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

• ### H.-J. Rossberg

Universität Leipzig, Germany
• ### M. Riedel

Universität Leipzig, Germany
• ### B. B. Ramachandran

New Delhi, India ## Abstract

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  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  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.

H.-J. Rossberg, M. Riedel, B. B. Ramachandran, 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), no. 1, pp. 191–200