Testing Tail Weight of a Distribution Via Hazard Rate
This work addresses the challenge of characterizing distribution tails for fields relying on data analysis, but it is incremental as it builds on existing distribution property testing frameworks.
The paper tackles the problem of distinguishing light-tailed from non-light-tailed distributions using a hazard rate-based definition, developing an algorithm that requires a polynomial number of samples to succeed with high probability. It also proves a hardness result showing the problem is unsolvable without assumptions.
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.