Advances and Applications in Statistics
Volume 10, Issue 1, Pages 85 - 109
(October 2008)
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EXACT DISTRIBUTIONS OF THE EMPIRICAL GENERALIZED SURVIVAL AND DISTRIBUTION FUNCTIONS
John J. Hsieh (U. S. A.)
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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. |
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