Doubly Robust Estimator — AIPW, Double Robustness, and EIF

Mathematical formulation

One notation system is used throughout: φ is the uncentered AIPW estimating signal, φ^c is that signal centered by its empirical mean, D^* is the true nonparametric efficient influence function, and IF^stack is the influence value of the finite-dimensional stacked M-estimator. These objects are related, but they are not interchangeable.

Graduate-level notation

O=(X,T,Y)∼P₀

Observed data. T∈{0,1} is the observed treatment; a∈{0,1} indexes a hypothetical intervention level.

e₀(x)=P₀(T=1|X=x)

True propensity score for treatment 1.

π_a,0(x)=P₀(T=a|X=x)

Treatment-level probability: π_a,0(x)=e₀(x)^a{1−e₀(x)}^1−a. Hence π_1,0=e₀ and π_0,0=1−e₀.

m_a,0(x)=E₀(Y|T=a,X=x)

True conditional outcome mean under observed treatment level a.

η₀=(e₀,m_0,0,m_1,0)

Collection of nuisance functions under the true observed-data law.

Pₙf=n⁻¹Σᵢf(Oᵢ)

Empirical average. The evaluation nuisance fit is η̂ for same-sample fitting and η̂^(−k(i)) for subject i under cross-fitting.

Causal target under consistency, exchangeability, and positivity

ψ_{a, 0} = E_{0} {Y (a)} = E_{0} {m_{a, 0} (X)}, Δ_{0} = ψ_{1, 0} - ψ_{0, 0}

AIPW estimating signal and estimator

\begin{matrix} π_{a} (x) \equiv P (T = a | X = x) = {e (x)}^{a} {1 - e (x)}^{1 - a} = a e (x) + (1 - a) {1 - e (x)}, a \in {0, 1} . \\ φ_{a} (O; η) = m_{a} (X) + \frac{𝟙 (T = a) {Y - m_{a} (X)}}{π_{a} (X)} . \\ {\hat{φ}}_{a, i} = \{\begin{cases} φ_{a} (O_{i}; \hat{η}), same-sample fitting, \\ φ_{a} (O_{i}; {\hat{η}}^{(- k (i))}), cross-fitting. \end{cases} \\ φ_{Δ} (O; η) \equiv φ_{1} (O; η) - φ_{0} (O; η), {\hat{ψ}}_{a} = P_{n} {\hat{φ}}_{a}, \hat{Δ} = P_{n} {\hat{φ}}_{Δ} . \end{matrix}

Reading rule: the entire indicator–residual product is in the numerator, and π_a(X) is the denominator. Thus π₁(X)=e(X) and π₀(X)=1−e(X); the signal requires π_a(X)>0 almost surely.

Treated-world signal

{\hat{φ}}_{1, i} = {\hat{m}}_{1} (X_{i}) + \frac{T_{i} {Y_{i} - {\hat{m}}_{1} (X_{i})}}{{\tilde{e}}_{i}}

The actual weighting score is ẽᵢ=Π_[ε,1−ε](êᵢ). With ε=0, ẽᵢ=êᵢ apart from a machine-precision floor.

Control-world signal

{\hat{φ}}_{0, i} = {\hat{m}}_{0} (X_{i}) + \frac{(1 - T_{i}) {Y_{i} - {\hat{m}}_{0} (X_{i})}}{1 - {\tilde{e}}_{i}}

The same empirical distribution of X is used for both treatment worlds.

ATE signal and estimator

\begin{matrix} {\hat{φ}}_{Δ, i} = {\hat{φ}}_{1, i} - {\hat{φ}}_{0, i}, \\ \hat{Δ} = \frac{1}{n} \sum_{i = 1}^{n} {\hat{φ}}_{Δ, i} . \end{matrix}

φ̂_Δ,i is an uncentered AIPW signal, sometimes called an AIPW pseudo-outcome. It is not itself the EIF.

Notation rule used everywhere below: φ denotes the uncentered AIPW signal; φ^c=φ−Pₙφ denotes its empirically centered version; D^* is reserved for the true nonparametric efficient influence function; IF^stack denotes the influence value from the finite-dimensional stacked estimating equations.

Nuisance evaluation rule: for same-sample fitting, φ̂_a,i=φ_a(O_i;η̂). Under K-fold cross-fitting, if i∈I_k, then φ̂_a,i=φ_a(O_i;η̂^(−k)). Thus the observation-specific evaluation fit is not treated as a new population nuisance function.

Exact population bias identity for fixed η=(e,m₀,m₁) with 0<e(X)<1 and integrable terms

E_{0} {φ_{Δ} (O; η)} - Δ_{0} = R_{2} (η, η_{0}) = E_{0} [{e (X) - e_{0} (X)} {\frac{m_{1} (X) - m_{1, 0} (X)}{e (X)} + \frac{m_{0} (X) - m_{0, 0} (X)}{1 - e (X)}}] .

This formula uses the same φ_Δ defined in the estimator tab. It gives the bias of the uncentered AIPW signal relative to the true ATE. The displayed sign convention is therefore fixed and unambiguous.

Treatment-level conditional bias identity

E_{0} {φ_{a} (O; η) | X} - m_{a, 0} (X) = \frac{π_{a} (X) - π_{a, 0} (X)}{π_{a} (X)} {m_{a} (X) - m_{a, 0} (X)} .

This identity is the direct algebraic check: the conditional bias is zero if either π_a=π_a,0 or m_a=m_a,0.

Propensity nuisance correct

e = e_{0} \Rightarrow R_{2} = 0

The outcome regressions may be misspecified, provided the untruncated propensity score is correct and positivity holds.

Outcome nuisances correct

m_{a} = m_{a, 0} for a=0,1 \Rightarrow R_{2} = 0

Then each weighted residual has conditional mean zero for any positive weighting score.

Both nuisance components wrong

e \neq e_{0}, m_{a} \neq m_{a, 0} \Rightarrow R_{2} need not vanish

Double robustness is a union-model property, not protection against simultaneous misspecification.

Product-error bound

| R_{2} | \leq K_{ε} {∥ e - e_{0} ∥}_{P_{0}, 2} ({∥ m_{1} - m_{1, 0} ∥}_{P_{0}, 2} + {∥ m_{0} - m_{0, 0} ∥}_{P_{0}, 2})

If ε≤e(X)≤1−ε, K_ε may be taken proportional to 1/ε. The errors enter as a product. Here ‖h‖_P₀,2={E₀[h(O)²]}^1/2.

Truncation distinction: the theorem above applies to the actual function e used inside φ_Δ. If a correct fitted ê is replaced by a nonvanishing clipped ẽ≠e₀ on a set of positive probability, the propensity-only branch is no longer exact. The outcome-correct branch remains valid.

Simulation labels are not diagnostics: “correct” and “misspecified” are known only because the data-generating law is programmed. They cannot be verified from one observational dataset.

1. Uncentered AIPW signal

φ_{Δ} (O; η) = φ_{1} (O; η) - φ_{0} (O; η)

Its mean estimates the ATE. It is not mean zero and therefore is not an influence function.

2. Nonparametric EIF

D_{Δ}^{*} (O; P_{0}) = φ_{Δ} (O; η_{0}) - Δ_{0}

This is mean zero under P₀ and attains the nonparametric efficiency bound.

3. Stacked-estimator IF

{\hat{IF}}_{Δ, i}^{stack} = - c^{T} {\hat{A}}^{- 1} U_{i} (\hat{θ})

This is the dashboard’s finite-dimensional M-estimation influence value. It equals the EIF asymptotically only under the corresponding efficiency conditions.

Efficient influence function: compact and expanded forms

\begin{array}{l} D_{a}^{*} (O; P_{0}) = m_{a, 0} (X) + \frac{𝟙 (T = a) {Y - m_{a, 0} (X)}}{π_{a, 0} (X)} - ψ_{a, 0}, a \in {0, 1} . \\ D_{Δ}^{*} (O; P_{0}) = D_{1}^{*} (O; P_{0}) - D_{0}^{*} (O; P_{0}) . \\ D_{Δ}^{*} (O; P_{0}) = m_{1, 0} (X) - m_{0, 0} (X) - Δ_{0} \\ + \frac{T}{e_{0} (X)} {Y - m_{1, 0} (X)} \\ - \frac{1 - T}{1 - e_{0} (X)} {Y - m_{0, 0} (X)} . \end{array}

Critical reading check: the treated residual is multiplied by T/e₀(X), not by T·e₀(X). Likewise, the control residual is multiplied by (1−T)/{1−e₀(X)}. These are the two inverse-probability residual corrections in the nonparametric EIF.

Empirically centered AIPW signal

{\hat{φ}}_{Δ, i}^{c} \equiv {\hat{φ}}_{Δ, i} - \hat{Δ}, P_{n} {\hat{φ}}_{Δ}^{c} = 0 .

φ̂^c_Δ,i is exactly centered in the sample. It is an empirical surrogate for D_Δ^* only when the nuisance estimates and the target estimate converge to their true counterparts.

EIF-style variance from the centered signal

\hat{SE} (\hat{Δ}) = \sqrt{\frac{\sum_{i = 1}^{n} {\hat{φ}}_{Δ, i}^{c}^{2}}{n (n - 1)}} .

No additional centering term is needed because Pₙφ̂^c_Δ=0 exactly. This estimates the EIF variance only when φ̂^c_Δ,i consistently approximates D_Δ^*.

Stacked sandwich actually used

\begin{matrix} \hat{A} = P_{n} \frac{\partial U (O; \hat{θ})}{\partial θ^{T}}, \hat{B} = P_{n} {U U^{T}}, \\ \hat{Var} (\hat{Δ}) = \frac{1}{n} c^{T} {\hat{A}}^{- 1} \hat{B} {\hat{A}}^{- T} c . \end{matrix}

Here θ stacks the nuisance-model coefficients with ψ₁ and ψ₀, and c extracts ψ₁−ψ₀.

Crucial distinction: under one-model misspecification, the centered signal φ̂^c_Δ,i=φ̂_Δ,i−Δ̂ is generally not the full influence function of the stacked working-model estimator. The dashboard therefore calculates and displays IF̂^stack_Δ,i separately.

Efficiency statement: when both nuisance functions are consistently estimated at suitable rates, no substantive truncation is active, and regularity conditions hold, IF̂^stack and the centered signal φ̂^c_Δ both approach the nonparametric EIF. The sign convention is treatment minus control throughout: +T/e₀(X) for the treated residual and −(1−T)/{1−e₀(X)} for the control residual.

Shared-fold nuisance estimates

\begin{matrix} {\hat{η}}^{(- k)} is fitted on I_{k}^{c}, \\ {\hat{φ}}_{Δ, i} = φ_{Δ} (O_{i}; {\hat{η}}^{(- k)}) for i \in I_{k} . \end{matrix}

The propensity and both outcome-regression predictions use the same fold partition.

Cross-fitted AIPW estimator

\hat{Δ} = \sum_{k = 1}^{K} \frac{n_{k}}{n} P_{n, k} {\hat{φ}}_{Δ}

Each observation is evaluated by nuisance fits that excluded its fold.

Product-rate condition for convergence to the nonparametric EIF

{∥ \tilde{e} - e_{0} ∥}_{P_{0}, 2} ({∥ {\hat{m}}_{1} - m_{1, 0} ∥}_{P_{0}, 2} + {∥ {\hat{m}}_{0} - m_{0, 0} ∥}_{P_{0}, 2}) = o_{p} (n^{- 1 / 2}) .

Here ẽ is the treatment probability actually used in the AIPW denominator (equal to ê when no substantive clipping is active). Together with L₂ consistency of ẽ and both outcome regressions, this product-rate condition yields convergence to the efficient influence function. Product smallness alone is not enough if one nuisance converges to a wrong limit.

Identification assumptions

Consistency: Y=Y(T).
Conditional exchangeability: {Y(1),Y(0)} ⟂ T | X.
Positivity: 0<e₀(X)<1 almost surely.
Regularity: finite variance and stable nuisance estimates.

Point consistency versus efficient inference: one correct nuisance can make R₂ vanish, but convergence to the nonparametric EIF generally requires both nuisance functions to converge. In the finite-dimensional union model used here, the block-stacked sandwich accounts for first-order nuisance-estimation terms when one working model is misspecified.

Cross-fitting does not repair identification. It reduces own-observation overfitting and empirical-process restrictions, but it cannot fix unmeasured confounding or structural positivity failure.

6. The double-robustness matrix

Click any cell to refit the dashboard. Green cells satisfy the theoretical union-model condition under the displayed, untruncated propensity score; amber indicates that nonvanishing truncation compromises propensity-only protection.

One cohort can still be noisy. The repeated-sampling experiment below distinguishes sampling variation from systematic misspecification bias.

Inspect subject-level nuisance predictions and AIPW contributions

Subject	Age	BMI	Income	T	Y	ê raw	ẽ used	m̂0	m̂1	OR contrast	T correction	Control correction	φ̂Δ,i signal	φ̂ᶜΔ,i=φ̂Δ,i−Δ̂	IF̂stack

Doubly Robust Estimator — Se Yoon Lee, Ph.D.