Gaussian EQMOM

This set of five moments is used for bi-Gaussian EQMOM with N = 2 (Chalons et at., 2010). 

The application of multivariate EQMOM to evaluate integrals is very similar to that of univariate EQMOM. For the multi-Gaussian EQMOM, we have... 

Note that, while the spread parameter crj is the same for all terms in the summation, the conditional parameter 0-2,0, can depend on a. Also, the functional form used for 6 1 need not be the same as that used for ( -g- 1 could use beta EQMOM, while 2 uses Gaussian EQMOM). Although the form in Eq. (3.134) is not as general as that in Eq. (3.124), we shall see that it leads to a direct method for moment inversion that is very similar to the one used in the CQMOM. The bivariate moments found from Eq. (3.134) have the form... 

As an example of univariate EQMOM, we will consider an NDE with (-co, -1-00) and define kernel density functions using Gaussian distributions (Chalons et al, 2010) ... 

In the literature (Chalons et al, 2010), only a bivariate EQMOM with four abscissas represented by weighted Gaussian distributions with a diagonal covariance matrix has been considered. However, it is likely that brute-force QMOM algorithms can be developed for other distribution functions. Using the multi-Gaussian representation as an example, the approximate NDF can be written as... 

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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