Bounded Statistics

TL;DR

This paper introduces an algorithm to estimate the similarity between two probability density functions based on their known moment bounds, enabling comparison without explicit functional fitting.

Contribution

It presents a novel method that quantifies PDF similarity using moment bounds and the concept of functions behaving similarly at specific length scales.

Findings

The algorithm provides a quantitative measure of PDF similarity.

It allows comparison of data sets with PDFs without fitting explicit models.

The method is applicable in data analysis for efficient data-PDF comparison.

Abstract

If two probability density functions (PDFs) have values for their first $n$ moments which are quite close to each other (upper bounds of their differences are known), can it be expected that the PDFs themselves are very similar? Shown below is an algorithm to quantitatively estimate this "similarity" between the given PDFs, depending on how many moments one has information about. This method involves the concept of functions behaving "similarly" at certain "length scales", which is also precisely defined. This technique could find use in data analysis, to compare a data set with a PDF or another data set, without having to fit a functional form to the data.