On optimal smoothing of density estimators obtained from orthogonal polynomial expansion methods

Marumo, Kohei and Wolff, Rodney C. (2016) On optimal smoothing of density estimators obtained from orthogonal polynomial expansion methods. Journal of Risk, 18 3: 47-76.

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Author Marumo, Kohei
Wolff, Rodney C.
Title On optimal smoothing of density estimators obtained from orthogonal polynomial expansion methods
Journal name Journal of Risk   Check publisher's open access policy
ISSN 1465-1211
Publication date 2016-02
Year available 2016
Sub-type Article (original research)
Volume 18
Issue 3
Start page 47
End page 76
Total pages 30
Place of publication London, United Kingdom
Publisher Incisive Media
Collection year 2017
Language eng
Formatted abstract
We discuss the application of orthogonal polynomials to the estimation of probability density functions, particularly with regard to accessing features of a portfolio’s profit/loss distribution. Such expansions are given by the sum of known orthogonal polynomials multiplied by an associated weight function. However, naive applications of expansion methods are flawed. The shape of the estimator’s tail can undulate under the influence of the constituent polynomials in the expansion, and it can even exhibit regions of negative density. This paper presents techniques to remedy these flaws and improve the quality of risk estimation.We show that by targeting a smooth density that is sufficiently close to the target density, we can obtain expansion-based estimators that do not have the shortcomings of equivalent naive estimators. In particular, we apply optimization and smoothing techniques that place greater weight on the tails than on the body of the distribution. Numerical examples using both real and simulated data illustrate our approach. We further outline how our techniques can apply to a wide class of expansion methods and indicate opportunities to extend to the multivariate case, where distributions of individual component risk factors in a portfolio can be accessed for the purpose of risk management.
Keyword Delta-Gamma-Vega-Normal model
Expected shortfall
Hermite polynomials
Historical simulation method
Value at risk
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: W.H. Bryan Mining Geology Research Centre
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Created: Wed, 26 Aug 2015, 13:51:57 EST by Rodney Wolff on behalf of WH Bryan Mining and Geology Centre