Solution to the practical problem of moments using non-classical orthogonal polynomials, with applications for probabilistic analysis

A method is presented for solving the “practical” problem of moments to produce probability density functions (PDFs) using non-classical orthogonal polynomials. PDFs are determined from given sets of moments by applying the Gram–Schmidt process with the aid of computer algebra. By selecting weighting functions of similar shape to desired PDFs, orthogonal polynomial series are obtained that are stable at high order and allow accurate approximation of tail probabilities. thod is first demonstrated by approximating a χ2 PDF with an orthogonal series based on a lognormal weighting function. More general orthogonal expansions, based on Pearson type I and Johnson transform distributions, are then demonstrated. These expansions are used to produce PDFs for maximum daily river discharge, concrete strength, and maximum seasonal snow depths, using limited data sets. In all three cases the moments of the high order series are found to closely match those of the data.