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On the supremum and its location of the standardized uniform empirical process

Dietmar Ferger

TL;DR

The paper analyzes the standardized uniform empirical process and proves that the maximizing location and the scaled supremum converge jointly to a limit where the location concentrates at the endpoints with equal probability and the scaled supremum follows a Gumbel distribution. It develops a maximal-inequality for the weighted process, shows boundary concentration of the maximizer, and leverages existing limit results to establish asymptotic independence between the location and the extremal value. This clarifies the asymptotic structure of weighted supremum statistics and has implications for inference in goodness-of-fit tests based on such processes.

Abstract

We show that the maximizing point and the supremum of the standardized uniform empirical process converge in distribution. Here, the limit variable (Z, Y ) has independent components. Moreover, Z attains the values zero and one with equal probability one half and Y follows the Gumbel-distribution.

On the supremum and its location of the standardized uniform empirical process

TL;DR

The paper analyzes the standardized uniform empirical process and proves that the maximizing location and the scaled supremum converge jointly to a limit where the location concentrates at the endpoints with equal probability and the scaled supremum follows a Gumbel distribution. It develops a maximal-inequality for the weighted process, shows boundary concentration of the maximizer, and leverages existing limit results to establish asymptotic independence between the location and the extremal value. This clarifies the asymptotic structure of weighted supremum statistics and has implications for inference in goodness-of-fit tests based on such processes.

Abstract

We show that the maximizing point and the supremum of the standardized uniform empirical process converge in distribution. Here, the limit variable (Z, Y ) has independent components. Moreover, Z attains the values zero and one with equal probability one half and Y follows the Gumbel-distribution.

Paper Structure

This paper contains 2 sections, 2 theorems, 34 equations.

Table of Contents

  1. Introduction and Main Result
  2. The proof

Key Result

Theorem 1

For all $x \in \mathbb{R} \setminus \{0\}$ and $y \in \mathbb{R}$ the following limit-theorem holds: where $\mathbb{P}(Z=1)=\mathbb{P}(Z=0)=\frac{1}{2}.$ In particularly, with $Z$ and $Y$ independent.

Theorems & Definitions (4)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof