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Keywords:
characterization; completeness property; convexity (concavity); exponential distribution; order statistics; record values
characterization; completeness property; convexity (concavity); exponential distribution; order statistics; record values
Summary:
The exponential distribution is one of the most widely used continuous distributions in reliability theory, failure time analysis, Poisson processes, and survival analysis. In this paper, we establish new characterizations of the exponential distribution based on the probability integral transformation. Characterizations derived from both distributional relationships and moment properties of certain functions of the underlying variables are presented. Furthermore, analogous results for distributional relationships are obtained for functions of order statistics and $k$-record statistics. Using the completeness properties of the sequences of functions $\{(1-u)^{i-1}u^{n-i},0<u<1,n \ge i \ge 1\}$ and $\{(-\log u)^{n-1}, 0<u<1,n \ge 1\}$, moment-based characterization results are also established for functions of order statistics and $k$-record statistics, respectively. Since ordinary record values are special cases of $k$-records, the corresponding results hold for usual records as well.
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