Limit theorems for functions of marginal quantiles

TL;DR

This paper develops limit theorems for functions of marginal quantiles in multivariate distributions, including a multivariate CLT and strong law, under broad conditions, extending Bahadur's representation.

Contribution

It introduces a limit theory for the mean of functions of order statistics, including a multivariate CLT and a strong law, generalizing existing results with broad applicability.

Findings

Established a multivariate central limit theorem for functions of marginal quantiles.

Proved a strong law of large numbers for these functions.

Demonstrated the general conditions are satisfied in many common scenarios.

Abstract

Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadur's representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that \[\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^n\phi\bigl(X_{n:i}^{(1)},...,X_{n:i}^{(d)}\bigr)-\bar{\gamma}\Biggr)=\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_{n,i}+\mathrm{o}_P(1)\] as $n\rightarrow\infty$, where $\bar{\gamma}$ is a constant and $Z_{n,i}$ are i.i.d. random variables for each $n$. This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations.