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Generalized Rényi statistics

Péter Kevei, László Viharos

TL;DR

The paper introduces generalized Rényi statistics $X_{k,n}=\sum_{j=1}^k \frac{Z_j}{n+1-j}$ with iid $Z_j$ of mean $\gamma>0$, and shows that a random permutation of indices yields asymptotically iid exponential limits, motivating a heavy-tailed data model $W_{k,n}=C e^{X_{k,n}}$ with strict Pareto tails of index $1/\gamma$. It develops inferential tools for $\gamma$, including a quantile-based estimator $\tilde{\gamma}_n(s)$ with weak consistency and an asymptotic normal distribution, a maximum-likelihood approach, and a Hill estimator that, in this model, reduces to the average of iid terms, yielding standard asymptotics without second-order regular variation or stringent tail conditions. The results establish a universality property: in the limit the generalized model behaves like an iid exponential model, while still closely approximating heavy-tailed data, enabling robust tail-index estimation and providing explicit large-deviation behavior. Practically, this work offers a near-iid framework for heavy-tailed inference that preserves tractable asymptotics for the Hill estimator across a wide class of underlying $Z$ distributions, and connects the tail behavior to a Pareto-type limit via log-spacings and $W_{k,n}$ representations.

Abstract

In Rényi's representation for exponential order statistics, we replace the iid exponential sequence with any iid sequence, and call the resulting order statistic generalized Rényi statistic. We prove that by randomly reordering the variables in the generalized Rényi statistic, we obtain in the limit a sequence of iid exponentials. This result allows us to propose a new model for heavy-tailed data. Although the new model is very close to the classical iid framework, we establish that the Hill estimator is weakly consistent and asymptotically normal without any further assumptions on the underlying distribution or on the number of upper order statistics used in the estimator.

Generalized Rényi statistics

TL;DR

The paper introduces generalized Rényi statistics with iid of mean , and shows that a random permutation of indices yields asymptotically iid exponential limits, motivating a heavy-tailed data model with strict Pareto tails of index . It develops inferential tools for , including a quantile-based estimator with weak consistency and an asymptotic normal distribution, a maximum-likelihood approach, and a Hill estimator that, in this model, reduces to the average of iid terms, yielding standard asymptotics without second-order regular variation or stringent tail conditions. The results establish a universality property: in the limit the generalized model behaves like an iid exponential model, while still closely approximating heavy-tailed data, enabling robust tail-index estimation and providing explicit large-deviation behavior. Practically, this work offers a near-iid framework for heavy-tailed inference that preserves tractable asymptotics for the Hill estimator across a wide class of underlying distributions, and connects the tail behavior to a Pareto-type limit via log-spacings and representations.

Abstract

In Rényi's representation for exponential order statistics, we replace the iid exponential sequence with any iid sequence, and call the resulting order statistic generalized Rényi statistic. We prove that by randomly reordering the variables in the generalized Rényi statistic, we obtain in the limit a sequence of iid exponentials. This result allows us to propose a new model for heavy-tailed data. Although the new model is very close to the classical iid framework, we establish that the Hill estimator is weakly consistent and asymptotically normal without any further assumptions on the underlying distribution or on the number of upper order statistics used in the estimator.
Paper Structure (7 sections, 10 theorems, 61 equations, 3 figures)

This paper contains 7 sections, 10 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Statistical inference
  3. Quantile estimator
  4. Maximum likelihood estimator
  5. Hill's estimator
  6. Comparison of model \ref{['eq:modelheavy']} with the classical iid model
  7. Proofs

Key Result

Theorem 1

Assume that $Z_1, Z_2, \ldots$ in eq:generalized are iid random variables such that $E (Z_1 ) = \gamma > 0$, and let $(\delta_1, \ldots, \delta_n)$ be a uniform random permutation of $(1,\ldots, n)$ independent of the $Z$'s. Then for any $k \geq 1$ where $Y_1, \ldots, Y_k$ are independent exponential random variables with mean $\gamma$.

Figures (3)

  • Figure 1: Left: The function $h(s)$ appearing in the asymptotic variance of $\widetilde{\gamma}_n(s)$. Right: Empirical variances of $\sqrt{n} (\widetilde{\gamma}_n(s) - \gamma) / \sqrt{ \operatorname{Var}(Z)}$, with $n= 1000$, for 1000 independent simulations. The common distribution of the $Z$'s in \ref{['eq:generalized']} is uniform, Bernoulli, and exponential, respectively.
  • Figure 2: The Hill estimator in the classical iid model (left) and in model \ref{['eq:modelheavy']} (right).
  • Figure 3: Coverage frequencies of the asymptotic confidence intervals for $\varepsilon = 0.1$ in the classical iid model (left) and in model \ref{['eq:modelheavy']} (right). In the iid model we used confidence interval \ref{['eq:confint-new']} and \ref{['eq:confint-Hill']}.

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Example 1
  • Example 2
  • Example 3
  • Theorem 5
  • Theorem 6
  • Corollary 7
  • ...and 3 more