Doubly Robust Confidence Intervals

• Doubly robust confidence intervals are statistical methods that combine propensity score and outcome regression models to ensure valid inference if at least one model is correctly specified.
• They use augmented inverse probability weighting and nonparametric smoothing to construct uniform confidence bands for function-valued targets like the conditional average treatment effect function.
• The approach avoids the curse of dimensionality by modeling high-dimensional covariates parametrically while focusing nonparametric estimation on low-dimensional effect modifiers.

Doubly robust confidence intervals are a class of statistical inference procedures designed to yield valid coverage for causal or semiparametric parameters such as the average treatment effect or the conditional average treatment effect function (CATEF), even when one of two underlying nuisance models—typically a model for the propensity score or a model for the outcome regression—is misspecified. These approaches achieve robustness against model misspecification for both point estimation and inferential uncertainty, with adaptations to high-dimensional, nonparametric, and function-valued (e.g., uniform band) settings. The development of doubly robust confidence intervals has facilitated valid inference in complex and high-dimensional applications while maintaining interpretability and theoretically justified error control.

1. Fundamental Principles of Doubly Robust Confidence Intervals

The doubly robust (DR) inference paradigm leverages two parallel nuisance models: a model for the propensity score $\pi(\mathbf{Z}) = P(D = 1 \mid \mathbf{Z})$ and a model for the conditional mean outcome $$m_j(\mathbf{Z}) = \mathbb{E}[Y \mid \mathbf{Z}, D = j]$$ for $j = 0,1$. A DR estimator combines these models, typically in an augmented inverse probability weighting (AIPW) form, to estimate a functional target $g(\mathbf{x})$ (e.g., the CATEF, $$g(\mathbf{x}) = \mathbb{E}[Y_{1} - Y_{0} \mid \mathbf{X} = \mathbf{x}]$$).

The essential property is that the estimator is consistent if at least one of the two models is correctly specified. If a parametric model for the propensity score or for the conditional mean is well specified, DR estimation produces asymptotically normal estimators, with confidence intervals constructed using variance estimates derived from efficient influence functions or empirical resampling. In function-valued settings, uniform confidence bands are constructed to ensure coverage over a range or interval of interest.

The local linear DR estimator for $g(\mathbf{x})$ is given by: $$\hat{g}(x) = e_1^\top \hat{\gamma}(x), \quad \hat{\gamma}(x) = \arg\min_{\gamma} \sum_{i} [\psi(\mathbf{W}_i, \hat\theta) - \gamma_0 - \gamma_1(x_i - x)]^2 K\left(\frac{x_i - x}{h_n}\right),$$ where $\psi(\mathbf{W}_i, \hat\theta)$ is the DR score, $K$ is a kernel function, and $h_n$ is a bandwidth parameter. The structure enforces that the influence of estimation error in the nuisance functions enters only at a second-order rate under regular conditions.

2. Construction of Uniform Confidence Bands

For function-valued targets such as the CATEF, uniform bands covering an interval $\mathcal{I}$ of the covariate of interest are constructed using a nonparametric regression of the DR score on $\mathbf{X}$. Pointwise standard errors are estimated at each $x$ as: $$SE(x) = \frac{\hat{s}(x)}{\sqrt{nh_n}}, \qquad \hat{s}(x) = \sqrt{ [n h_n \hat{f}_X(x)]^{-1} \int K^2(u) du \cdot \hat{\sigma}^2(x) },$$ where $\hat{f}_X(x)$ is a kernel density estimator and $\hat{\sigma}^2(x)$ estimates the conditional variance of the DR score at $x$.

The uniform band is constructed as: $$\hat{g}(x) \pm c_{1-\alpha} \cdot SE(x), \qquad \forall x \in \mathcal{I},$$ where $c_{1-\alpha}$ is a critical value derived from the supremum of a Gaussian process approximating the rescaled estimation error process: $$c_{1-\alpha} = \sqrt{a_n^2 - 2\log\{\log((1-\alpha)^{-1/2})\}},$$ with $a_n$ a function of bandwidth, domain length, and kernel specifics.

These uniform bands accommodate the sampling distribution's tail behavior, controlling for multiple comparisons over the continuum of $x$.

3. Avoidance of the Curse of Dimensionality

The proposed DR procedure avoids the curse of dimensionality by decomposing the set of covariates $$\mathbf{Z} = (\mathbf{X}^\top, \mathbf{V}^\top)^\top$$ into a (typically low-dimensional) set of interest $\mathbf{X}$ (for CATEF heterogeneity analysis) and a high-dimensional set $\mathbf{V}$ (for confounding adjustment). Nuisance functions $\pi(\mathbf{Z})$, $\mu_j(\mathbf{Z})$ are modeled parametrically to avoid high-dimensional smoothing, while the nonparametric estimation is confined to the low-dimensional space of $\mathbf{X}$.

Consequently, the nonparametric step inherits favorable convergence rates and mitigates sample size inefficiency, enabling feasible inference even when the number of adjustment covariates is large.

4. Empirical Applications and Monte Carlo Results

The DR uniform confidence band framework has been applied to assess heterogeneous treatment effects, notably in evaluating the effect of maternal smoking on birth weight. In such settings, $Y$ is birth weight, $D$ is an indicator of smoking, $\mathbf{Z}$ are extensive controls, and $\mathbf{X}$ (e.g., maternal age) is the covariate of interest for effect heterogeneity.

Empirical results for datasets (from Pennsylvania and North Carolina) display that the constructed confidence bands reveal statistically significant negative effects of smoking on birth weight in specific age intervals (e.g., 20–30 years). This supports the practical utility of the DR band for identifying regions of robust effect heterogeneity.

Monte Carlo simulations, conducted for $p = 10, 30$ and $n = 500, 2000$, compare scenarios of correct and incorrect model specification. When at least one model is correct, the DR estimator is approximately unbiased with small root-mean-squared errors and correct confidence band coverage (approaching 95% and 99%). Estimators relying solely on IPW or regression adjustment are more vulnerable to model misspecification. Compared to pointwise or Gumbel-based uniform bands, the proposed critical value yields bands closer to nominal coverage and avoids excessive conservatism.

5. Regularity Conditions and Limitations

The methodology is justified under several key assumptions:

• Unconfoundedness: $(Y_0, Y_1) \perp D \mid \mathbf{Z}$,
• CATEF smoothness: $g(\mathbf{x})$ is at least twice continuously differentiable over $\mathcal{I}$,
• Kernel function $K(\cdot)$ is symmetric, well-behaved, and differentiable,
• Bandwidth $h_n$ is chosen for undersmoothing (ensuring asymptotic bias is negligible),
• The dimension of $\mathbf{X}$ is less than 4.

Limitations include vulnerability to bias if both the propensity score and regression models are misspecified, and the coverage of the confidence band relies on asymptotic properties that may be affected by sample size and violation of technical conditions. The theoretical uniform coverage relies on advanced Gaussian process approximations and an undersmoothing bias regime.

6. Mathematical Formulation

The estimator’s core components are: $$\begin{aligned} \psi_1(\mathbf{W},\alpha_1, \beta) &= \frac{D Y}{\pi(\mathbf{Z},\beta)} - \frac{D - \pi(\mathbf{Z},\beta)}{\pi(\mathbf{Z},\beta)}\,\mu_1(\mathbf{Z},\alpha_1), \ \psi_0(\mathbf{W},\alpha_0, \beta) &= \frac{(1-D) Y}{1-\pi(\mathbf{Z},\beta)} + \frac{D - \pi(\mathbf{Z},\beta)}{1-\pi(\mathbf{Z},\beta)}\,\mu_0(\mathbf{Z},\alpha_0), \ \psi(\mathbf{W},\theta) &= \psi_1(\mathbf{W},\alpha_1,\beta) - \psi_0(\mathbf{W},\alpha_0,\beta), \ g(\mathbf{x}) &= \mathbb{E}\left[\psi(\mathbf{W},\theta_0) \mid \mathbf{X} = \mathbf{x}\right]. \end{aligned}$$

The uniform confidence band is

$$\hat{g}(x) \pm c_{1-\alpha} \cdot SE(x), \quad \forall x \in \mathcal{I},$$

where

$$SE(x) = \frac{\hat{s}(x)}{\sqrt{n h_n}}, \quad \hat{s}(x) = \sqrt{[n h_n \hat{f}_X(x)]^{-1}\int K^2(u) du \cdot \hat{\sigma}^2(x)},$$

and

$$c_{1-\alpha} = \sqrt{a_n^2 - 2\log\{\log((1-\alpha)^{-1/2})\}},$$

$$a_n = \left[2 \log(h_n^{-1}(b - a)) + 2 \log\left(\frac{\lambda^{1/2}}{2\pi}\right)\right]^{1/2}, \quad \lambda = -\frac{\int K(u)K''(u)du}{\int K^2(u) du}.$$

7. Implications and Extensions

The uniform doubly robust inference approach, by combining parametric adjustment for high-dimensional controls with nonparametric smoothing only over the low-dimensional effect modifiers, establishes a practical and theoretically principled route to uniform confidence bands for heterogeneous treatment effects. This paradigm is particularly valuable for observational studies where complex confounding adjustment is necessary, but the main scientific interest centers on interpretable, low-dimensional covariates.

The methodological innovation provides a template that can be extended to alternative functionals or to alternative semiparametric inference problems. Future work may focus on further relaxing modeling assumptions, bandwidth/data-driven tuning, and extensions to broader classes of treatment effect functionals or more challenging data regimes.