Risk Tails and General Orthonormal Polynomials

Abstract

In order to characterize a statistical probability distribution p(x) of a variable x, the moments of the distribution are used; the first two of which are the mean and standard deviation. The z-score is often used to characterize data points of x (e.g. outliers with large z-scores). Polynomials with respect to p(x) as the measure in the orthogonality relation for the polynomials can be constructed. These generalize the ubiquitous z-score. These polynomials (which we call GONPOMs) can be useful to refine the characterization of data. Specifically they can be used in a targeted way to characterize the change in shape of a distribution, e.g. for risk tails. It turns out that the GONPOMs are (non-standard) polynomials first described by Chebyshev. Our purpose here is to describe the theory and give a simple prototype numerical example.