The Use of Polynomial Splines and Their Tensor Products in Multivariate Function Estimation

By C. Stone. 1994

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Double Reinforcement Learning for Efficient Off-Policy Evaluation in Markov Decision Processes (2019, background): “…for π̂b,(j) can be obtained from standard results for nonparametric regression, such as for kernel and sieve estimators (Newey and Mcfadden, 1994; Stone, 1994) or nonparametric estimators suited for high dimensions and non-smooth models (Bibaut and van der Laan, 2019a; Imaizumi and Fukumizu,…”
Noise Estimation Using Density Estimation for Self-Supervised Multimodal Learning (2020, methodology): “Existing density estimation algorithms can be divided into two categories: (a) parametric or semi-parametric approaches such as Gaussian Mixture models (McLachlan and Krishnan 2007; Wang and Wang 2015) and probabilistic graphical models (Koller and Friedman 2009); and (b) non-parametric approaches s”
Direct generalized additive modeling with penalized likelihood (1998, background): “Stone (1994) considers multidimensional splines where the knot placement problem becomes even larger.”
Spline-backfitted kernel smoothing of nonlinear additive autoregression model (2006, methodology): “ametric estimation and testing of interaction in additive models. Econometric Theory 18197–251. [24] Stone, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689–705. [25] Stone, C. J. (1994). The use of polynomial splines and their tensor products in multivariate fu”
Functional Coefficient Regression Models for Non‐linear Time Series: A Polynomial Spline Approach (2004, unknown): “There has been substantial recent development of asymptotic theory for the spline method for i.i.d. data (see e.g. Stone, 1994; Huang, 1998, 2001).”
POLYNOMIAL SPLINE CONFIDENCE BANDS FOR REGRESSION CURVES (2009, background): “The polynomial spline estimators, on the other hand, are global, see Stone (1985, 1994) and Huang (2003).”
DIMENSION REDUCTION BASED ON CANONICAL CORRELATION (2002, background): “In recent years, advances in multivariate function estimation (see, e.g., Stone (1994)), in neural nets (see, e.g., Ripley (1994)) and in tree-based regression and classification methods (see, e.g., Breiman, Friedman, Olshen and Stone (1984)) have made it possible to quantify a highly nonlinear…”
Projection estimation in multiple regression with application to functional ANOVA models (1998, methodology): “Stone 1994 studied the L 2 rates of convergence for functional ANOVA models in the settings of regression, generalized regression, density estimation and conditional density estimation, where univariate splines and their tensor products were used as building blocks for the approximating spaces.”
Visualization and exploration of high-dimensional functions using the functional anova decomposition (1996, background): “Stone (1994) describes a decomposition similar to the one given here, in whichthe components are orthogonal with respect to expectation.”
General empirical Bayes wavelet methods and exactly adaptive minimax estimation (2005, background): “…and asymptotic minimaxity and adaptivity have been commonly used to judge the overall performance of estimators; see comprehensive reviews in Stone (1994), Donoho, Johnstone, Kerkyacharian and Picard (1995) and Barron, Birgé and Massart (1999), and recent books by Efromovich (1999) and…”
Wavelet threshold estimation for additive regression models (2003, background): “In a series of studies, Stone (1985, 1986, 1994) proved that an additive component can be estimated by the one-dimensional optimal convergence rate under the assumption that all additive components have the same degree of “homogeneous” smoothness in Hölder space by using a linear estimator of the…”
ON DERIVATIVE ESTIMATION IN SPLINE REGRESSION (2000, background): “Stone (1985, 1994) has extended this result to additive and multivariate models, among other things.”
Institutions in International Relations: Understanding the Effects of the GATT and the WTO on World Trade (2007, background): “…that can be approximated using country and time fixed effects+ Moreover, Baier and Bergstrand forthcoming conclude that, for research on trade agreements, the best approach to dealing with endogenous unit effects is either fixed effects or differencing+ 30+ Stone 1994+ ht tp s: // do i.o rg /1 0.”
Penalized triograms: total variation regularization for bivariate smoothing (2004, methodology): “Knot selection strategies adapted from the regression spline methods of Stone (1994) and Stone et al. (1997) are proposed to control the degree of smoothness of the estimates.”
GENERALIZED ADDITIVE PARTIAL LINEAR MODELS WITH HIGH-DIMENSIONAL COVARIATES (2013, background): “…for nonparametric regression in the literature, including Andrews (1991), Andrews and Whang (1990), Chen (2007), Donald and Newey (1994), Huang (1998), Newey (1997), Stone (1986, 1994), and Xue and Yang (2006), many of which considered general series expansion where spline is a special case.”
Generalized Functional ANOVA Diagnostics for High-Dimensional Functions of Dependent Variables (2007, methodology): “For even moderate d , this is not feasible using the tensor-products of piecewise polynomials studied in Stone (1994).”