Summary
SupposeX is a non-negative random variable with an absolutely continuous (with respect to Lebesgue measure) distribution functionF (x) and the corresponding probability density functionf(x). LetX 1,X 2,...,X n be a random sample of sizen fromF andX i,n is thei-th smallest order statistics. We define thej-th order gapg i,j(n) asg i,j(n)=X i+j,n−Xi,n′ 1≤i<n, 1≤n≦n−i. In this paper a characterization of the exponential distribution is given by considering a distribution property ofg i,j(n).
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References
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Ahsanullah, M. A characterization of the exponential distribution by higher order gap. Metrika 31, 323–326 (1984). https://doi.org/10.1007/BF01915219
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DOI: https://doi.org/10.1007/BF01915219