Optimal Rates of Convergence for Nonparametric Estimators

By C. Stone. 1980

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k-NN Regression Adapts to Local Intrinsic Dimension (2011, background): “Formally stated, the curse of dimension is the fact that for any nonparametric regressor there exists a distribution in R D such that, given a training size n , the regressor converges at a rate no better than n − 1 /O ( D ) (see e.g. [1, 2]).”
Escaping the Curse of Dimensionality with a Tree-based Regressor (2009, methodology): “This phenomenon is quantifiable in th e case of regression algorithms: as initially shown by Stone [ Sto80 , Sto82], if we only assume that the regression function f(x) is Lipschitz 1 in RD, then no non-parametric estimator can achieve a convergence rate faster thann−2/(2+D).”
A Moran coefficient-based mixed effects approach to investigate spatially varying relationships (2016, background): “56 Stone (1980) and Fan (1993) show that locally weighted regression, including 57 GWR, maximizes the rate of asymptotic convergence to a true function that is given by a 58 local linear smoother of y, and the smoothness of g(si, sj) is required to identify the true 59 function.”
Random Forests and Adaptive Nearest Neighbors (2006, background): “On the other hand, it is well known (Stone 1980) that the optimal convergence rate of MSE in regression problems is n−2m/(2m+d), where m is the order of smoothness of the regression function g.”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, background): “Note that an extra logarithmic factor appears in the L∞-rate in agreement with the corresponding minimax rate for the problem (see [32, 33]).”
Finite-Time Bounds for Fitted Value Iteration (2008, background): “Hence known lower bounds for regression must apply to RL, as well (see Stone, 1980, 1982 and Chapter 3 of Györfi, Kohler, Krzy˙ zak, and Walk, 2002 for such bounds).”
Simultaneous Confidence Bands and Hypothesis Testing in Varying‐coefficient Models (2000, background): “It was systematically studied by Stone (1977, 1980) and Cleveland (1979).”
Sharp Adaptive Estimation of Linear Functionals (2001, background): “…f at a fixed point of Rd, or, more generally, the estimation of some linear functional T f with values in R. Ibragimov and Hasminskii (1981, 1984), Stone (1980), Sacks and Ylvisaker (1981), Donoho and Liu (1991), Donoho and Low (1992), Donoho (1994b) obtained optimal rates of convergence and…”
Threshold Regression with Endogeneity (2017, unknown): “A popular loss function (e.g., Stone, 1980) is the 0–1 loss L (̂ θ, θ ) = 1 {⏐⏐̂θ − θ ⏐⏐ > ϵ 2 } for some fixed ϵ > 0, which will be used in this paper.9 Under this loss, R (̂ θ,P ) is the maximum probability that θ̂ is not in the ϵ/2 neighborhood of θ .”
Functional nonparametric model for time series: a fractal approach for dimension reduction (2002, background): “4.4) gives the optimal rate of convergence in the d-dimensional case given by Stone (1980) (resp.”
Nonparametric independent component analysis (2004, background): “The lower bound with the same rate n-s/(2s+l) is a trivial consequence of the one-dimensional lower bound (Stone 1980) applied successively to d subclasses of densities such that only one component density in the right-hand side of (1) is allowed to vary, all others being fixed.”
Estimation of a semiparametric transformation model (2008, methodology): “Stone (1980, 1982) and Ibragimov and Hasminskii (1980) showed that the optimal rate for estimating m is n−ℓ/(2ℓ+d), with ℓ a measure of the smoothness of m.”