Testing Tail Weight of a Distribution Via Hazard Rate

TL;DR

This paper presents a new algorithm for testing whether a distribution has a light tail based on hazard rate, requiring a polynomial number of samples under certain assumptions, with proven hardness results for the general case.

Contribution

The paper introduces a novel bucketing-based algorithm for hazard rate tail testing and establishes sample complexity bounds under natural assumptions.

Findings

Algorithm distinguishes light-tailed from non-light-tailed distributions efficiently.

Sample complexity is polynomial in problem parameters.

Hardness results show the necessity of assumptions for the problem.

Abstract

Understanding the shape of a distribution of data is of interest to people in a great variety of fields, as it may affect the types of algorithms used for that data. We study one such problem in the framework of distribution property testing, characterizing the number of samples required to to distinguish whether a distribution has a certain property or is far from having that property. In particular, given samples from a distribution, we seek to characterize the tail of the distribution, that is, understand how many elements appear infrequently. We develop an algorithm based on a careful bucketing scheme that distinguishes light-tailed distributions from non-light-tailed ones with respect to a definition based on the hazard rate, under natural smoothness and ordering assumptions. We bound the number of samples required for this test to succeed with high probability in terms of the parameters of the problem, showing that it is polynomial in these parameters. Further, we prove a hardness result that implies that this problem cannot be solved without any assumptions.