Semiparametric Minimax Rates

By James M. Robins, E. T. Tchetgen Tchetgen, Lingling Li, A. van der Vaart. 2009

64 citations recorded by the source corpus.

We consider the minimax rate of testing (or estimation) of non-linear functionals defined on semiparametric models. Existing methods appear not capable of determining a lower bound on the minimax rate of testing (or estimation) for certain functionals of interest. In particular, if the semiparametric model is indexed by several infinite-dimensional parameters. To cover these examples we extend the approach of [1], which is based on comparing a “true distribution” to a convex mixture of perturbed distributions to a comparison of two convex mixtures. The first mixture is obtained by perturbing a first parameter of the model, and the second by perturbing in addition a second parameter. We apply the new result to two examples of semiparametric functionals:the estimation of a mean response when response data are missing at random, and the estimation of an expected conditional covariance functional.

Subfields: Statistics and Probability

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Citations in the corpus

The hardness of conditional independence testing and the generalised covariance measure (2018, methodology): “Estimators of the expected conditional covariance functional requiring minimal possible smoothness conditions may be derived using the theory of higher order influence functions [42, 43, 34, 44]; these estimators are however significantly more complicated.”
Deep Neural Networks for Estimation and Inference: Application to Causal Effects and Other Semiparametric Estimands (2018, background): “For more on the results and limitations of Schmidt-Hieber (2019), see the published discussions (Ghorbani et al.”
Higher Order Inference On A Treatment Effect Under Low Regularity Conditions (2011, methodology): “In that case Robins et al. (2009b) proved that a lower bound for the minimax rate is n − 2(βb+βp)/d 1+2(βb+βp)/d ≫ n−1/2.”
Higher Order Tangent Spaces and Influence Functions (2014, background): “) is a restrictive requirement, whose validity is questionable for many applications. Higher order inﬂuence functions allow to construct better estimators than the linear estimator (1.4). As shown in [5, 9, 10, 12, 15], there are two cases: • (α+ β)/2 ≥ d/4. In this case estimation at rate n−1/2 is ”
Semiparametric Counterfactual Density Estimation (2021, unknown): “Estimation and inference for parameters like (9) is relatively well-understood [Robins et al., 2009, 2017], which allows exploiting existing theory and methods in the density estimation context.”
MINIMAX ESTIMATION OF A FUNCTIONAL ON A STRUCTURED (2016, background): “In [25], this rate is shown to be the fastest possible in the minimax sense, for the model in which a and b range over balls in C(Z) and C(Z), and g being known.”
Cross-fitting and fast remainder rates for semiparametric estimation (2017, methodology): “Consequently, this estimator will be root-n consistent and semiparametric eﬃcient when s γ + s α > r/ 2 and K is chosen appropriately, which is the minimal condition of Robins et al. (2009).”
Towards optimal doubly robust estimation of heterogeneous causal effects (2020, background): “…s/d is slower than the usual functional estimation rate n − 4 s/ (4 s + d ) , which is minimax optimal for the ATE if the co-variate density is smooth enough [Robins et al., 2009b], and also occurs for simpler functionals like the expected density [Bickel and Ritov, 1988, Birgé and Massart, 1995].”
The Balancing Act in Causal Inference (2021, background): “…rate, as well as in the range between these extremes in which both are to some degree smooth but neither estimable at n−1/4 rate; these asymptotics are called optimally double robust because it is known that these requirements cannot be improved (Mukherjee et al., 2017; J. Robins et al., 2009).”