Gaussian quadrature univariate

The theory of Gaussian quadrature applies only to univariate distributions. However, in practical cases, the study of distributions with multiple internal coordinates is often necessary. In these cases the closure problem generally assumes the following form ... 

As has already been mentioned, Eq. (3.36) is not a Gaussian quadrature approximation its degree of accuracy is not known a priori (and strongly depends on the choice of moments on which the formula is constructed) and the algorithms for its derivation, from the moments of the NDF, are not well known (unlike for the univariate case). 

The Gaussian quadrature algorithm introduced in Section 3.1.1 is equivalent to approximating the univariate NDF by a sum of Dirac delta functions ... 

By extending the concepts of the previous section, it is possible to derive an approximation (which is no longer a real Gaussian quadrature) for the bivariate system. The resulting Appoint quadrature approximation transforms the definition of the general bivariate moment into (Wright et at, 2001b) m = Za=i where, as for the univariate case, Wa... 

Chapter 3 provides an introduction to Gaussian quadrature and the moment-inversion algorithms used in quadrature-based moment methods (QBMM). In this chapter, the product-difference (PD) and Wheeler algorithms employed for the classical univariate quadrature method of moments (QMOM) are discussed, together with the brute-force, tensor-product, and conditional QMOM developed for multivariate problems. The chapter concludes with a discussion of the extended quadrature method of moments (EQMOM) and the direct quadrature method of moments (DQMOM). 

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