Slide Statistics And Financial Returns

TL;DR

This paper introduces slide statistics, a new method based on differential entropy variants, to analyze financial returns, revealing that common models like stable distributions may not accurately represent actual market data.

Contribution

It presents a novel set of statistics derived from genial entropy, applicable to any metric space, and demonstrates their effectiveness in analyzing financial return distributions.

Findings

Slide statistics suggest stable distributions are poor models for financial returns.

The first slide statistic converges to Pi/4 in normal distributions.

For some variables, the statistic converges to the reciprocal of the Hausdorff dimension.

Abstract

We introduce a new approach to financial returns based on an infinite family of statistics called slide statistics. The evidence these statistics provide suggests that certain distributions such as the stable distributions are not good models for the financial returns from various securities and indexes. The slide statistics are derived from a variant of differential entropy called the genial entropy and can be computed for any sample in a metric space. We give explicit formulas for the first two of these statistics that are easily evaluated by a computer and make this theory particularly suitable for applications. In simulations with a normal random variable, the first slide statistic appears to converge to Pi/4 and for certain other random variables it appears to converge to the reciprocal of the Hausdorff dimension.