Novel Doubly Robust Estimator

• The paper introduces a novel estimator that remains consistent if either the treatment model or the outcome model is correctly specified.
• It employs nonparametric kernel smoothing with cross-validation to optimally balance bias and variance in estimating continuous dose–response curves.
• Simulation studies and real-world applications demonstrate that the estimator achieves low bias and competitive mean squared error compared to traditional methods.

A novel doubly robust estimator is a statistical method designed to estimate causal effects—such as average treatment effects or continuous treatment curves—while achieving consistency and valid inference if at least one of two nuisance models (typically, an outcome regression and a treatment model, or their analogs) is correctly specified. Recent advances in this area have produced estimators that merge nonparametric or machine learning-based fitting for nuisance functions with kernel or local smoothing, cross-fitting, and other modern adaptive techniques. These approaches are particularly valuable in scenarios featuring continuous or high-dimensional treatments, complex covariate relationships, or data-adaptive nuisance estimation. The seminal contribution of (Kennedy et al., 2015) is a nonparametric kernel-smoothing procedure for dose–response estimation that is doubly robust, nonparametric, theoretically tractable, and immediately compatible with flexible machine learning estimators of the nuisance functions.

1. Mathematical Formulation and Core Estimator

The central target is the continuous dose–response curve

$\theta(a) = E\{\, E[Y \mid L, A = a]\,\}.$

The method defines a pseudo–outcome transformation

$$\xi(Z;\pi,\mu) = \frac{Y - \mu(L, A)}{\pi(A \mid L)}\ \varpi(A) + m(A),$$

where

$$\varpi(a) = \int \pi(a \mid l) dP(l), \ \ m(a) = \int \mu(l,a) dP(l),$$

with $\mu$ an outcome regression, $\pi$ a conditional treatment density, and $Z = (L, A, Y)$. This transformation ensures that even if either $\pi$ or $\mu$ is inconsistently estimated, the conditional mean $E\{\xi(Z;\pi,\mu) \mid A = a\}$ equals $\theta(a)$.

The estimator of $\theta(a)$ is constructed via local linear kernel regression of the estimated pseudo-outcome $\hat\xi(Z;\hat\pi,\hat\mu)$ on $A$, specifically: $\hat\theta_h(a) = g_{h,a}(a)^\top\, \hat\beta_h(a)$ where $g_{h,a}(t) = \big(1, (t-a)/h\big)^\top$ encodes the local linear basis and $\hat\beta_h(a)$ is obtained by minimizing the kernel-weighted squared loss,

$$\hat\beta_h(a) = \arg\min_{\beta\in\mathbb{R}^2} \mathbb{P}_n \left[K_{h,a}(A) \left\{ \hat\xi(Z;\hat\pi,\hat\mu) - g_{h,a}(A)^\top \beta \right\}^2\right].$$

This construction is compatible with data-adaptive, high-dimensional, or nonparametric estimators for the nuisance functions $\pi$ and $\mu$, provided they admit adequate convergence rates.

2. Assumptions and Theoretical Guarantees

The estimator is built under the following key assumptions:

• Consistency and ignorability: Standard conditions for causal identification, with positivity requiring $\pi(a|l)\geq \pi_{min}>0$ across the support.
• Smoothness: The effect curve $\theta(a)$ must be twice continuously differentiable, and the marginal density $\varpi(a)$ and conditional density of the pseudo–outcome must be continuous in $a$.
• Complexity control: The functions $\pi$ and $\mu$, and their estimators, belong to function classes with bounded envelopes and finite entropy integrals. Inverse propensity scores are assumed uniformly bounded.

Under these and mild additional regularity conditions, the following properties hold:

• Consistency and Rate:

$$|\hat\theta_h(a) - \theta(a)| = O_p\left(\frac{1}{\sqrt{nh}} + h^2 + r_n(a)s_n(a)\right)$$

where $h$ is the bandwidth, $r_n(a)$ and $s_n(a)$ are local convergence rates for $\hat\pi$ and $\hat\mu$. If one nuisance estimator is consistent (rate $o(1)$), the product is $o_p(1)$, ensuring double robustness. With $h \sim n^{-1/5}$, the optimal nonparametric rate $n^{-2/5}$ is achieved.

• Asymptotic Normality:

$$\sqrt{nh} \Big\{\,\hat\theta_h(a) - \theta(a) + b_h(a) \,\Big\} \xrightarrow{d} N\Big(0, \ \frac{\sigma^2(a)\int K(u)^2\,du}{\varpi(a)} \Big)$$

where $$b_h(a) = (h^2 / 2)\, \theta''(a) \int u^2 K(u)\,du + o(h^2)$$ is the standard kernel bias, and $\sigma^2(a)$ derives from the conditional variance of the efficient influence function. If one nuisance estimator achieves $n^{-1/2}$ rate locally, the nonparametric term dominates and doubly robust inference and confidence intervals are valid.

3. Kernel Smoothing and Bandwidth Selection

The local linear kernel regression uses a symmetric kernel $K$ (e.g. Gaussian or Epanechnikov) and bandwidth $h$. To select $h$, the estimator adopts a leave–one–out cross-validation procedure, minimizing the error in predicting the pseudo–outcome: $$\hat h_{opt} = \arg\min_{h \in \mathcal{H}} \sum_{i=1}^n \left( \, \frac{\hat\xi(Z_i; \hat\pi, \hat\mu) - \hat\theta_h(A_i)}{1 - \hat W_h(A_i)} \,\right)^2$$ with $\hat W_h(A_i)$ the $i$th diagonal entry of the smoother’s hat matrix. This enables automatic calibration of the bias–variance tradeoff inherent to kernel smoothing, ensuring good empirical performance.

4. Advantages Compared to Traditional Estimators

Double Robustness: The estimator is consistent for $\theta(a)$ if either the treatment model $\pi$ or the outcome model $\mu$ is correct, but not necessarily both. This attribute is pivotal in practical scenarios with model uncertainty or high-dimensional confounders.

Nonparametric Flexibility: Unlike approaches imposing a specified functional form on $\theta(a)$, this estimator only assumes smoothness, enabling recovery of complex continuous effect curves (e.g., nonlinear dose–response relationships) that would be missed by parametric specifications.

Data-adaptivity: Nuisance functions can be estimated with highly flexible machine learning methods (e.g., Super Learner, random forests), provided their local convergence rates are sufficient, integrating recent advances in data-adaptive estimation with nonparametric effect curve estimation.

Bias-Variance Tradeoff: The main error term is governed by the product of local nuisance convergence rates, which can be negligible; the principal challenge thus becomes control of the standard nonparametric bias and variance, addressed through informed bandwidth tuning.

5. Simulation Studies and Real-World Application

Simulation Evidence: Comparisons of three estimators—a regression-based plug-in, an IPW estimator, and the proposed DR estimator—demonstrate that the DR method attains small bias across settings where either the treatment or outcome model is correct, whereas single-model-reliant estimators incur large bias under model misspecification. Precision (mean squared error) is competitive or superior to alternatives, including in small samples ($n=100$).

Applied Case Study: In an empirical analysis relating nurse staffing levels to hospital readmissions penalties (2,976 US hospitals), the estimator recovers a flexible effect curve. Estimated penalty rates remain constant below 5 hours of nurse time, then decline sharply, with the lowest penalty rates at staffing levels near 11 hours. These findings, enabled by the nonparametric flexibility and double robustness, elucidate critical regions of the dose–response curve, guiding informed intervention.

Estimator	Bias (good nuisance)	Bias (bad nuisance)	Variance (MSE)
Regression (plugin)	small	large	moderate
IPW	small	large	high (if $\pi$ near zero)
Doubly Robust	small	small	competitive

6. Implementation and Practical Considerations

Implementation Steps:

1. Estimate nuisance functions: Fit $\pi(a|l)$ and $\mu(l,a)$, e.g., using machine learning regression or classification.
2. Compute pseudo–outcomes: For each observation $Z_i$, calculate $\hat\xi(Z_i; \hat\pi, \hat\mu)$.
3. Kernel regression: Fit a local linear regression of $\hat\xi$ on $A$ via kernel smoothing at each $a$ of interest.
4. Bandwidth selection: Use leave–one–out cross-validation to choose an optimal $h$.

Computational Requirements: The estimator is scalable, as local kernel regressions are efficiently implemented and can rely on batching or vectorization in most scientific programming environments.

Limitations: The method, while robust to misspecification of either nuisance, requires at least one nuisance estimator to achieve a reasonable convergence rate near each $a$. In practice, extremely weak data support (for particular $a$, i.e., sparse regions of the treatment distribution) may limit finite-sample performance. Bandwidth or kernel mis-specification, and high variance of IPW terms (when propensity $\pi \approx 0$) may pose further challenges.

Extensions and Research Directions: The framework is naturally extendable to more complex longitudinal, multistate, or clustered data structures, as well as to semiparametric settings involving high-dimensional covariates or regularized estimation. Kernel-based DR estimators can be combined with cross-fitting and sample splitting for enhanced theoretical guarantees when using complex machine learning algorithms.

7. Summary

The novel doubly robust estimator for continuous treatment effects introduced in (Kennedy et al., 2015) provides a kernel-smoothing approach whereby a pseudo–outcome with the double robustness property is regressed nonparametrically on the treatment variable. The method achieves consistency and asymptotic normality if at least one preliminary model (treatment or outcome) is correctly specified, and supports the use of flexible, machine learning-based nuisances. Bandwidth selection is handled via cross-validation, and empirically, the method yields low bias and competitive mean squared error, outperforming estimators that rely solely on treatment or outcome modeling. Its construction marks a significant methodological advance, enabling robust analysis of causal dose–response relationships in observational studies without restrictive parametric constraints and accommodating modern data-adaptive regression tools.