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Problems of calculating the reliability of instruments and systems and the development of measures to increase efficiency and reduce operational costs confronted physicists and mathe maticians at the end of the '40's and the beginning of the '50's in connection with the unrelia bility of electro-vacuum instruments used in aviation. Since then steadily increasing demands for the accuracy, reliability and complexity required in electronic equipment have served as a stimulus in the development of the theory of reliability. From 1950 to 1955 Epstein and Sobel [67,68] and Davis [62], in an analysis of statistical data of the operating time of an instrument up to failure, showed that the distribution is exponential in many cases. Consequently, the ex ponential distribution became basic to research associated with experiments on life expectancy. Further research has shown that there are a whole series of problems in reliability theory for which the exponential distribution is inapplicable. However, it can practically always be used as a first approximation. The ease of computational work due to the nice properties of the exponential distribution (for example, the lack of memory property, see Section 1) is also a reason for its frequent use. AB a rule, data on the behavior of the failure rate function are used to test the hypothesis that a given distribution belongs to the class of exponential distributions, and order statistics are used to estimate the parameter of the exponential distribution.
List of contents
1 Introduction.- 1.1 Characterization of the exponential distribution by the lack of memory property.- 2 Characterization Problems in the Class of Distributions with a Monotone Failure Rate.- 3 Some Properties of Order Statistics.- 4 Characterization of the Exponential Distribution by an Independent Function of its Order Statistics.- 5 Characterization of the Exponential Distribution by a Property of its Order Statistics.- 6 Characterizations of the Distributions by Moment Properties of Order Statistics.- 7 Characterization Problems of the Geometric Distribution.- 8 Characterization of the Exponential Distribution using the Geometric Distribution.- 9 Multivariate Exponential Distributions and their Characterizations.- Comments.- Author Index.
