A COMPARATIVE STUDY BASED ON BAYES ESTIMATION UNDER DIFFERENT CENSORING CRITERION

: The censoring arises when exact lifetimes are only partially known, and it is useful in life testing experiments for time and cost restrictions. Especially, when some sample values at either or both extremes might have been adulterated. In present article, the Bayes estimation for unknown parameter of Gompertz distribution has been addressed based on three different censoring criterions. The performances of the procedures are illustrated by a simulation technique.


Introduction
first introduced the underlying model, and its distribution and probability density function are defined as where the parameter  is known as scale parameter.
Gompertz probability distribution has many useful applications in areas of technology, medical, biological, and natural sciences (especially in failure and survival analysis). This distribution also widely used in the model of human mortality and fit in the actuarial tables.
Based on progressive first-failure censoring plan Soliman et al. (2012) studied the Bayes and frequentist estimators for two-parameter Gompertz distribution. Ismail (2010) discussed the Bayes estimation for unknown parameters of Gompertz distribution and acceleration factor under partially accelerated life tests under Type-I censoring. They applied Bayesian approach for estimation problem in case of step stress partially accelerated life tests for two stress levels and under Type-I censoring. Wu, et al. (2003) discussed about point and interval estimations for Gompertz distribution under Progressive Type-II censoring. Jaheen (2003) discussed about the Bayesian analysis of record statistic from Gompertz model. The maximum likelihood (ML) estimation for mixtures of two Gompertz distributions when censoring occurs has derived by Gordon (1990). Based on human mortality model, some parameter estimation of Gompertz population was discussed by Chen (1997). Anands et al. (1996) derived some adaptive Bayes estimators for the parameters of Gompertz survival model. They present Bayes estimators for the parameters in terms of single numerical integrations.
Bayesian analysts have pointed out that there is no clear-cut way from which one can conclude that one prior is better than the other. It is more frequently the case that, a prior is selected to restrict attention to a given natural family of priors, and one is chosen from that family, which seems to match best with one's personal beliefs. However, if one has adequate information about the parameter, one should use informative prior; otherwise it is preferable to use noninformative prior. The Gamma distribution is considered here as a conjugate family of prior for unknown parameter ,  having probability density function The Jeffrey's prior is the example of non-informative prior which materializes the use of Bayesian estimation methods when no prior information is available. In present case, the second prior is considered as the non-informative (Jeffrey's) prior and is defined as The selection of loss function may be crucial in Bayesian analysis also. If most commonly used loss function, squared error loss function (SELF) is taken as a measure of inaccuracy then the resulting risk is often too sensitive to assumptions about the behavior of tail of probability distribution. Also the use of SELF in Bayesian estimation may not be appropriate in case when positive and negative errors have different consequences. In addition, in some estimation problems overestimation is more serious than underestimation, or vice-versa. A useful and flexible class of asymmetric loss function (LINEX loss function (LLF)) is defined for such cases as (1.4) Here, ˆ be any estimate corresponding to unknown parameter  and ' 'a be the shape parameter of LLF. Negative (positive) value of , ' ' a gives more weight to overestimation (underestimation) and its magnitude reflect the degree of asymmetry. It is seen that, for , 1  a LLF is quite asymmetric with overestimation being more costly than underestimation. For small values of |, | a LLF is almost symmetric and is not far from SELF.
In many life testing experiments, the experimenter may not be observed the lifetimes of all inspected units in life test. This may be because of time limitation and/or cost or material resources on data collection. Also, the trimmed samples are widely utilized when some sample values at either or both extremes adulterated.
The aim of present paper is to present a comparative study based on Bayes estimation for unknown parameter under different censoring plans. For this we consider here three different censoring schemes wiz, Item-Failure (Type-II), Doubly Type-II and Progressive Type-II right censoring. Bayes estimation has been addressed for Gompertz model under LINEX loss function.
The performances of the procedures are illustrating by a simulation technique. A real data (King et al. (1979)) example is also provided to illustrate the proposed methods. A brief conclusion is presented at the end.

Bayes Estimation of Unknown
Based on Bayes theorem, the posterior density is now defined as Using (2.1) and (1.2) in (2.2), the posterior density is obtain as The Bayes estimator IC ˆ for unknown parameter  under LLF is obtain by simplifying following equality

Doubly Type-II Censoring
The plan of experiments with aim of reducing total duration or number of failures leads naturally to Type-I & Type-II censoring scheme. The main disadvantage of these censoring schemes is that they do not allow removal of units at points other than termination point of an experiment.
In addition, sometime, some data may not be observed, a known number of observation in an ordered sample are missing at both ends in failure censored experiments, observations are smallest and largest, are random then data collected will be. In such case doubly censoring criterion is useful.
Consider a life-test experiment in which n identical units are placed on a test from the (1.1). The first r units may be left censored due to negligence or problems at beginning of the experiment, and experiment terminates as soon as the th s unit failed. Then the data be the observed failure items, then the joint probability density function for these order statistics is defined as ) ( f and ) ( F be the corresponding probability density and distribution function respectively. Simplifying (2.5), we get Applying Bayes theorem, the posterior density is obtained as The Bayes estimator DC ˆ for unknown parameter  under LLF is given as

Progressive Type-II Censoring
The progressive censoring appears to be a great importance in planned duration experiments in reliability studies. In many industrial experiments involving lifetimes of machines or units, experiments have to be terminated early and the number of failures must be limited for various reasons.
Let us suppose an experiment in which n independent and identical units The progressive normalizing constant p C is defined as The Bayes estimator PC ˆ for the parameter  under LLF is given as (2.12)

Item Failure Censoring
Using joint probability density function (2.1) and equation (2.2), the posterior density corresponding to the Jeffrey's prior (1.3) is obtained as The Bayes estimator IC ˆ for unknown parameter  under LLF is obtain by simplifying following equality

Doubly Type-II Censoring
Using joint probability density function (2.6) and equation (

Simulation Study
A simulation is used in order to compare the performances of the proposed Bayes estimators in terms of risk magnitude under different censored plans. We carry out this comparison by taking The Gompertz distribution is a flexible distribution that can be skewed to the right and to the left. The Gompertz distribution is often applied to describe the distribution of adult lifespans by actuaries and demographers. Computer scientists have also started to model the failure rates of computer codes by the Gompertz distribution. Related fields of science such as biology and gerontology also considered the Gompertz distribution for the analysis of survival. More recently, In Marketing Science, it has been used as an individual-level simulation for customer lifetime value modeling. All the said applications, overestimation is more serious than underestimation.
Hence, the positive value of shape parameter ) 0 ( a is considered here and it shows the loss function is quite asymmetric about 0 with overestimation being more costly than underestimation.

Item-Failure Censoring Scheme
The test is terminated when   15 , 10 , 5  r , as it is supposed. The risk magnitude is increase as prior parameter increase (except for  ) , ( c b (10, 3.16)). A decreasing trend has seen in the risk magnitude when censored sample size r increases when other parametric values are fixed (Table 1). Further an opposite trend in the risk magnitude also seen when ' 'a increase.
It is noted that, when the value of parameter  increases the risk magnitude first decrease and then increase. However, the magnitude of risk under the LLF is nominal (Table 1).

Doubly Censoring Scheme
The Risk of the Bayes estimators DC ˆ and DC ˆ have been obtained under LLF, for the similar set of considered parametric values as discussed above and presented in Table 2. The left censoring rate is assumed to be 20%, 30%, 40%; and the right censoring rate accordingly assume for the fixed non-censored items 5, 10, 15 respectively. For example if the left censoring rate is 20% and the non-censored items is prefixed into 5, then right censoring rate will be 55%. All the behaviors have been seen similar corresponding to the Bayes estimators obtained under Item-Failure criterion. The gain in risk magnitude is noted here for doubly censoring criterion as compared to previous one. However, the gain in magnitude is robust.  Table 7 and the risk of Bayes estimators under LLF in Table 3.

Progressive Censoring Scheme
Again it is seen that all the properties of Bayes estimators are similar as discussed above, for both censoring criterion. Further, it is noted also here that the magnitude of risk are smaller than compared to Item Failure or doubly censoring criterion.

Studies with Generated Values of
For observing the effect of the unknown parameter ,  we generate the value of parameter  by using b and c from prior density (1.2). Using these generated values of  and other parametric values considered as above, the risk of Bayes estimators obtained and presented in Tables 4-6 under all three considered censoring criterion. All the properties have been seen similar as discussed above. It is also noted that the risk magnitude of Bayes estimators is robust when compared to pre-assume values of parameter  case, for all considered censoring plans.

A Real Data Example
We illustrate the use of estimation methods given in this paper by real data considered by King et al. (1979) for a tumor-free days of 30 rats fed with unsaturated diet. These data are also presented by Lee (1992) and studied by Chen (1997) and Wu et al. (2003). They assumed a Gompertz distribution for tumor-free times.
Under the different censoring plans, the censored sample size r is taken as 18 from 30  n . The risks magnitude of Bayes estimators under LLF are presented in Table 8, with others parametric values as considered in previous section. All the properties as discussed above have been seen similar. Also, the risks magnitude obtained here are least than compared to simulation study. This pattern is similar under each censoring plans.

Conclusion
The censoring arises when exact lifetimes are only partially known and it is useful in life testing experiments for time and cost restrictions. Especially when some sample values at either or both extremes might have been adulterated. The aim of article is to present a comparative study based on Bayes estimation under different censoring plans. For this we consider here Item failure censoring, Doubly and Progressive Type-II right censoring. The Bayes estimation has been addressed for underlying model under the LINEX loss function.
Here, two different criterions for selection of values of unknown parameter has been taken, in which one is pre assumed value of unknown parameter and second one is the generated values of the unknown parameter. From the above sections we have already seen that all properties of Bayes estimators under all censoring plans are seen to be similar. But, one remarkable point is that the risk magnitude of Bayes estimators under LLF is nominal when it is taken from Progressive Type-II censoring. In comparison of censoring schemes, the result under Progressive Type-II censoring seems to be more efficient over Item-failure or Doubly censoring scheme.
It has been seen also that, when the shape parameter of LINEX loss ' 'a increase the risk first increase and then stable. For higher values of , ' ' a the changes in magnitude of the risk is robust.
Hence, small values of shape parameter are suitable.