Invariant estimation with application to linear models
The method of invariant estimation proposed in this dissertation relies on defining a group of transformations on the sample space such that i) the group structure is isomorphic to the parameter space and"carries" the estimation problem in a natural manner (thus defining an “carries” the estimation problem), and ii) the group structure generates orbits on the parameter space and the problem is to estimate the orbit in which the parameter lies (thus defining an invariant estimation problem). If the group of transformations can be expressed as the semi-direct product of two subgroups, one a"nuisance" group which is a normal subgroup, then an estimator of orbits under the nuisance group in the invariant estimation problem can be naturally obtained from the best estimator in the equivariant estimation problem.
The primary application is to the invariant estimation of the parameters in the general linear model under the (nuisance) group of scale changes on the dependent and independent variables. The invariant estimator of the regression coefficient is found to be a"standardized regression coefficient,'' but this standardized regression coefficient is not the same as the typical one ("beta coefficient") found in elementary statistics texts and social science research.
Comparison of the proposed estimator to the usual estimator, in the case in which the input matrix is nonstochastic, shows the proposed estimator to be superior to the usual estimator in terms of such criteria as consistency, unbiasedness, and simplicity of distribution. In the case in which the input matrix is stochastic, some justification can be found for the use of the usual estimator.
Application of the proposed method of invariant estimation to the problem of obtaining estimators invariant under nonsingular transformations is straightforward, although the estimator obtained is difficult to interpret.