Characterization of the exponential distribution by properties of the difference \(X_{k+s:n}-X_{k:n}\) of order statistics
Summary: 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 \[ P(X_{k+s:n}- X_{k:n}\geq x)= P(X_{s:n-k}\geq x)\qquad (x\geq 0),\tag{0} \] where \(n\), \(k\) and \(s\) are fixed integers with \(k+s\leq n\). It is an old question if condition (0) implies that \(F\) is of exponential type. The first and second authors [Metrika 41, No. 1, 1-19 (1994; Zbl 0793.62011)] 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 the above quoted paper, 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.
scientific article; zbMATH DE number 1011109