On the uniform convergence of empirical norms and inner products, with application to causal inference

By Sara A. van de Geer. 2013

38 citations recorded by the source corpus.

Uniform convergence of empirical norms - empirical measures of squared functions - is a topic which has received considerable attention in the literature on empirical processes. The results are relevant as empirical norms occur due to symmetrization. They also play a prominent role in statistical applications. The contraction inequality has been a main tool but recently other approaches have shown to lead to better results in important cases. We present an overview including the linear (anisotropic) case, and give new results for inner products of functions. Our main application will be the estimation of the parental structure in a directed acyclic graph. As intermediate result we establish convergence of the least squares estimator when the model is wrong.

Subfields: Computational Mechanics, Statistics and Probability

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Citations in the corpus

CAM: Causal Additive Models, high-dimensional order search and penalized regression (2013, background): “Regarding the latter issue about order selection, a recent analysis in [37] extends our low-dimensional consistency result for the (nonrestricted) maximum likelihood estimator to the scenario where the number of variables can grow with sample size, in the best case essentially as fast as p = p(n) = ”
Doubly penalized estimation in additive regression with high-dimensional data (2019, background): “The spline result for m ≥ 1 follows easily from Proposition 1 in Mammen & van de Geer (1997). Consider the case m = 1.”
The Partial Linear Model in High Dimensions (2013, background): “For a further treatment, see van de Geer (2014).”
Penalized Estimation in Additive Regression with High-Dimensional Data (2017, background): “The following result can be obtained from Guedon et al. (2007) and, in its present form, van de Geer (2014), Theorem 2.1.”
Causal inference in partially linear structural equation models (2016, methodology): “Various estimation methods have been introduced for additive nonlinear SEMs to infer the underlying DAG [14, 18, 26].”