On a discrete Hill's statistical process based on sum-product statistics and its finite-dimensional asymptotic theory

TL;DR

This paper introduces a new class of sum-product statistics generalizing Hill's estimator, analyzes their asymptotic behavior, and describes the covariance structure of their limiting Gaussian process, extending known results in extreme value theory.

Contribution

It develops a finite-dimensional asymptotic theory for a new family of sum-product statistics based on order statistics, generalizing classical tail index estimators.

Findings

Limiting laws of the process are derived.

Covariance function of the Gaussian limit is explicitly described.

Asymptotic normality and laws of the iterated logarithm are extended to these estimators.

Abstract

The following class of sum-product statistics T_n(p)=\frac{1}{k}\sum_{h=1}^p \sum_{(s_1...s_h)\in P(p,h)} \sum_{i_1=l+1}^{i_0} ... \sum_{i_h=l+1}^{i_{h-1}} i_h \prod_{i=i_1}^{i_h} \frac{(Y_{n-i+1,n}-Y_{n-i,n})^{s_i}}{s_i!} (where $l,$ $k=i_{0}$ and n are positive integers, $0<l<k<n,$ $P(p,h)$ is the set of all ordered parititions of $\ p>0$ into $\ h$ positive integers and $Y_{1,n}\leq ...\leq Y_{n,n}$ are the order statistics based on a sequence of independent random variables $Y_{1},$ $Y_{2},...$with underlying distribution $\mathbb{P}(Y\leq y)=G(Y)=F(e^{y})$), is introduced. For each p, $T_{n}(p)^{-1/p}$ is an estimator of the index of a distribution whose upper tail varies regularly at infinity. \ This family generalizes the so called Hill statistic and the Dekkers-Einmahl-De Haan one. We study the limiting laws of the process ${T_{n}(p),1\leq p<\infty}$ and completely describe the covariance function of the Gaussian limiting process with the help of combinatorial techniques. Many results available for Hill's statistic regarding asymptotic normality and laws of the iterated logarithm are extended to each margin $T_{n}(p,k)$, for $p$ fixed, and for any distribution function lying in the extremal domain. In the process, we obtain special classes of numbers related to those of paths joining the opposite coins within a parallelogram.