Asymptotic Representations of Statistics in the Functional Empirical process : A portal and some applications

TL;DR

This paper develops a comprehensive asymptotic representation framework for a wide class of statistics using functional Gaussian processes, enabling joint distribution analysis and comparison of statistical measures.

Contribution

It introduces the General Representation of Statistics (GRI) in the functional empirical process, providing a unified approach for asymptotic analysis of various statistics, including L-statistics.

Findings

Framework covers a broad class of statistics including L-statistics.

Allows joint distribution analysis of multiple statistics.

Provides tools for comparing statistical measures and their decomposability.

Abstract

In this research monograph, we deal with a very general asymptotic representation for statistics named GRI expressed in the functional empirical process, both one-dimensional and multidimensional, and another call residual empirical process. Most of statistics in form of combination of L-statistics are covered by the asymptotic theory dealt here. This treatise is conceived to be a kind of \textbf{spaceship} on which modules are hanged. The spaceship is a functional Gaussian process and each module is the asymptotic representation of one statistic in terms of that Gaussian process. In that way, it is possible to navigate from one module to another, that is, to find the joint distribution of any pair of statistics, to compare them with respect to the areas and the times. In order to be able to do so, we should have a broad conception at the beginning. Within the constructed frame, the asymptotic joint law of any finite number of other statistics is automatically given as well as the joint distribution of its spatial variation or temporal variation, in absolute or relative values. We also deal with the general problem of decomposability of statistics by comparing statistical decomposability, a new view we introduce, versus functional decomposability. A general result only based on the GRI is provided. \noindent This monograph is also the portal of a handbook of GRI that will cover the largest number possible of statistics. In prevision of that, we treat three important examples as show cases. It is expected that this portal and the handbook will attract the attention of researchers working in the asymptotic area and will furnish useful tools to scientists who are interested in application of asymptotic tests, completed by computer packages.