On a discrete Hill's statistical process based on sum-product statistics and its finite-dimensional asymptotic theory
Gane Samb Lô · arXiv (Cornell University) · 2012
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, $00$ 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