Custom-made Gauss quadrature for statisticians

Abstract: The theory and computational methods for custom-made Gauss quadrature have been described in Gautschi's 2004 monograph. Gautschi has also provided Fortran and MATLAB code for the implementation and illustration of these methods. We have written an R package, implemented in the high-precision arithmetic provided by the R package Rmpfr, that uses a moment-based method via moment determinants to compute a Gauss quadrature rule, with up to 33 nodes, provided that the moments can be computed to arbitrary precision using the standard mathematical functions provided by the Rmpfr package. Our hope is that the provision of our free R package and the numerical results that we present will encourage other statisticians to also consider the custom-made construction of Gauss quadrature rules.