Univariate Linear Normal Models: Optimal Equivariant Estimation

In this paper, we establish the existence and uniqueness of the minimum intrinsic risk equivariant (MIRE) estimator for univariate linear normal models. The estimator is derived under the action of the subgroup of the affine group that preserves the column space of the design matrix, within the framework of intrinsic statistical analysis based on the squared Rao distance as the loss function. This approach provides a parametrization-free assessment of risk and bias, differing substantially from the classical quadratic loss, particularly in small-sample settings. The MIRE is compared with the maximum likelihood estimator (MLE) in terms of intrinsic risk and bias, and a simple approximate version (a-MIRE) is also proposed. Numerical evaluations show that the a-MIRE performs closely to the MIRE while significantly reducing the intrinsic bias and risk of the MLE for small samples. The proposed intrinsic methods could extend to other invariant frameworks and connect with recent developments in robust estimation procedures.