Generalized Fisher-Darmois-Koopman-Pitman Theorem and Rao-Blackwell Type Estimators for Power-Law Distributions

Core Concepts

The author explores the concept of sufficiency beyond maximum likelihood estimation, focusing on power-law distributions and likelihood functions.

Abstract

The content delves into the generalization of sufficiency in estimation problems, introducing new concepts like minimal sufficient statistics. It discusses the Fisher-Darmois-Koopman-Pitman theorem and its implications for power-law families. The paper also extends the Rao-Blackwell theorem to power-law distributions, providing insights into robust inference methods.

Stats

A statistic T is sufficient for estimating θ if the conditional distribution of the sample given T is independent of θ.
A set of probability distributions E is an exponential family if it can be expressed in a specific form involving functions w, f, and h.
Student distributions fall within a power-law extension of exponential families.
The Cauchy-Schwarz likelihood function introduces a new approach to estimation with polynomial roots involved in finding estimators.
Jones et al. likelihood function provides a robust alternative to maximum likelihood estimation.

Quotes

"The entire sample is trivially a sufficient statistic with respect to any estimation."
"Minimal sufficient statistics not only help in data reduction but also lead to better estimators than the given one."
"Jones et al. likelihood function offers a robust alternative to ML estimation by down-weighting outliers' effects."

Key Insights Distilled From

Generalized Fisher-Darmois-Koopman-Pitman Theorem and Rao-Blackwell Type Estimators for Power-Law Distributions

by Atin Gayen,M... at arxiv.org 02-29-2024

https://arxiv.org/pdf/2205.00530.pdf

Deeper Inquiries

How does the generalized notion of sufficiency impact traditional statistical estimation methods

The generalized notion of sufficiency expands the traditional statistical estimation methods by allowing for a broader range of estimators beyond maximum likelihood. By considering estimation problems based on alternative likelihood functions like Jones et al. and Basu et al., this concept broadens the scope of sufficiency in statistics. It provides a framework to identify sufficient statistics that may not align with maximum likelihood estimation, opening up new avenues for efficient parameter estimation.

What are the practical implications of extending classical sufficiency concepts to power-law distributions

Extending classical sufficiency concepts to power-law distributions has significant practical implications in data analysis. Power-law distributions are commonly found in various real-world phenomena such as network connectivity, income distribution, and earthquake magnitudes. By establishing sufficiency within these distributions, researchers can develop more robust and accurate estimators tailored to handle heavy-tailed data characteristic of power-law distributions. This extension enhances the reliability and efficiency of statistical inference when dealing with complex datasets exhibiting power-law behavior.

How can these findings be applied beyond mathematical theory into real-world data analysis scenarios

The findings regarding generalized sufficiency for power-law distributions have direct applications in real-world data analysis scenarios across diverse fields such as social sciences, economics, biology, and physics. In network analysis, where power-law degree distributions are prevalent, understanding the minimal sufficient statistics can lead to improved modeling techniques for network structures. In financial markets characterized by fat-tailed returns following a power law, utilizing these concepts can enhance risk management strategies and asset pricing models. Moreover, in biological systems displaying scale-free networks governed by power laws, applying these theoretical advancements can aid in deciphering complex interactions among biological entities accurately.