EXACT DISTRIBUTIONS OF THE EMPIRICAL GENERALIZED SURVIVAL AND DISTRIBUTION FUNCTIONS

John J. Hsieh (U. S. A.)

Received May 13, 2008; Revised July 8, 2008

Abstract

This article introduces the generalized residual lifetime and failure time as well as their expectations, the generalized survival and distribution functions, respectively, with their sample means corresponding to the empirical generalized survival and distribution functions. Both cohort and stationary population interpretations of the generalized survival and distribution functions and the ratios of consecutive orders of the survival function and of the distribution function are, respectively, described in terms of the steady-state distribution of a population renewal process. The distributions of these generalized residual sojourn times are derived in terms of the original life distribution. The exact distributions of the empirical generalized survival and distribution functions are then derived from the distributions of the generalized residual sojourns. The extension of the results to cover double truncation and censoring is also described for the statistical analysis of lifetime and failure time data. We have also proposed in this article several new functions for describing a distribution, including trap function, micro failure time expectancy, macro life expectancy and macro failure time expectancy, and provided their interpretations. The present work can also be considered as a statistical generalization of life table.

Keywords and phrases: residual life, failure time, distribution theory, mixed distribution, generalized function, renewal process, stationary population, life table, reliability, survival analysis.