Distribution Based on Record Values

This article discusses Bayesian and non-Bayesian estimations problems of the unknown parameters for the two-parameter bathtub-shaped lifetime distribution based on upper record values. The ML and the Bayes estimates based on record values are derived for the two unknown parameters as well as hazard function. When the Bayesian approach is considered, under the assumption that both parameters are unknown, the Bayes estimators cannot be obtained in explicit forms. An approximation form due to Soland (Soland (1969)) is used for obtaining the Bayes estimates based on a conjugate prior for the first shape parameter and a discrete prior for the second shape parameter of this model. This is done with respect to the squared error loss and LINEX loss functions. The estimation procedure is then applied to real data set and simulation data.


Introduction
proposed a new two-parameter lifetime distribution with bathtubshaped or increasing failure rate (IFR) function.Some probability distributions have been proposed with models for bathtub-shaped failure rates, such as Hjorth (1980), Mudholkar and Srivastava (1993) and Xie and Lai (1996).The new twoparameter life time distribution with bathtub-shaped or increasing failure rate function compared with other models has some useful properties.First, it has only two parameters to model the bathtub-shaped failure rate function.Second, it holds some nice properties on the classical inferential front, where the confidence intervals for the shape parameter and the joint confidence regions for the two parameters have closed form.For more details, see Chen (2000), Wang (2002), Wu et. al. (2004Wu et. al. ( , 2005) ) and Lee et al. (2007).
A new two-parameter bathtub-shaped lifetime distribution has a cumulative distribution function of the form (Chen 2000) and hence the probability density function (pdf) is given by The reliability and hazard (failure rate) functions of this distribution are given, respectively, by and The failure rate function of this distribution has a bathtub shape when and has increasing failure rate function when (see, Chen (2000)).
Record data arise in several real-life problems including industrial stress testing, meteorological analysis, hydrology, seismology, athletic events, and oil and mining surveys.The formal study of record value theory probably started with the pioneering paper by Chandler (1952) This article is concerned with the Bayesian and non-Bayesian estimations based on upper record values for the two unknown parameters of the new twoparameter bathtub-shaped lifetime distribution and its hazard (failure rate) function.In Section 2, the maximum likelihood estimators are derived.In Section 3, the Bayes estimators of the parameters and hazard function are derived based on the squared error and LINEX loss functions.The estimation procedure is then applied to real data set and simulation data in Section 4. Finally, conclusions appear in Section 5.

Let
be the first m upper record values from the new two-parameter bathtub-shaped lifetime distribution with pdf as given in (1.2), for simplicity of notation, we will use instead of .The likelihood function (LF) is given by (see Ahsanullah (1995)) where By substituting Eqs.(1.1) and (1.2) in Eq. ( 2.

1) we obtain
The natural logarithm of the likelihood function (2.2) is given by

MLE with known
Under the assumption that the parameter is known.The maximum likelihood estimator (MLE) of , denoted by , can be derived from (2.3) as follows

MLE with unknown λ and
Assuming that both parameters and λ are unknown.The maximum likelihood estimators (MLE) of denoted by can be shown to be Where is the MLE of the parameter which, can be obtained as a solution of the following non-linear equation Using the invariance property, the corresponding MLE of the hazard rate function H(t) are obtained from (1.4) after replacing and by their MLEs and .

Known shape parameter
Under the assumption that the parameter is known, we consider the natural conjugate prior distribution for is a gamma prior density function with pdf Combining the likelihood function (2.2) and the prior density function (3.1) and applying the Bayes theorem, we get the posterior density function of as follows where

Bayes estimator based on squared error loss function
Assuming the commonly used squared error loss function, , the Bayes estimator of (i.e., the value that minimizes the posterior expected loss) is the posterior mean.Then, the Bayes estimates of and based on the square error loss function can be derived, respectively from (3.2) as follow and

Bayes estimator based on LINEX loss function
Under the Linex loss function, the Bayes estimator of a function is given by From (3.2) and (3.6), the Bayes estimator for the parameter is Similarly, the Bayes estimator of H(t), is

Unknown two parameters and
Under the assumption that both the parameters and are unknown, specifying a general joint prior for and may leads to computational complexities.In an attempting to solve this problem and simplify the Bayesian analysis, we can use the Soland's method.Soland (1969)  Suppose that the parameter is restricted to a finite number of values with respective prior probabilities such that and , that is .Further, suppose that a conditional prior distribution for given has a natural conjugate prior with distribution having a gamma with pdf Where and are chosen to reflect prior beliefs on given that .
Combining the likelihood function in (2.2) and the conditional prior in (3.9), we get the conditional posterior of as follows where The marginal posterior probability distribution of obtained by applying the discrete version of Bayes' theorem, is given by where is a normalized constant given by and

Estimators based on squared error loss function
The Based on these six upper record values, the hyper-parameters and and the values of are obtained by the following steps: 1.By using the nonparametric approach of the reliability function, we set and in (4.2), we obtain and .
2. Based on these six upper record values, the MLE of the parameter from (2.6), is .Therefore, we suppose that takes ten values around 0.43 (0.01) 0.52, each has probability 0.1.
3. The values of the hyper-parameters and for each given are obtained numerically from (4.1), using the Newton-Raphson method.
Table 1 shows the values of the hyper-parameters and the posterior probabilities derived for each .Table 2 contains the MLEs and the Bayes estimates of and which are computed from data in Table 1.Based on these seven upper record values, the maximum likelihood and Bayes estimates of and are obtained by the following steps: 1. We approximate the prior for over the interval (0.675, 0.9) by the discrete prior with taking the 10 values 0.675 (0.025) 0.9, each with probability 0.1.
2. By using the nonparametric procedure in (4.2), we assume that the reliability function for times and are, respectively, and .
3. Substituting the two values of obtained in step 2 into equation (4.1).The values of the hyper-parameters and for each given , are obtained numerically by using the Newton-Raphson method.The values of the hyper-parameters and the posterior probabilities for each are displayed in Table 0.

Based on the entries of
considered a family of joint prior distributions that places continuous distributions on the scale parameter and discrete distributions on the shape parameter to achieve the Bayesian analysis of Weibull distribution.This approximation was used for obtaining the Bayes estimates by several authors such as,Soliman et al. (2006),Sultan (2008) andPreda et al. (2010).

Table 0 : Prior information, Hyper-parameters of the gamma and the posterior probabilities
In this article, we present the maximum likelihood and Bayes estimates of the two unknown parameters and hazard function for the new two-parameter lifetime model based on record values.Bayes estimators, under squared error loss and LINEX loss functions, are derived in approximate forms by using Soland's method.The comparisons between different estimators are made based on simulation study and a real record values set.It has been noticed from Tables 4 that, the Bayes estimates based on squared error loss and LINEX loss functions are perform better than the maximum likelihood estimates.The Bayes estimates of the parameters that are obtained based on the LINEX loss function tend to the corresponding estimates which are obtained based on squared error loss when tends to zero.