Abstract:
Maximum likelihood estimation of parameter structures in the exponential class is considered
by using a procedure for maximum likelihood estimation under constraints. This procedure does not
require maximization of the likelihood function or derivation of the likelihood equations. Constraints
are imposed on the expected values of the canonical statistics and not specifically on the parameters
themselves. It provides a solution to maximum likelihood estimation in a restricted parameter space.
Maximum likelihood estimation under constraints provides a flexible procedure for the simultaneous
modelling of specific structures of means and variances - in both the univariate and multivariate case.
In the univariate case, it includes linear models for means and variances, heteroskedastic linear regression,
analysis of variance (the balanced and unbalanced case where the assumption of homoscedasticity
is not met), gamma regression and linear mixed models.
Heteroscedastic and/or autocorrelated disturbances in econometric applications are dealt with
by treating correlated observations as a single multivariate observation.
The same procedure is used for maximum likelihood estimation of parameter structures in the case
of multivariate normal samples. The procedure provides a new statistical methodology for the analysis
of specific structures in mean vectors and covariance matrices - including the case where the sample
size is small relative to the dimension of the observations. Special cases include different variations of
the Behrens-Fisher problem, proportional covariance matrices and proportional mean vectors.
Maximum likelihood estimation under constraints as a general approach to estimation in a variety
of problems concerning covariance matrices, is also considered. Estimation of Wishart matrices is a
special case of the general procedure.
Specific structures are illustrated with real data examples. Maximum likelihood estimation under constraints provides a convenient method to obtain maximum
likelihood estimates of mean and (co)variance structures directly - also in problematic situations
where the likelihood equations need to be solved numerically.
