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Characterization of the exponential distribution function by properties of the difference \(X_{k+s:n}- X_{k:n}\) of order statistics. (English) Zbl 0793.62011

Summary: We consider \(n\) independent and identically distributed random variables with common continuous distribution function \(F\) concentrated on \((0,\infty)\). Let \(X_{1:n} \leq X_{2:n} \cdots \leq X_{n:n}\) be the corresponding order statistics. Put \[ d_ s(x) =P (X_{k+s:n}-X_{k:n} \geq x)-P(X_{s:n-k} \geq x),\quad x \geq 0, \] and \[ \delta_ s(x,\rho) = P (X_{k+s:n}-X_{k:n} \geq x)-e^{-\rho(n-k)x}, \quad \rho>0,\;x \geq 0. \] For \(s=1\) it is well known that each of the conditions \(d_ 1(x)=0\) \(\forall x \geq 0\) and \(\delta_ 1(x,\rho)=0\) \(\forall x \geq 0\) implies that \(F\) is exponential; but the analytic tools in the proofs of these two statements are radically different. In contrast to this we present a rather elementary method which permits us to derive the above conclusions for some \(s\), \(1 \leq s \leq n-k\), using only asymptotic assumptions (either for \(x \to 0\) or \(x \to \infty)\) on \(d_ s(x)\) and \(\delta_ s (x,\rho)\), respectively.

MSC:

62E10 Characterization and structure theory of statistical distributions
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References:

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