Statistical commutators with applications to estimation, inference, and regularization tuning

Abstract We develop a classical operator framework, the statistical commutator , to study estimation and regularization through the non-commutativity of statistical operators. For a dominated parametric model, the score-transport operator $$\Lambda _\theta g:= \partial _\theta g + g\,S_\theta $$ Λ θ g : = ∂ θ g + g S θ encodes differentiation under the integral sign via $$\partial _\theta \mathbb {E}_\theta [g]=\mathbb {E}_\theta [\Lambda _\theta g]$$ ∂ θ E θ [ g ] = E θ [ Λ θ g ] . Its commutator with multiplication by an estimator T satisfies $$\left[ \Lambda _\theta ,\textsf{M}_T \right] g=(\partial _\theta T)\,g$$ Λ θ , M T g = ( ∂ θ T ) g , clarifying precisely what is (and is not) captured by commutator terms: for standard estimators $$T=T(X)$$ T = T ( X ) with no explicit $$\theta $$ θ -dependence, the transport commutator vanishes, and efficiency/variance lower bounds arise from score-covariance geometry rather than from a nonzero commutator. This leads to a clean geometric Cramér–Rao inequality and an angle interpretation of efficiency in $$L^2(P_\theta )$$ L 2 ( P θ ) . We give explicit conditions under which the transport identity holds, treat the biased and parameter-dependent-support cases (so that the classical unbiased bound $$\textrm{Var}_\theta (T)\,I(\theta )\ge 1$$ Var θ ( T ) I ( θ ) ≥ 1 appears as a special case), connect the geometry to asymptotic efficiency of the MLE, and state the multiparameter (vector) version of all key identities. Our main applied contribution is a commutator-based, non-resampling criterion for selecting regularization strength in quadratic and generalized linear model (GLM) settings. A naive commutator between the “data influence” and “penalty influence” operators degenerates identically because the two influences sum to the identity. We therefore introduce two nontrivial commutator families, $$\left[ \textbf{A