arXiv subjects

William J. Ralph

Publications and source records attributed to William J. Ralph.

2 recordsLinked to original sources

arXiv2015

Slide Statistics And Financial Returns

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.

math.ST↗

arXiv2014

The Slide Dimension of Point Processes

We associate with any finite subset of a metric space an infinite sequence of scale invariant numbers $ρ_1,ρ_2,\dots$ derived from a variant of differential entropy called the genial entropy. 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. We use the $ρ_n$ to define a new notion of dimension called the slide dimension for a special class of point processes on metric spaces. The slide calculus is developed to define $ρ_n$ and an explicit formula is derived for the calculation of $ρ_1$. For a uniform random variable X on $[0,1]^n$, evidence is given that $ρ_1(X) =1/n$ and $ρ_2(X) =-π^2/(6n^2)$ and simulations with a normal variable $Z$ suggest that $ρ_1(Z) =4/π$ and $ρ_2(Z) =-1$. Some potential applications to spatial statistics are considered.

math.PR↗