General method of moments

In selecting the model, a practitioner will select the market variables that are incorporated in the model these can be directly observed such as zero-coupon rates or forward rates, or swap rates, or they can be indeterminate such as the mean of the short rate. The practitioner will then decide the dynamics of these market or state variables, so, for example, the short rate may be assumed to be mean reverting. Finally, the model must be calibrated to market prices so, the model parameter values input must be those that produce market prices as accurately as possible. There are a number of ways that parameters can be estimated the most common techniques of calibrating time series data such as interest rate data are general method of moments and the maximum likelihood method. For information on these estimation methods, refer to the bibliography. 

Hansen, L., 1982. Large sample properties of generalized method of moments estimators. Econometrica 50, 1029-1055. 

Method of Moments The first step in the analysis of chromatographic systems is often a characterization of the column response to sm l pulse injections of a solute under trace conditions in the Henry s law limit. For such conditions, the statistical moments of the response peak are used to characterize the chromatographic behavior. Such an approach is generally preferable to other descriptions of peak properties which are specific to Gaussian behavior, since the statisfical moments are directly correlated to eqmlibrium and dispersion parameters. Useful references are Schneider and Smith [AJChP J., 14, 762 (1968)], Suzuki and Smith [Chem. Eng. ScL, 26, 221 (1971)], and Carbonell et al. [Chem. Eng. Sci., 9, 115 (1975) 16, 221 (1978)]. 

For fluid particles that continuously coalesce and breakup and where the bubble size distributions have local variations, there is still no generally accepted model available and the existing models are contradictory [20]. A population density model is required to describe the changing bubble and drop size. Usually, it is sufficient to simulate a handful of sizes or use some quadrature model, for example, direct quadrature method of moments (DQMOM) to decrease the number of variables. 

The most familiar estimation procedure is to assume that the population mean and variance are equal to the sample mean and variance. More generally, the method of moments (MOM) approach is to equate sample moments (mean, variance, skewness, and kurtosis) to the corresponding population. Software such as Crystal Ball (Oracle Corporation, Redwood Shores, CA) uses MOM to fit the gamma and beta distributions (see also Johnson et al. 1994). Use of higher moments is exemplified by fitting of the... 

Generalization of the method of moments of coupled-cluster equations to excited electronic states Exact formalism... 

Theoretical analysis has always been an important part of MCD spectroscopy. The parameters Aj, Bj, and Cj can be extracted from an experimental spectrum by a fit to a suitable set of functions or through the method of moments (27-28). The interpretation of these parameters is generally not a trivial task. For smaller, symmetrical molecules group theory has been used to good effect to extract information from an MCD spectrum (11). In recent years, quantum chemical calculation has proven a very useful aid in the interpretation of the often-complicated spectra of larger, nonsymmetric molecules. 

The earliest work employed the method of moments, and Fraissard et al (192) were the first to consider in detail the second moment of the proton NMR resonance in nonspinning solids. In general, when the line shape is determined solely by the I-I and I-S magnetic dipole-dipole interaction, the Van Vleck second moment, M2, is given by (193)... 

Bemardi and Boys49 have examined the problem of the accuracy of the energy and other variables in this method, and give explicit formulae for improving the calculations. The original formulation of the method to cover the calculation of expectation values was given by Handy and Epstein in 1970.50 Armour 51 has examined the method of moments and the transcorrelated wavefunction method (which is a particular form of the method of moments) in some detail. Several expectation values were evaluated in the course of applications of the former method to H2, and in general fairly accurate results were obtained, but numerical problems can occur, and further study is needed. 

In the method of moments (Cotterman et ai, 1985 Luks et ai, 1990), as applied for example to a flash calculation, a functional form for X (jc) and X "(x) is assumed. This will in general contain one or more parameters. One then writes an appropriate model for the chemical potentials in the two phases, and one calculates the values of the parameters by requiring higher and higher moments of the distributions to match the equilibrium condition (equality of chemical potentials in the two phases). 

The method of moments therefore remains applicable to Equation 56. When L = 0, Equations 55 and 56 reduce to Equations 46 and 47, respectively. A preliminary numerical examination shows that increasing the L-value from zero generally improves the approximation of In b u), but going much beyond L = 5 does not seem worthwhile. A comparison of these approximations of In b u) will be shown and fully discussed later in this section. 

Yoon, C. McGraw, R. 2004a Representation of generally mixed multivariate aerosols by the quadrature method of moments I. Statistical foundation. Journal of Aerosol Science 35, 561-576. 

Using the method of moments, Aris (1956) was able to generalize Eq. [25] for a straight tube with an aperture of arbitrary cross-section. His expression for the dimensionless dispersion coefficient is ... 

The second most important general method of determination of dipole moments is microwave spectroscopy. Many reliable values were obtained from the frequencies of the lines in rotational spectra by calculating the three principal moments of inertia of a molecule with respect to the axes x, y, and z and using them to evaluate the geometric parameters of a particular structure [43,44]. All polar molecules give pure rotational spectra whereas molecules with no dipole moments give no such spectra. An external electric field is applied and its intensity is determined by calibration (commonly with carbonyl sulfide, COS) [45]. 

The kinetic equation (11.1) is a nonlinear integro-differential equation, general theory of which does not exist. Its known exact solutions are based on the use of operational methods with reference to a case of linear dependence K(V,w) on each of drop volumes [2]. To solve the equation (11.1) with more general kernel, the approximate methods are used - parametric methods and method of moments, and also numerical methods. Parametric methods and method of moments are based on transforming the kinetic equation into a system of equations for the moments of drop distribution over volumes. However, the resulting system of equations is, as a rule, incomplete, since, apart from the integer moments,... 

The practical application of these results shows that the method of moments provides an accurate description of the kinetics of coalescence only at its initial stage. With some restrictions imposed on the form of the kernel of the kinetic equation and on the initial distribution [6], a self-similar solution of the kinetic equation for a longer time range can be obtained. In the general case, the solution can be obtained by numerical methods. 

We will start with a very simple homopolymerization model that includes only initiation, propagation, transfer to hydrogen, -hydride elimination and imimolecular catalyst deactivation, as depicted in Table 2.4. From our previous discussion of the standard model for polymerization with coordination catalysts, it is known that several steps are not included in Table 2.4. It will be shown, however, that general expressions for population balances and the methods of moments starting with this simplified mechanism can be developed and later they can be extended, rather easily, to include more polymerization steps. 

Generally, if the dipoles in an array are not parallel or coUinear, the computation for self- and mutual impedances is more difficult, and using the method of moments code is a more suitable approach. 

Thus (4.4.8) shows how any term in the population balance equation involving the number density can be expressed purely in terms of the first M integral moments. While the use of generalized Laguerre polynomials with additional parameters provided some flexibility for the method of moments, Hulburt and Akiyama have reported difficulties with this method. 

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