The Equivariance Criterion in a Linear Model for Random-$X$ Cases

Abstract

Equivariance is increasingly used in machine learning and statistics, often without systematic justification. In a companion article, the equivariance criterion was applied to the normal linear model with a fixed design matrix (fixed-$X$), yielding the minimum risk equivariant (MRE) estimators of the coefficient vector and of the condensed diagonal covariance matrix under a multivariate invariant location--scale group. We extend these results to the random-$X$ case, with covariates sampled from a population. The extension hinges on a distinction vacuous for fixed-$X$ but fundamental for random-$X$: whether risk and unbiasedness are evaluated conditionally on the realized design or after averaging over the design distribution. Under conditional evaluation, the fixed-$X$ group applies given $X$: least squares remains the best equivariant estimator of the coefficient vector, and the MRE estimators of the population variances keep their fixed-$X$ forms with population sizes at the realized design. Under absolute evaluation with an i.i.d.\ design, the picture changes qualitatively: the natural scale group acting jointly on $(Y,X)$ fixes the coefficient vector, the induced parameter-space action is intransitive, equivariant risks are constant only along orbits indexed by the signal-to-noise ratio $\rho =\Vert \beta {\Vert }^2/{\sigma }^2$, and no uniformly minimum risk equivariant estimator exists. In the scalar case the optimal equivariant weight is the oracle shrinkage factor $w^{\ast }(\rho )=\rho /(\rho +E[T^{-1}])$, with least squares recovered as the infinite-signal limit $\rho \to \infty$---explaining and refining the known failure of the Gauss--Markov theorem with random regressors. For a centered design under location--scale transformations, least squares remains optimal within the natural invariant-contrast class, and the MRE estimator $S^2/(n-p+2)$ of the error variance is valid under both modes.