A Non-parametric Approach to Inference about the Tail of a Continuous or a Discrete Distribution

TL;DR

The paper tackles non-parametric tail inference for both discrete and continuous distributions by introducing a tail profile concept grounded in information-theoretic quantities from domains of attraction on countable alphabets. It constructs an unbiased estimator $T_v$ for the tail profile $\tau_v$ using a U-statistic $Z_v$ and proves $T_v/\tau_v \xrightarrow{a.s.}1$, enabling a visual, plot-based classifier via entropic plots to distinguish power, sub-exponential, and near-exponential tails. The methodology extends to continuous data through a binning result and demonstrates practical utility via Amazon stock returns, US city population data, and extensive simulations, with an accompanying R package TailClassifier. Overall, the approach provides a model-free pre-screening tool for tail-type selection and parameter estimation in heavy-tailed contexts, with robust finite-sample performance and clear guidelines for tail-profile selection and interpretation.

Abstract

This article introduces a non-parametric information-theoretic approach to inference about the tail of a continuous or a discrete distribution. Leveraging a new concept named tail profile -- a set of information-theoretic quantities developed from results of domains of attraction on countable alphabets -- theoretical evidence supports the identification of specific discrete distributional tail types through a sequence of plots. The approach discerns tail types by bench-marking against exponential, and three thicker-than-exponential families: near-exponential, sub-exponential, and power-law (zipf, Pareto). For tails thicker-than-exponential, the approach also provides point and interval estimates for some of the underlying distribution parameters. While primarily designed to streamline the selection of discrete parametric models for detailed statistical analysis, a supporting theorem enables the method's extension use to continuous data, stating that binning continuous data with a common width preserves the tail decay rate under certain conditions. Simulations are presented to demonstrate the method's performance across various scenarios.

Figures (7)

• Figure 1: Entropic Plots: The solid curve is for power decay, the dashed line is sub-exponential, and the long dashed curve is for near-exponential.
• Figure 2: Entropic plots for AMZN left-tail minute log-return data.
• Figure 3: Entropic plots for AMZN right-tail minute log-return data.
• Figure 4: Entropic Plots for a random sample with size 4000 that was generated from a Power decaying tail distribution with parameter 1.5.
• Figure 5: Entropic Plots for a random sample with size 10000 that was generated from a sub-exponentially decaying tail distribution with parameter 0.5.

Theorems & Definitions (19)

• Definition 1: Tail index and tail profile
• Theorem 1
• Proposition 1: Theorem 6.3 in zhang2016statistical
• proof : Proof of Theorem \ref{['th1']}
• Lemma 1
• Theorem 2
• Lemma 2
• proof : Proof of Lemma \ref{['lemmaNEsolution']}
• proof : Proof of Theorem \ref{['th2']}
• Lemma 3: Corollary 2.1 in Ref. zhang2016statistical