Residual variance model parameter estimation using maximum

RESIDUAL VARIANCE MODEL PARAMETER ESTIMATION USING MAXIMUM LIKELIHOOD... 

An independent method to identify the stochastic errors of impedance data is described in Chapter 21. An alternative approach has been to use the method of maximum likelihood, in which the regression procedure is used to obtain a joint estimate for the parameter vector P and the error structure of the data. The maximum likelihood method is recommended under conditions where the error structure is unknown, but the error structure obtained by simultaneous regression is severely constrained by the assumed form of the error-variance model. In addition, the assumption that the error variance model can be obtained by minimizing the objective function ignores the differences eimong the contributions to the residual errors shown in Chapter 21. Finedly, the use of the regression procedure to estimate the standard deviation of the data precludes use of the statistic... 

Carroll and Ruppert (1988) and Davidian and Gil-tinan (1995) present comprehensive overviews of parameter estimation in the face of heteroscedasticity. In general, three methods are used to provide precise, unbiased parameter estimates weighted least-squares (WLS), maximum likelihood, and data and/or model transformations. Johnston (1972) has shown that as the departure from constant variance increases, the benefit from using methods that deal with heteroscedasticity increases. The difficulty in using WLS or variations of WLS is that additional burdens on the model are made in that the method makes the additional assumption that the variance of the observations is either known or can be estimated. In WLS, the goal is not to minimize the OLS objective function, i.e., the residual sum of squares,... 

A determinant criterion is used to obtain least-squares estimates of model parameters. This entails minimizing the determinant of the matrix of cross products of the various residuals. The maximum likehhood estimates of the model parameters are thus obtained without knowledge of the variance-covariance matrix. The residuals e, , and correspond to the difference between predicted and actual values of the dependent variables at the different values of the Mth independent variable (m = to to u = tn), for the ith, 7th, and kth experiments (A, B, and C), respectively. It is possible to constmct an error covariance matrix with elements v,y ... 

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