Intrinsically Optimal Equivariant Estimator for the Univariate Linear Normal Models

In this paper we prove the existence and unicity of the minimum risk equivariant estimator (MIRE) for the univariate linear normal models, under the action of the subgroup of the affine group which leaves the column space of the design matrix invariant and using the framework of the intrinsic analysis of statistical estimation which uses the square of the Rao distance as the loss function, which expectation behaves very different to the corresponding of the square error loss when we work with small samples, and supplying also an intrinsic bias measure for any equivariant estimator. This estimator is studied and compared with the standard maximum likelihood estimator (MLE) in terms of its intrinsic risk and bias, showing explicitly its differences, apparent with small samples, fortunately small for large sample sizes. Moreover, an approximate and very simple estimator is also suggested, which reduces the majority of bad behaviour of MLE, of intrinsic risk and bias, for small samples.