Mathematical Foundations of Infinite-Dimensional Statistical Models

By Evarist Gin'e, Richard Nickl. 2015

745 citations recorded by the source corpus.

1. Nonparametric statistical models 2. Gaussian processes 3. Empirical processes 4. Function spaces and approximation theory 5. Linear nonparametric estimators 6. The minimax paradigm 7. Likelihood-based procedures 8. Adaptive inference.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7acb2f72bdfb505233fec1e9521d9a7d70ddcde4

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Citations in the corpus

Value-Aware Loss Function for Model-based Reinforcement Learning (2017, result): “For many examples of the metric entropy results, refer to [van de Geer, 2000; Györfi et al., 2002; Giné and Nickl, 2015].”
Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions (2015, background): “3 in [13] for the usual definitions and basic properties of Besov spaces.”
Distributed Statistical Estimation and Rates of Convergence in Normal Approximation (2017, background): “It follows from results in (Pinelis, 2016) (in particular, equation (5.5)) that, whenever conditions (1)-(5) hold, assumption 1 is satisﬁed with σ n = ( nI ( θ ∗ )) − 1 / 2 , where I ( θ ∗ ) is the Fisher information, and and the claim follows from Theorem 1.”
Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality (2018, background): “It is known that the minimax rate to estimate f o is lower bounded by n − 2 s/ (2 s + d ) , (Gin´e & Nickl, 2015).”
Iterative Value-Aware Model Learning (2018, result): “…the behaviour of the metric entropy is p log( 1 ε ) , which can be seen as having α → 0 with a certain rate, For many examples of the covering number and metric entropy results, refer to van de Geer [2000], Györﬁ et al. [2002], Zhou [2003], Steinwart and Christmann [2008], Giné and Nickl [2015].”
Empirical process results for exchangeable arrays (2019, methodology): “6 in Giné and Nickl (2015). This observation implies that for every r = 1, .”
Likelihood approximation with hierarchical matrices for large spatial datasets (2017, background): “Theorem 2.2.7 in Gine and Nickl (2015) provides estimates for a norm of a Gaussian vector, ∥Z∥, for instance for supt∈T |Z(t)|, Pr {⏐⏐⏐⏐sup t∈T |Z(t)| − M ⏐⏐⏐⏐ > u} ≤ 2(1 − Φ ( uσ )) ≤ e− u22σ2 , M > 0.”
Sequential estimation of quantiles with applications to A/B testing and best-arm identification (2019, methodology): “Our methods for deriving time-uniform, iterated-logarithm CDF and quantile bounds are closely related to the class of methods known as “chaining” in probability theory (Dudley, 1967; Talagrand, 2006; Giné and Nickl, 2015; Boucheron et al., 2013), and similar bounds can be derived using existing chai”
Hypothesis Testing For Densities and High-Dimensional Multinomials: Sharp Local Minimax Rates (2017, methodology): “This might seem counter-intuitive from the perspective of the upper bound since we create many low-probability cells which are likely to be empty in a small finite-sample, and indeed this construction is in some sense opposite to the quantile transformation suggested by previous work [12, 15].”
Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, methodology): “4 in [25] and arguing as in the proof of Theorem 2.”
Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression (2018, methodology): “Inequality (1) above can be found in Exercise 3.3.4(a) of Giné and Nickl (2016) and inequality (2) follows from Theorem 1.1 of Jameson (2015).”
Plugin estimation of smooth optimal transport maps (2021, methodology): “For concreteness, we describe these constructions in terms of the compactly-supported N -th Daubechies scaling and wavelet functions ζ0, ξ0 ∈ Cr(Rd), where r ≥ 0.18(N − 1) for an integer N ≥ 2 (Daubechies (1988); Giné and Nickl (2016), Theorem 4.2.10).”
Convergence rates of least squares regression estimators with heavy-tailed errors (2017, background): “Here, the reasons for our focus on the different structure—models with entropy conditions, are twofold: • Entropy is a standard and well-understood notion for the complexity of a large class of models; see examples in [23, 66].”
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “184) in [21] for the case when the Hölder functions are defined over [0,1]d, and this bound applies to our setting by a standard extension arguments (and regarding M,X as subsets of [0,1]d, with d= 2 in the former case).”
Current status linear regression (2016, methodology): “393 in [8], with m = 1, p = 2 and h n−1/5.”