Warp statistics and financial returns

Abstract

We present a new approach to financial returns based on an infinite family of statistics called slide statistics that we introduce. 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 or indexes like the S&P 500 and the Dow Jones. Formally, we associate with any finite subset of a metric space an infinite sequence of scale invariant numbers ρ 1 , ρ 2 , . . . derived from a variant of differential entropy called the genial entropy. We give explicit formulas for ρ 1 and ρ 2 that are easily evaluated by a computer and make this theory particularly suitable for applications. As statistics for point processes, these numbers often appear to converge in simulations and we give examples where 1/ρ 1 converges to the Hausdorff dimension and we prove that ρ 1 ≥ 0. For a uniform random variable X on [a, b] n , the evidence from simulations suggests that ρ 1 (X) = 1/n and ρ 2 (X) = −π 2 /(6n 2 ) which yields new tests for spatial randomness. The slide statistics describe continuous random variables in an entirely new way. For example, if Z is any normal variable then simulations suggest that ρ 1 (Z) = 4/π and ρ 2 (Z) = −1 which provides new goodness of fit tests for normality.