A Prediction Sufficient Statistic for Censoring Data in the K-parameter Exponential Family

摘要

In the theory of estimation, the sufficient statistic summarizes and exhausts in itself all the relevant information on the parameter that is contained in the sample. It plays a key role in the problem of estimating the parameter of the underlying probability distribution. A similar concern arises in the problem of predicting a future (that is, not yet observed) random variable on the basis of some existing observable random variables when the parameter of the underlying probability distribution does not concern us directly. Likewise, we are looking for a statistic that is exhaustive of all the relevant information on the future random variable that is available in the current observable random variables. The notion of prediction sufficient statistics initiated by Fisher and Skibinsky would deal with this concern. It has been extensively investigated in literatures such as [1, 2, 7,10-12]. In life testing, a statistic that plays a central role is the total time on test. In this article, we make a comprehensive study showing that the total time on test is a function of the prediction sufficient statistic for predicting the lifetime of a censored observation when the data is assumed to have a distribution from the k-parameter exponential family and the test is terminated after the r-th failure.