Estimation.Orthogonal­Moments.Auto­Debias

For a regression-nuisance moment system, a regression nuisance function, a candidate representer on the covariate space, a scalar target value, and an observation, the automatically debiased score is the baseline moment contribution plus the candidate representer evaluated at the observation's covariates times the observed-outcome residual from the regression target.

For a regression-nuisance moment system, a regression nuisance function, a candidate representer on the covariate space, and a scalar target value, the automatically debiased population moment is the expectation of the automatically debiased score under the system's observation distribution.

For a regression-nuisance moment system and a candidate regression function in its nuisance class, the general Riesz loss is the covariate-distribution mean of the squared regression target of that function minus twice the directional derivative of the population moment in that function's direction.

For a regression-nuisance moment system, a sample-indexed auxiliary sample space, observation-valued sample data, sample-indexed target estimates, sample-indexed nuisance estimates, finite-difference scales, a candidate nuisance-direction function, sample-index sets, a sample size, and a realized auxiliary sample point, the finite-difference representer loss is the empirical second moment of the candidate's regression target on the specified index set minus twice the centered finite-difference approximation to the moment's nuisance derivative at the corresponding estimates.

For a linear regression-functional system, the population linear functional maps each regression function in that system's admissible class to the integral of its linear moment function under the system's observation measure.

For a linear regression-functional system, a regression function in its admissible class, a candidate pairing function on the covariate space, a scalar target value, and an observation, the observation-level pairing score adds the observed linear functional to the pairing-weighted regression residual and subtracts the target.

For a linear regression-functional system, its mean-pairing representation, and an observation, the full automatic-debiasing influence function is the observed target fluctuation m(W;g₀)-θ₀ plus the representer-weighted regression residual α₀(X)(Y-g₀(X)).

For a linear regression-functional system and a candidate regression function in its admissible class, the linear Riesz loss is the covariate-distribution mean of the squared regression target of that function minus twice the system's population linear functional evaluated at it.

For a linear regression-functional system, sample data, a candidate representer index, training indices, a sample-size index, and a sample realization, the empirical linear Riesz loss averages the candidate's squared value minus twice its observed linear functional over the training indices.

Given a linear regression-functional system, sample data, a candidate class, training indices, and a sample-indexed fitted representer, the empirical-Riesz-minimizer property says that each fit belongs to the class and has no larger empirical Riesz loss than any other class member.

For a linear regression-function system, the joint nuisance object for feasible automatic debiasing pairs a regression-function index with a candidate pairing function on the covariate space.

Given an auxiliary sample-space measure, a linear regression-functional system, its mean-pairing representation, a nonnegative nuisance-neighborhood radius, and measurability of every observation-level score, the affine automatic-debiasing moment has coefficient minus one and observed constant term m(W;γ) + α(X)(Y-γ(X)).

Under the same inputs as the affine automatic-debiasing moment, the corresponding general moment forgets only the affine decomposition.

For a linear regression-functional system, its mean-pairing representation, and a joint nuisance candidate, the integrability and representer-measurability package used by the bilinear remainder contains the finite population moments needed to compare observed and population scores; the regression equation then supplies residual orthogonality.

Given a linear regression-functional system, its mean-pairing representation, and score regularity inputs, an i.i.d. sample, a K-fold split, and fold-specific regression fits and Riesz fits, the feasible cross-fitted automatic-debiasing estimator solves the fold-averaged empirical score equation.

For a finite measurable covariate space, the ATE regression nuisance type consists of the two unrestricted real regression functions indexed by the treatment arm.

For a back-door ATE system and an observed covariate--arm pair, the signed inverse-propensity Riesz weight is 1/e(X) on the treated arm and -1/(1-e(X)) on the control arm.

Given a finite-covariate back-door estimation system satisfying the standard identification assumptions, the ATE regression-nuisance moment system has covariate object (X,D), regression target g(D,X), population moment ∫(g(1,x)-g(0,x))dP_X-θ, and its observation-level counterpart.

Given a finite-covariate back-door estimation system satisfying the standard identification assumptions, the ATE linear regression-functional system reuses the observation law, regression target, pushforward identity, and residual equation of ateRegNuisanceMomentSys, with observation-level linear functional g(1,X)-g(0,X).

Given a finite-covariate back-door system, the back-door assumptions, and pointwise strict overlap, the ATE automatic-debiasing mean representation uses the signed inverse-propensity weight on (X,D).

For a finite-covariate back-door system, under the back-door assumptions and strict overlap at the positive level ε, the signed inverse-propensity weight gives a mean-pairing representation for the ATE regression-functional system.

For a regression-nuisance moment system, the joint nuisance type used by the general automatic-debiasing DML bridge pairs a regression nuisance with a candidate representer on the covariate space.

Given an auxiliary sampling measure, a regression-nuisance moment system, its automatic-debiasing representer, a nonnegative neighborhood radius, measurability of every debiased score, an affine decomposition of the original score, and a nonzero population mean of the truth coefficient, the associated linear moment uses the joint nuisance (g, α), the automatically debiased observation-level score, and L² regression-target and representer error gauges.