Maximum-likelihood estimation, numerical

Cobelli, C.. and Saccomani, M. P. (1990). Unappreciation of a priori identifiability in software packages causes ambiguities in numerical estimates. Am. J. Physiol. 2S8, E10S8-E10S9. Davidian, M., and Gallant, A. R. (1992). Smooth nonparametric maximum likelihood estimation for population pharmacokinetics, with application to quinidine. J. Pharmacokinet. Biopharm. 20, 529-556. 

Both a and /S are unknown and must be estimated from data. For unknown ju., Ni has a negative binomial distribution. Pooling all triggers allows the use of all available data, from which MLE for a and /S can be obtained. Maximum likelihood estimation is more computationally demanding, but it has been shown to provide better results than method of moments estimates (Elvik 2008). The MLE equations do not exist in closed form, but numerical methods can be employed to provide estimates. Estimates for a and P were obtained using the Newton-Raphson method with initial values ... 

Roughly, about 250 data points are required to fit the generalized hyperbolic distributions. However, about 100 data points can offer reasonable results. Although maximum-likelihood estimation method can be used to estimate the parameters, it is very difficult to solve such a complicated nonlinear equation system with five equations and five unknown parameters. Therefore, numerical algorithms are suggested such as modified Powell method (Wang, 2005). Kolmogorov-Smirnov statistics can also be used here for the fitness test. 

We also show the derivation of the Maximum Likelihood Estimates (MLE) of the model parameters as an alternative inference approach to the moment method. While MLE s provide more accurate estimates for this model there is not a closed form expression, so numerical methods are required to obtain a solution. 

Stanley and Guichon [151] have recently proposed the expectation-maximization (EM) method for numerical estimation of adsorption energy distributions. This method does not require prior knowledge of the distribution function or any analytical equation for the total isotherm. Moreover, it requires no smoothing of the adsorption isotherm data and coverages with high stability toward the maximum-likelihood estimate. 

Numerous criteria for supervised classifications exist, see for example [5]. In this work a maximum likelihood criterion is applied for classification. Maximum likelihood estimators of the Gaussian parameters and Ec within each class c = 1, 2,..., no are first obtained based on the training data... 

Assumptions may be made or models adopted (often by implication) about a system being measured that are not consistent with reality. The selection of the method of data reduction may be partly on the basis of the model adopted and partly on the basis of features such as computation time and simplicity. Kelly classified data processing methods as direct, graphical, minmax, least squares, maximum likelihood, and bayesian. Each method has rules by which computations are made, and each produces an estimate (or numerical result) of reality. 

In the previous section, we discussed various modes of inference about the parameters in the models that we specify for the data. We now wish to discuss the actual numerical computations necessary to obtain the parameter estimates. It should be noted that while maximum likelihood and Bayesian inference are modes of estimation, the choice of the numerical algorithm is a separate decision. 

Having generated the data, we then estimate a based on the data and the 4 values using numerical maximum likelihood methods. 

When the assumption of error-free x-values is not valid, either in method comparisons or, in a conventional calibration analysis, because the standards are unreliable (this problem sometimes arises with solid reference materials), an alternative comparison method is available. This technique is known as the functional relationship by maximum likelihood (FREML) method, and seeks to minimize and estimate both x- and y-dlrection errors. (The conventional least squares approach can be regarded as a special and simple case of FREML.) FREML involves an iterative numerical calculation, but a macro for Minitab now offers this facility (see Bibliography), and provides standard errors for the slope and intercept of the calculated line. The method is reversible (i.e. in a method comparison it does not matter which method is plotted on the x-axis and which on the y-axis), and can also be used in weighted regression calculations (see Section 5.10). 

The maximum-likelihood parameter estimates for an MA process can be obtained by solving a matrix equation without any numerical iterations. 

In the Fisher approach, parameter estimates can be obtained by nonlinear least squares or maximum likelihood together with their precision, such as, a measure of a posteriori or numerical identifiabihty. Details and references on parameter estimation of physiologic system models can be found in Carson et al. [1983] and Landaw and DiStefano [1984]. Weighted nonlinear least squares is mostly used, in which an estimate 9 of the model parameter vector 0 is determined as... 

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