Derivative of Reduced Cumulative Distribution Function and Applications

Kevin Maritato () and Stan Uryasev
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Kevin Maritato: Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA
Stan Uryasev: Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA

JRFM, 2023, vol. 16, issue 10, 1-24

Abstract: The reduced cumulative distribution function (rCDF) is the maximal lower bound for the cumulative distribution function (CDF). It is equivalent to the inverse of the conditional value at risk (CVaR), or one minus the buffered probability of exceedance (bPOE). This paper introduces the reduced probability density function (rPDF), the derivative of rCDF. We first explore the relation between rCDF and other risk measures. Then we describe three means of calculating rPDF for a distribution, depending on what is known about the distribution. For functions with a closed-form formula for bPOE, we derive closed-form formulae for rPDF. Further, we describe formulae for rPDF based on a numerical bPOE when there is a closed-form formula for CVaR but no closed-form formula for bPOE. Finally, we give a method for numerically calculating rPDF for an empirical distribution, and compare the results with other methods for known distributions. We conducted a case study and used rPDF for sensitivity analysis and parameter estimation with a method similar to the maximum likelihood method.

Keywords: reduced probability density function (rPDF); reduced cumulative distribution function (rCDF); sensitivity analysis; buffered probability density function (bPDF); buffered cumulative distribution function (bCDF); maximum likelihood estimation (MLE); reduced maximum likelihood estimation (rMLE); buffered probability of exceedance (bPOE); conditional value at risk (CVaR); expected shortfall (ES) (search for similar items in EconPapers)
JEL-codes: C E F2 F3 G (search for similar items in EconPapers)
Date: 2023
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