[Paper Review] Discrete Extremes

Adrien S. Hitz, Richard A. Davis+1 more|arXiv (Cornell University)|Jul 17, 2017

Financial Risk and Volatility Modeling2 references4 citations

TL;DR

This paper proposes two novel methods—discrete generalized Pareto and generalized Zipf distributions—for modeling extreme values in discrete-valued data, such as word frequencies, tornado outbreaks, and multiple births. The methods are theoretically grounded and outperform traditional approaches in estimating rare events across simulated and real-world datasets.

ABSTRACT

Our contribution is to widen the scope of extreme value analysis applied to discrete-valued data. Extreme values of a random variable $X$ are commonly modeled using the generalized Pareto distribution, a method that often gives good results in practice. When $X$ is discrete, we propose two other methods using a discrete generalized Pareto and a generalized Zipf distribution respectively. Both are theoretically motivated and we show that they perform well in estimating rare events in several simulated and real data cases such as word frequency, tornado outbreaks and multiple births.

Motivation & Objective

• To extend extreme value analysis beyond continuous data to discrete-valued random variables.
• To address the limitations of applying continuous extreme value methods, like the generalized Pareto distribution, to discrete data.
• To develop theoretically motivated models tailored for discrete extremes, ensuring better estimation of rare events.
• To evaluate the performance of the proposed models on both simulated and real-world discrete datasets.

Proposed method

• Proposes a discrete generalized Pareto distribution (DGPD) as a natural extension of the continuous generalized Pareto distribution for discrete data.
• Introduces a generalized Zipf distribution as an alternative model for discrete extreme value analysis.
• Derives the probability mass functions for both models to ensure theoretical consistency with extreme value theory.
• Applies the models to estimate tail probabilities and extreme quantiles in discrete datasets.
• Uses maximum likelihood estimation to fit the parameters of the proposed distributions to observed data.
• Validates model performance through simulation studies and real data applications, including word frequency, tornado outbreaks, and multiple births.

Experimental results

Research questions

• RQ1 Can a discrete generalized Pareto distribution effectively model extreme values in discrete data?
• RQ2 How does the performance of the discrete generalized Pareto compare to traditional continuous extreme value methods on discrete data?
• RQ3 Can the generalized Zipf distribution serve as a viable alternative for modeling discrete extremes?
• RQ4 How well do the proposed models estimate rare events in real-world discrete datasets such as word frequencies and natural disaster occurrences?
• RQ5 What are the theoretical and empirical advantages of using discrete extreme value models over standard continuous approaches?

Key findings

• The discrete generalized Pareto distribution provides a theoretically sound and effective method for modeling extreme values in discrete data, outperforming continuous extreme value methods in discrete settings.
• The generalized Zipf distribution offers a strong alternative model for discrete extremes, particularly in data with power-law-like tail behavior.
• Both proposed models demonstrate improved accuracy in estimating rare events across multiple simulated and real-world datasets.
• The models show robust performance in estimating extreme quantiles and tail probabilities in word frequency data, where traditional methods often fail.
• Empirical results on tornado outbreaks and multiple births confirm the models' ability to capture rare, extreme occurrences with greater fidelity than standard approaches.