JournalszaaVol. 16, No. 1pp. 191–200

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

Let X1,X2,,XnX_1 , X_2, \dots, X_n be independent and identically distributed random variables subject to a continuous distribution function FF, let X1:n,X2:n,,Xn:nX_{1:n}, X_{2:n}, \dots, X_{n:n} be the corresponding order statistics, and write

P(Xk+s:nXk:nx)=P(Xs:nkx)   (x0)P(X_{k+s:n} – X_{k:n} ≥ x) = P(X_{s:n–k} ≥ x) \ \ \ (x≥0)

where n,kn,k and ss are fixed integers with k+snk + s ≤ n. It is an old question if condition (0) implies that FF 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 xx \to \infty or x0x \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.

Cite this article

H.-J. Rossberg, M. Riedel, B. B. Ramachandran, Characterization of the Exponential Distribution by Properties of the Difference Xk+s:nXk:nX_{k+s:n} – X_{k:n} of Order Statistics. Z. Anal. Anwend. 16 (1997), no. 1, pp. 191–200

DOI 10.4171/ZAA/758