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Zeitschrift für Analysis und ihre Anwendungen


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Volume 16, Issue 1, 1997, pp. 191–200
DOI: 10.4171/ZAA/758

Published online: 1997-03-31

Characterization of the Exponential Distribution by Properties of the Difference $X_{k+s:n} – X_{k:n}$ of Order Statistics

H.-J. Rossberg[1], M. Riedel[2] and B. B. Ramachandran[3]

(1) Universität Leipzig, Germany
(2) Universität Leipzig, Germany
(3) New Delhi, India

Let $X_1 , X_2, \dots, X_n$ be independent and identically distributed random variables subject to a continuous distribution function $F$, let $X_{1:n}, X_{2:n}, \dots, X_{n:n}$ be the corresponding order statistics, and write $$P(X_{k+s:n} – X_{k:n} ≥ x) = P(X_{s:n–k} ≥ x) \ \ \ (x≥0)$$ where $n,k$ and $s$ are fixed integers with $k + s ≤ n$. It is an old question if condition (0) implies that $F$ is of exponential type. In [8] we showed among others that condition (0) can be greatly relaxed; namely, it can be replaced by asymptotic relations (either as $x \to \infty$ or $x \downarrow 0$) to derive this very result. Using a theorem on integrated Cauchy functional equations and in essential way a result of [8] we find now a more elegant and deeper theorem on this subject. The case of lattice distributions is also considered and some new problems are stated.

Keywords: Exponential distributions, integrated Cauchy functional equations

Rossberg H.-J., Riedel M., Ramachandran B.: Characterization of the Exponential Distribution by Properties of the Difference $X_{k+s:n} – X_{k:n}$ of Order Statistics. Z. Anal. Anwend. 16 (1997), 191-200. doi: 10.4171/ZAA/758