### Abstract

We determine the joint limiting distribution of adjacent spacings around a central, intermediate, or an extreme order statistic $$X_{k:n}$$Xk:n of a random sample of size $$n$$n from a continuous distribution $$F$$F. For central and intermediate cases, normalized spacings in the left and right neighborhoods are asymptotically i.i.d. exponential random variables. The associated independent Poisson arrival processes are independent of $$X_{k:n}$$Xk:n. For an extreme $$X_{k:n}$$Xk:n, the asymptotic independence property of spacings fails for $$F$$F in the domain of attraction of Fréchet and Weibull ($$\alpha \ne 1$$α≠1) distributions. This work also provides additional insight into the limiting distribution for the number of observations around $$X_{k:n}$$Xk:n for all three cases.

Original language | English |
---|---|

Pages (from-to) | 515-540 |

Number of pages | 26 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 67 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2015 |

### Fingerprint

### Keywords

- Central order statistics
- Extremes
- Intermediate order statistics
- Poisson process
- Spacings
- Uniform distribution

### Cite this

*Annals of the Institute of Statistical Mathematics*,

*67*(3), 515-540. https://doi.org/10.1007/s10463-014-0466-9