Collaborative targeted inference from continuously indexed nuisance parameter estimators
This work addresses a methodological challenge in statistical inference for researchers dealing with high-dimensional data or model misspecification, though it is incremental as it builds on existing TMLE frameworks.
The paper tackles the problem of constructing valid confidence intervals for parameters when nuisance parameter estimators converge too slowly, by proposing a collaborative targeted minimum loss estimator (C-TMLE) that adapts to an oracle tuning parameter, and demonstrates its effectiveness in simulations for average treatment effect estimation under various conditions.
We wish to infer the value of a parameter at a law from which we sample independent observations. The parameter is smooth and we can define two variation-independent features of the law, its $Q$- and $G$-components, such that estimating them consistently at a fast enough product of rates allows to build a confidence interval (CI) with a given asymptotic level from a plain targeted minimum loss estimator (TMLE). Say that the above product is not fast enough and the algorithm for the $G$-component is fine-tuned by a real-valued $h$. A plain TMLE with an $h$ chosen by cross-validation would typically not yield a CI. We construct a collaborative TMLE (C-TMLE) and show under mild conditions that, if there exists an oracle $h$ that makes a bulky remainder term asymptotically Gaussian, then the C-TMLE yields a CI. We illustrate our findings with the inference of the average treatment effect. We conduct a simulation study where the $G$-component is estimated by the LASSO and $h$ is the bound on the coefficients' norms. It sheds light on small sample properties, in the face of low- to high-dimensional baseline covariates, and possibly positivity violation.