Characterization of parameters with a mixed bias property

By Andrea Rotnitzky, Ezequiel Smucler, James M. Robins. 2019

81 citations recorded by the source corpus.

We study a class of parameters with the so-called mixed bias property. For parameters with this property, the bias of the semiparametric efficient one-step estimator is equal to the mean of the product of the estimation errors of two nuisance functions. In nonparametric models, parameters with the mixed bias property admit so-called rate doubly robust estimators, i.e., estimators that are consistent and asymptotically normal when one succeeds in estimating both nuisance functions at sufficiently fast rates, with the possibility of trading off slower rates of convergence for the estimator of one of the nuisance functions against faster rates for the estimator of the other nuisance function. We show that the class of parameters with the mixed bias property strictly includes two recently studied classes of parameters which, in turn, include many parameters of interest in causal inference. We characterize the form of parameters with the mixed bias property and of their influence functions. Furthermore, we derive two functional loss functions, each being minimized at one of the two nuisance functions. These loss functions can be used to derive loss-based penalized estimators of the nuisance functions.

Subfields: Statistics and Probability

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Citations in the corpus

Efficiently Breaking the Curse of Horizon in Off-Policy Evaluation with Double Reinforcement Learning (2019, background): “Because of the mixing bias property (Rotnitzky et al., 2019) of the inﬂuence function, the rate is multiplicative in the two estimators’ convergence rate.”
Double Reinforcement Learning for Efficient Off-Policy Evaluation in Markov Decision Processes (2019, background): “In addition to efficiency, DRL enjoys a double robustness guarantee (Rotnitzky and Vansteelandt, 2014; Rotnitzky et al., 2019).”
Semiparametric doubly robust targeted double machine learning: a review (2022, background): “First, every functional has a somewhat unique remainder term R 2 in the expansion (5), leading to corresponding unique bias term T 2 in the decomposition (10) (though see Rotnitzky et al. [2019] for some unifying properties).”
A unifying approach for doubly-robust $\ell_1$ regularized estimation of causal contrasts (2019, methodology): “In what follows we ignore regularity conditions, see Rotnitzky et al. (2019) for the precise conditions.”
Efficiently Breaking the Curse of Horizon: Double Reinforcement Learning in Infinite-Horizon Processes (2019, background): “This kind of double robustness is sometimes called model double robustness (Smucler et al., 2019) because the rates needed essentially correspond to parametric estimation and therefore the conditions refer to whether these parametric models are well-speciﬁed.”