Cumulative Distribution Functions and UPM/LPM Analysis

Abstract

We show that the Cumulative Distribution Function (CDF) is represented by the ratio of the lower partial moment (LPM) ratio to the distribution for the interval in question. The addition of the upper partial moment (UPM) ratio enables us to create probability density functions (PDF) for any function without prior knowledge of its characteristics. We are able to replicate discrete distribution CDFs and PDFs for normal, uniform, poisson, and chi-square distributions, as well as true continuous distributions. This framework provides a new formulation for UPM/LPM portfolio analysis using co-partial moment matrices which are positive symmetrical semi-definite, aggregated to yield a positive symmetrical definite matrix.

Keywords: Partial moments, CDF, PDF, Normal Distribution, Continuous Variable Estimates

JEL Classification: C13, C14