# Spacings Around An Order Statistic

## TL;DR

This paper analyzes the asymptotic distribution of spacings around various order statistics in a sample, revealing exponential behavior for central and intermediate cases and complex dependencies for extremes, with implications for understanding local data structure.

## Contribution

It characterizes the joint limiting distribution of spacings around order statistics, including independence results and failure of independence for extreme cases in certain distribution domains.

## Key findings

- Spacings around central and intermediate order statistics are asymptotically exponential and independent.
- Independence of spacings and Poisson processes holds for central and intermediate cases.
- For extreme order statistics, independence fails in certain distribution domains.

## Abstract

We determine the joint limiting distribution of adjacent spacings around a central, intermediate, or an extreme order statistic $X_{k:n}$ of a random sample of size $n$ from a continuous distribution $F$. 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 $X_{k:n}$. For an extreme $X_{k:n}$, the asymptotic independence property of spacings fails for $F$ in the domain of attraction of Fr\'{e}chet and Weibull ($\alpha \neq 1$) distributions. This work also provides additional insight into the limiting distribution for the number of observations around $X_{k:n}$ for all three cases.