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

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.