Universal Least Favorable Models and Their Higher-Order Approximations in One-Step Targeted Maximum Likelihood Estimation

Music City Center 

Targeted Maximum Likelihood Estimation (TMLE) is a plug-in estimator for a target estimand that is asymptotically efficient under regularity conditions. A key component of TMLE is the targeting step, which maps an initial estimator $P_n^0$ of the true data distribution $P_0$ to an updated estimator $P_n^*$ that solves the score equation $P_nD(P_n^*) \approx 0$, where $D$ denotes the canonical gradient (also called the efficient influence curve, EIC). The targeting step is specified by choosing a least-favorable model (LFM) $\{P_t\}$ with an offset $P$--for example, the initial estimator of the data distribution--and performing iterative maximum-likelihood updates along this path until the EIC score equation is satisfied. A local LFM is a parametric model that spans a score at the offset ($t=0$) equal to the canonical gradient of the pathwise derivative of the target estimand. A local LFM often exists in closed form but is not unique, and it generally requires iterative MLE updates to ensure that the EIC score equation is solved to the desired precision.

Van der Laan \& Gruber (2016) introduced the universal least favorable model (ULFM), defined as the integral curve of the vector field $\partial_t\log p_t(x)=D(P_t)(x)$, whose score remains proportional to the EIC for all $t$. They observed that a single MLE update along this model exactly solves the EIC score equation, hence makes iterative targeting unnecessary, and makes this one-step TMLE a statistically robust choice. However, because explicit ULFMs are rarely available, the one-step TMLE can be computationally demanding--especially when evaluating the canonical gradient at a given measure $P$ is challenging: the one-step approach may require many such evaluations, whereas iterative TMLE typically requires only a few. Thus, there is a trade-off between the statistical performance of the ULFM-based one-step TMLE and the computational simplicity of the iterative TMLE based on an explicit local LFM. To navigate this trade-off, we construct a hierarchy of $k$th-order local paths whose Taylor expansions match that of the ULFM up through order $k$. A single targeting step along such a path cancels successive terms in the expansion of the first-order bias and yields $P_nD(P_n^*)=o_p(\|P_n^0-P_0\|^{k+1})$, and thereby sharpening the convergence rate of the efficient score compared to a TMLE based on a standard local LFM.

One contribution here is to supply--under mild $L^{2}$ continuity of the map $P\mapsto D(P)$--general existence and uniqueness results by formulating the ULFM PDE as an ODE in a Banach space, and to supply an explicit ODE representation for the $k$th-order local paths that directly underpins their numerical implementation. Pointwise error estimates show that these truncated paths approximate the ULFM to order $k+1$ for sufficiently small $t$. We illustrate this construction with closed-form examples for population moments, smooth density functionals, and the average treatment effect. Empirical studies with the quadratic functional $\Psi(P)=\int p^2(x) dx$ show that replacing a first-order local path with its second-order approximation reduces the EIC score error by at least a factor of the convergence rate of the initial estimator, producing noticeable finite-sample gains when the true density is sharply curved, even though the ultimate convergence rate remains governed by the von Mises remainder.

Finally, we analyze the limitations of $k$th-order approximations when the efficient influence curve $D(P)$ lacks smoothness in $P$--for example, in estimating the median--showing that our Banach-space existence and error bounds can break down for the truncated ODE. To address this, we introduce a simple smoothing of the parameter map that restores the mild $L^2$ continuity required for well-posedness of the $k$th-order local path. These results provide an explicit, implementable ODE formulation for one-step TMLE updates using higher-order local approximations of the ULFM, and make transparent exactly how the truncation order controls the first-order bias term in semiparametric inference.

Targeted Maximum Likelihood Estimation

Semiparametric Efficiency

Efficient Influence Function

Causal Inference

Perturbation Analysis

Convex Optimization 

Section on Nonparametric Statistics