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

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

where $n,k$ and $s$ are fixed integers with $k+s\le n$. 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.