## Abstract

Let X_{1}, . . . , X_{n}, be a sequence of independent and identically distributed random variables with a continuous distribution function F and let X_{n-kn+1} be the corresponding k_{n}th largest order statistics. Gather and Tomkins (J. Statist. Plann. Inference (1995), 175-183) proved that the condition (C) lim sup_{x→ωr} (1 - F (x + ε))/(1 - F (x)) < 1, where ωr := sup{x: F (x) < 1}, is sufficient for every sequence of upper-intermediate order statistics being absolutely stable. In this paper we prove that the condition (C) is also necessary, which solves an open problem by Gather and Tomkins (J. Statist. Plann. Inference (1995), 175-183). As an application of this result we give a criterion for every sequence of order statistics being absolutely stable.

Original language | English (US) |
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Pages (from-to) | 21-25 |

Number of pages | 5 |

Journal | Journal of Statistical Planning and Inference |

Volume | 64 |

Issue number | 1 |

State | Published - Oct 30 1997 |

Externally published | Yes |

## Keywords

- Order statistics
- Stability