Average Life Prediction Based on Incomplete Data *

The two-parameter exponential distribution can often be used to describe the lifetime of products for example, electronic components, engines and so on. This paper considers a prediction problem arising in the life test of key parts in high speed trains. Employing the Bayes method, a joint prior is used to describe the variability of the parameters but the form of the prior is not specified and only several moment conditions are assumed. Under the condition that the observed samples are randomly right censored, we define a statistic to predict a set of future samples which describes the average life of the second-round samples, firstly, under the condition that the censoring distribution is known and secondly, that it is unknown. For several different priors and life data sets, we demonstrate the coverage frequencies of the proposed prediction intervals as the sample size of the observed and the censoring proportion change. The numerical results show that the prediction intervals are efficient and applicable.


Introduction
Prediction problem has been very often and useful in many fields of applications.The general prediction problem can be regarded as that of using the results of previous data to infer the results of future data from the same population.The lifetime of the second round sample is an important index in life testing experiments and in many situations people want to forecast the lifetimes of these samples as well as the system composed of these samples (See [1,2] and among others).For more details on the history of statistical prediction, analysis and application, see [3,4].
As we know, many quality characteristics are not normally distributed, especially the lifetime of products for example, electronic components, engines and so on.Assume that the lifetime of a component follows the twoparameter exponential distribution whose probability density function (pdf) given by where > 0  and 0   are called the scale parameter and the location parameter, respectively, and   I A denotes the indicator function of the set A .The readers are referred to [5,6] for some practical applications of the two-parameter exponential distribution in real life.The recent relevant studies on the two-parameter exponential distribution can be found in [7][8][9], etc.
In this paper, we adopt the following testing scheme: for n groups of components, which come from n different manufacture units possessing the same technology and regulations, we sample m components from each group and put them to use at time = 0 t and to practice economy the experiment will be terminated if one of the m components is ineffective, where m is a predetermined integer.Denote the lifetime of the ineffective component by is the life of the j -th component of the i -th group.Hence, we obtain n lifetime data 1 2 , , , .
 , where > 0 a is a known constant, then we again sample one component from the i -th group and denote its unknown lifetime by i Y .In this paper, our interest is to predict the average life of the second round sample, i.e., approximately describes the average lifetime of a system of k components, based on the samples of the second round, is connected in active-parallel which fails only when all k components fail.
*Sponsored by the Scientific Foundation of BJTU (2007XM046) Normally, there are two different views on prediction problems, the frequentist approach and the Bayes approach.The Bayesian viewpoint has received large attention for analyzing data in past several decades and has been often proposed as a valid alternative to traditional statistical perspectives (see [10][11][12], etc.).A main different point between the Bayes approach and the frequentist approach is that in Bayesian analysis we use not only the sample information but also the prior information of the parameter.
To adopt the Bayes approach, we regard the parameters  and  as the realization value of a random        be independent and identically distributed (i.i.d.) with the prior distribution   , G   , and conditionally on   has the pdf (1.1) which will be denoted by   , Our problem is how to construct a function to predict S .As we know, many statistical experiments result in incomplete sample, even under well-controlled situations.This is because individuals will experience some other competing events which cause them to be removed.In life testing experiments, the experimenter may not always be in a position to observe the lifetimes of all components put on test due to time limitation or other restrictions(such as money, material resources, etc.) on data collection (see [9,13] and others).Hence, censored samples may arise in practice.In this paper we assume furthermore that , we define a predictive statistic for S and simulate its prediction results under the condition that the censoring distribution W is known.In Section 3, when the censoring distribution W is unknown, we obtain a similar result for a corresponding predictive statistic of S as well as demonstrate the prediction performances.Some conclusions and remarks are presented in Section 4 and Section 5 deals with the proofs of the main theorems.

Predictive Statistic for S with Known W
Note that ij X has the conditional pdf   , where U   denotes the support of the prior distribu- By Fubini's theorem, we know Equation (2.4) (Below) and Note that conditionally on   where we have Equation (2.8) (Below) and Hence, the statistics < , < ;  and p  denotes convergence in probability.
Clearly, S can be used as a predictive statistic for S in this case.
Especially, when there is no censorship S and 2 S turn into, respectively Consequently, we use 0 Normally, we choose Gamma prior for the parameters   ,   , however, it is easy to see that Theorem 1 does not depend on any specific prior distribution.This shows that for any a prior distribution satisfying the conditions of Theorem 1, the conclusion of Theorem 1 will hold.So in the simulation study, we let the prior distribution of parameters and the censoring distribution be where = 0.0001, 0.0002 c and 0.00025 can be used to describe the censorship proportion (CP)   > P X V , which denotes the probability that X is larger than V .In the censorship model, if the probability   > P X V gets larger, then more ' i V s are likely to be observed other than ' i X s .Also, let = 3 m .Under the above assumptions, it is not difficult to check that the conditions (i), (ii) and (iii), defined in Theorem Secondly, by Equations (2.1) and (2.15) we obtain . Thirdly, let the predetermined constant a be equal to the MTTF, we compute the frequencies of the event   < S S   for = 200  and = 100  with = 0.0001, 0.0002, 0.00025 c .Repeating the process for 5000 times, the results are reported in Table 1.
where PCP denotes the practical CP obtained from the simulation data.From Table 1, firstly, we find that when the PCP is fixed, the frequencies of   < S S   generally increase as the sample size n and  get larger, respectively.Secondly, as it can be expected, the frequencies tend to decrease as the PCP increases.As a contrast, we report the frequencies of   0 < S S   in Table 2 when there is no censorship, which are uniformly better than those of Table 1.
In Tables 3 and 4, we change the value of the constant a and present the frequencies of   < S S   with a = 0.5MTTF and a = 1.5MTTF, respectively.
Although it is difficult to describe how S and the frequencies depend on the constant a , there is a trend that the frequencies of   < S S   are getting larger as the constant a becomes smaller.The reason is that S denotes the average values of i Y 's, as we know, if a gets smaller, obviously, both will become larger, then the S is more like the average life other than denoting several or a few samples, hence, the proposed method works better.That is why the frequencies in Table 3 perform the best among the above tables, especially for larger n .From the above, we see that for the three different priors of   ,   , which have the same prior means for  and  , respectively, under the condition that their PCPs are almost the same, the more concentrative the prior values, the better the performances of the frequencies.This numerical evidence means that the proposed prediction intervals are in accordance with practice and applicable.

Predictive Statistic for S with Unknown W
Note that the censoring distribution (.) W is unknown, hence the predictive statistic S is unavailable to use.This leads us to adopt the product limit estimator, which introduced to statistical problems by [14], to propose a corresponding predictive statistic for S in this case.Define where and the product limit estimator   ˆn W t is given by Under the same conditions as Theorem 1, we have ˆ0.

p S S
  Obviously, Ŝ can be used as a predictive statistic for S when the censoring distribution W is unknown.
Remark 2. Note that in the case that the censoring distribution W is unknown, it is impossible to check whether the conditions of Theorem 2 are satisfied or not.Hence, we first need to propose a distribution function W to fit the data 1 2 , , , n V V V  .Consider the following several data sets, which come from [15].We take them as the the censoring variables As we know the Weibull distribution is widely applied to life testing and reliability analysis.Some studies on it have been quickly developed in recent years (see [16] and [17], etc).The cumulative distribution function (CDF) of the three-parameter Weibull distribution is where  is the shape parameter,  is the location parameter and  is the scale parameter.
Employing the method proposed by [18], we use the three-parameter Weibull distribution to fit the above four groups data and test the fitting by Kolmogorov-Smirnov Firstly, to estimate the location parameter  , we follow the principle of golden-section to find a value which located in the interval     1 0,t to maximize the absolute value of the correlation coefficient defined by Secondly, adopting the least square method and regarding every group data as 1 2 , , , n t t t  , we report the fitting results and the estimators of the parameters as well as the Kolmogorov-Smirnov test values in Figures 2-5 and Table 6.Where the K-S value denotes the Kolmogorov-Smirnov test value.
Assume that the parameters   ,   have priors similar to (2.13) and (2.14), for example , and = 3 m .Obviously, under the condition that the censoring distribution W is Weibull distribution with the above estimated parameters, it is easy to check that the conditions of Theorem 2 are satisfied.From Tables 7-9, firstly, it is the same as before, we find that for the fixed PCP, the frequencies of However, this may not be the case all the time.One reason is that i X 's and i V 's are different even for each the same sample size n and hence this makes the comparison more complicated.Thirdly, consistently, whether W is known or not the performances of the

Conclusions and Remarks
In this paper, assume the observed lifetimes of components are rightly censored, we define a prediction statistic to predict the average value of some untested components, firstly, under the condition that the censoring distribution is known and secondly, that it is unknown.In the case that the censoring distribution is unknown, we first fit the data 1 2 , , , n V V V  with a distribution function, say   W t , and test the fitting by Kolmogorov-Smirnov as being distributed according to   W t and check whether the conditions of Theorem 2 are satisfied or not.The numerical evidences show that the proposed prediction intervals are in accordance with practice and applicable.Also, it is easy to see that the proposed prediction method can be extended to many important survival models such as Erlang distribution, Gompertz distribution and so on.Furthermore, we may consider the same prediction problem in any a pdf, say     , >0 f x I x  , which may be a finite mixture of any two life distributions, which occurs when two different causes of failure are present (see [19] and among others).

The Proof of Theorem 1
Proof.In order to obtain the conclusion of Theorem 1, we first prove Firstly, it is easy to see Equation (5.3) (Below) Secondly, we have belowing Equation (5.4) Thirdly, 1 S can be represented as where (5.7) and (5.8) Along with Equations (5.3)-(5.4)and (5.6)-(5.8),we have (5.9) Combining Equation (5.9) and using the following facts: (5.10) and also by Cauchy-Schwarz inequality, we easily know that under the conditions 1), 2) and 3),   Then by Markov's inequality, we conclude that (5.1) holds.

The Proof of Theorem 2
Proof.To prove Theorem 2, it is enough to show that ˆ0.
By Equation (5.16) and the following result (see [20]),     It is well-known that the prior distribution   , G   reflects the past experience about the parameter   ,   in Bayesian analysis.During the process of our simulation, we find the fact that the performance of the frequencies of   < S S   depends on the prior distribution.In what follows, we generate three kinds data of

Figure 1 .
Figure 1.The values of   ,   n n for three different priors.
We simulate the frequencies of   ˆ< S S   as the PCP changes and present the results in what follows.Also, we refer to the performances of the frequencies of

bles 1 , 3 and 4 ,
the Constant a gets small.Secondly, compared with Tafor the same sample size n , the frein this case for given  S is more concentrated in the vicinity of S .
as the PCP increases.

Table 3 . a = 0.5 MTTF.
different prior and report the corresponding performances of the frequencies in Figure1and Table5.At the same time, we simulate the performance of the frequencies of   0 S S  as a contrast.