Processus empiriques des rapports de $m$-espacements uniformes disjoints (Non-overlapping uniform $m$-spacings-ratio empirical processes)

TL;DR

This paper studies an empirical process based on ratios of non-overlapping $m$-spacings from independent samples, showing convergence to a mean-centered Brownian bridge under uniform distribution assumptions.

Contribution

It introduces a new empirical process involving $m$-spacings ratios and proves its convergence to a specific Brownian bridge process in the uniform case.

Findings

Convergence to a mean-centered Brownian bridge for the empirical process.

Explicit form of the limiting process involving Beta distribution and Brownian bridge.

Applicable to uniform distributions with potential extensions to other distributions.

Abstract

We consider an empirical process based upon ratio of selected pair of the non-overlapping $m$-spacings generated by independent samples of arbitrary sizes. As a main result, we show that when both samples are uniformly distributed on intervals of equal lengths, this empirical process converges to a mean-centered Brownian bridge of the form $\ \displaystyle (B\circ H_{m})_{C}(v)=B(H_{m}(v))-2(2m+1)C\ \frac{(2m-1)!}{m((m-1)!)^2}(v(1-v))^m\int_{0}^{1}B(H_{m}(s))\ud s,\ $ for $\ \displaystyle 0\leq v\leq 1,\ $ where $\ \displaystyle B(.)\ $ denotes a Brownian bridge, $\ \displaystyle H_{m},\ $ the distribution function of the Beta distribution with parameters $m$ and $m,\ $ and $C,\ $ a constant.