Gaussian quadrature for non-Gaussian distributions

Many problems of operations research or decision science involve continuous probability distributions, whose handling may be sometimes unmanageable; in order to tackle this issue, different forms of approximation methods can be used. When constructing a k-point discrete approximation of a continuous random variable, moment matching, i.e., matching as many moments as possible of the original distribution, is the most popular technique. This can be done by resorting to the so-called Gaussian quadrature procedure (originally developed by Gauss in the nineteenth century) and solving for the roots of an orthogonal polynomial or for the eigenvalues of a real symmetric tridiagonal matrix. The moment-matching discretization has been widely applied to the Gaussian distribution and more generally to symmetric distributions, for which the procedure considerably simplifies. Despite the name, Gaussian quadrature can be theoretically applied to any continuous distribution (as far as the first 2k − 1 raw moments exist), but not much interest has been shown in the literature so far. In this work, we will consider some examples of asymmetric distributions defined over the positive real line (namely, the gamma and the Weibull, for which expressions for the integer moments are available in closed form) and show how the moment-matching procedure works and its possible practical issues. Comparison with an alternative discretization technique is discussed.