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

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 Λ _θ g:= ∂ _θ g + g\,S_θ Λ θ g : = ∂ θ g + g S θ encodes differentiation under the integral sign via ∂ _θ E_θ [g]=E_θ [Λ _θ g] ∂ θ E θ [ g ] = E θ [ Λ θ g ] . Its commutator with multiplication by an estimator T satisfies [ Λ _θ ,MT ] g=(∂ _θ 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 θ 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 L²(P_θ ) 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 Var_θ (T)\,I(θ )≥ 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, $$[ {A