Risk Tails and General Orthonormal Polynomials

12 Pages Posted: 12 Jul 2016 Last revised: 22 Jul 2016

See all articles by Jan Dash

Jan Dash

Bloomberg LP

Harvey J. Stein

Bloomberg L.P.; Columbia University - Department of Mathematics

Mario Bondioli

Bloomberg L.P.

Date Written: July 11, 2016

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.

Keywords: risk tails, probability distribution, moments, GONPOMs, Chebyshev, generalize, z-score

JEL Classification: C63, F65, G01, G1

Suggested Citation

Dash, Jan and Stein, Harvey J. and Bondioli, Mario, Risk Tails and General Orthonormal Polynomials (July 11, 2016). Available at SSRN: https://ssrn.com/abstract=2808160 or http://dx.doi.org/10.2139/ssrn.2808160

Jan Dash (Contact Author)

Bloomberg LP ( email )

731 Lexington Ave
New York, NY 10022
United States

Harvey J. Stein

Bloomberg L.P. ( email )

731 Lexington Avenue
New York, NY 10022
United States
212 617 3059 (Phone)

Columbia University - Department of Mathematics ( email )

New York, NY
United States

Mario Bondioli

Bloomberg L.P. ( email )

731 Lexington Avenue
New York, NY 10022
United States

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