Higher order influence functions and minimax estimation of nonlinear functionals

By James M. Robins, Lingling Li, E. T. Tchetgen Tchetgen, A. van der Vaart. 2008

265 citations recorded by the source corpus.

We present a theory of point and interval estimation for nonlinear functionals in parametric, semi-, and non-parametric models based on higher order influence functions (Robins (2004), Section 9; Li et al. (2004), Tchetgen et al. (2006), Robins et al. (2007)). Higher order influence functions are higher order U-statistics. Our theory extends the first order semiparametric theory of Bickel et al. (1993) and van der Vaart (1991) by incorporating the theory of higher order scores considered by Pfanzagl (1990), Small and McLeish (1994) and Lindsay and Waterman (1996). The theory reproduces many previous results, produces new non-$\sqrt{n}$ results, and opens up the ability to perform optimal non-$\sqrt{n}$ inference in complex high dimensional models. We present novel rate-optimal point and interval estimators for various functionals of central importance to biostatistics in settings in which estimation at the expected $\sqrt{n}$ rate is not possible, owing to the curse of dimensionality. We also show that our higher order influence functions have a multi-robustness property that extends the double robustness property of first order influence functions described by Robins and Rotnitzky (2001) and van der Laan and Robins (2003).

Subfields: Statistics and Probability

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Citations in the corpus

Deep Neural Networks for Estimation and Inference: Application to Causal Effects and Other Semiparametric Estimands (2018, methodology): “Furthermore, Theorem 3 can be generalized straightforwardly to provide inference that is uniformly valid over the classes data-generating processes for which our assumptions hold uniformly, following the approach of Romano (2004), exactly as in Belloni, Chernozhukov, and Hansen (2014) or Farrell (20”
Generalized random forests (2016, background): “A number of proposals have emerged for estimating the variance of a regression forest, including work by Sexton and Laake (2009), Mentch and Hooker (2016) and Wager et al.”
A unifying approach for doubly-robust $\ell_1$ regularized estimation of causal contrasts (2019, background): “This key result extends to our general class of parameters, a similar one by Robins et al. (2008) for parameters in their class.”
Double/debiased machine learning for treatment and structural parameters (2017, unknown): “Robins et al. (2008) and Robins et al. (2017) use sample splitting in the construction of higher-order influence function corrections in semiparametric estimation.”
Semiparametric Estimation and Inference Using Doubly Robust Moment Conditions (2013, methodology): “We view these results as an important contribution to a recent literature that aims at improving the accuracy of inference in semiparametric models (e.g. Robins et al., 2008; Cattaneo et al., 2013a,b).”
Cross-fitting and fast remainder rates for semiparametric estimation (2017, methodology): “The double robustness of the inﬂuence function for the functionals we consider is shown in Chernozhukov et al.(2016), where the doubly robust estimators of Scharfstein, Rotnitzky, and Robins (1999), Robins, Rotnitzky, and van der Laan (2000), Robins et. al.”
From Real‐World Patient Data to Individualized Treatment Effects Using Machine Learning: Current and Future Methods to Address Underlying Challenges (2020, methodology): “…evaluation, Alaa and van der Schaar 77 pro-pose a method for assessing causal inference models by using influence functions (a technique in robust statistics and efficiency theory 78,79 ) to estimate the loss of machine learning models for causal inference without requiring counterfactual data.”
Bias-Reduced Doubly Robust Estimation (2015, background): “James Robins, Andrea Rotnitzky, Eric Tchetgen Tchetgen, and a referee noted that Robins et al. (2008) use similar estimating equations like (3) and (4) in an intermediate simplifying step in the construction of higher order influence functions, but with a different objective.”
Stacked Generalization: An Introduction to Super Learning (2017, unknown): “This concern can be alleviated through the use of doubly robust estimators [34–36, 37, 38] or higher-order inﬂuence function based estimators [39, 40].”
Higher Order Inference On A Treatment Effect Under Low Regularity Conditions (2011, methodology): “1 of Robins et al. (2009b) that the width of the confidence set  τ : 0 ∈ Cmeff ,k(meff ) (τ )  for τ (θ) shrinks with increasing n at the same rate  1 n k(meff ) n meff−11/2 as does the confidence”
Limits of Estimating Heterogeneous Treatment Effects: Guidelines for Practical Algorithm Design (2018, methodology): “Thus, one can construct the “most” efﬁcient estimator of ( ^ T ) using the most efﬁcient inﬂuence function of ( ^ T ) as follows ( Robins et al., 2008; Robins, 2004): The derivation of the estimator above can be found in Theorem 9 in (Robins, 2004) and Section 5 in (Robins et al., 2008).”
Locally Robust Semiparametric Estimation (2016, result): “These results are distinguished from Newey, Hsieh, and Robins (1998, 2004) and Robins et al. (2008, 2009) in the conclusion of Theorem 2 and in providing, for any first step key regularity conditions used in the asymptotic theory of Section 6.”
MINIMAX ESTIMATION OF A FUNCTIONAL ON A STRUCTURED (2016, methodology): “The mathematical contributions of the paper are to provide a heuristic for constructing minimax estimators in semiparametric models, and to apply this to a concrete model, which is a template for a number of other models (see [23, 33]).”
Characterization of parameters with a mixed bias property (2019, methodology): “If the parameter χ (η) was in the class of Robins et al. (2008) for some functions a∗ (L) and b∗ (L) which are non-constant in L, then, there would exist statistics Sab, Sa, Sb and S0 such that a∗ (L) = − Eη (Sb|L) Eη (Sab|L) , b∗ (L) = −− Eη (Sa|L) Eη (Sab|L) and such that χ1η = Saba ∗ (L) b∗ (L)…”
How to make causal inferences using texts (2018, unknown): “Although we use specific models to facilitate discovery, STM for text as outcome and sIBP for text as treatment, the process we describe here is general to any method for discovering g from data.”