Orthonormal Polynomial Expansions and Lognormal Sum Densities

Abstract

Approximations for an unknown density $g$ in terms of a reference density $f_\nu$ and its associated orthonormal polynomials are discussed. The main application is the approximation of the density $f$ of a sum $S$ of lognormals which may have different variances or be dependent. In this setting, $g$ may be $f$ itself or a transformed density, in particular that of $\log S$ or an exponentially tilted density. Choices of reference densities $f_\nu$ that are considered include normal, gamma and lognormal densities. For the lognormal case, the orthonormal polynomials are found in closed form and it is shown that they are not dense in $L_2(f_\nu)$, a result that is closely related to the lognormal distribution not being determined by its moments and provides a warning to the most obvious choice of taking $f_\nu$ as lognormal. Numerical examples are presented and comparisons are made to established approaches such as the Fenton--Wilkinson method and skew-normal approximations. Also extensions to density estimation for statistical data sets and non-Gaussian copulas are outlined.