Some uniform in bandwidth functional results for the tail uniform empirical and quantile processes

TL;DR

This paper studies the asymptotic behavior of a local uniform empirical process for uniform data, establishing uniform convergence results over a range of bandwidths that shrink with sample size.

Contribution

It provides new uniform limit theorems for the local empirical process over a range of bandwidths, extending previous results to more general settings.

Findings

Established uniform convergence of the local empirical process over bandwidths

Derived asymptotic distributions valid for a range of bandwidths

Extended classical results to local processes with shrinking bandwidths

Abstract

For fixed $t\in [0,1)$ and $h>0$, consider the local uniform empirical process $$\DD_{n,h,t}(s):=n^{-1/2}\coo\sliin 1_{[t,t+hs]}(U_i)-hs\cff,\;s\in [0,1],$$ where the $U_i$ are independent and uniformly distributed on $[0,1]$. We investigate the functional limit behaviour of $\DD_{n,h,t}$ uniformly in $\wth_n\le h\le h_n$ when $n\wth_n/\log\log n\rar \infty$ and $h_n\rar 0$.