Edgeworth series expansion of the truncated Cauchy function and its effectiveness in the study of atomic heterogeneity

Abstract

The effectiveness of a near Gaussian distribution, namely the Edgeworth Series representation of a truncated Cauchy distribution in studying heterogeneous cases has been studied for space group P¯1. Though moments do not exist for a Chauchy distribution a truncated Cauchy distribution has finite moments whose value depends on the truncation limit. A number of real examples with varying degrees of heterogeneity have been considered and the effect of truncation at various cut-off values has been studied. This approach has been compared with that of the exact approach method for which expression for cumulative probability for equal and heterogeneous cases (considering two seperate parameters, p= N

Quantifying superparamagnetic signatures in nanoparticle magnetite: a generalized approach for physically meaningful statistics and synthesis diagnostics

“…1, eqn (1)) and probability density function (PDF; eqn (2)) are modified by the location parameter, x 0 , and the scale parameter, γ . 29 As a stable distribution, the Cauchy distribution has applications in chemistry and physics, 30–32 geology, 33 engineering, 34 and economics. 35 …”