Abstract
The asymptotic behavior of the residual lifetime of the system and its characteristics are studied for the main distributions of reliability theory. Sufficiently precise and simple conditions for the domain of attraction of the exponential distribution are proposed, which are applicable for a wide class of distributions. This approach allows us to take into account important information about modeling the failure-free operation of equipment that has worked reliably for a long time. An analysis of the domain of attraction for popular distributions with “heavy tails” is given.
Highlights
The exponential distribution is widely used in reliability theory
It is well known that exponential distribution and the distribution of the residual lifetime have the property of being memoryless
It is of considerable interest to have convenient sufficient conditions for distribution functions that belong to the domain of residual lifetime attraction of the limit distribution
Summary
The exponential distribution is widely used in reliability theory. It is well known that exponential distribution and the distribution of the residual lifetime have the property of being memoryless. We derive asymptotics for the coefficient of variation and prove the sufficient condition for describing the domain of residual lifetime attraction. Which provides the necessary and sufficient conditions for determining the domain of the residual lifetime attraction of the limiting exponential distribution. These formulas and criterion (2) provide the necessary and sufficient conditions to belong to the domain of residual lifetime attraction for Weibull–Gnedenko distribution for all α > 0, β > 0. It is of considerable interest to have convenient sufficient conditions for distribution functions that belong to the domain of residual lifetime attraction of the limit distribution. We will prove that the region of attraction of the residual lifetime is characterized by the derivatives of the logarithms of the distribution tail and distribution density
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