Nonparametric Least Squares Estimation of a Multivariate Convex Regression Function

By E. Seijo, Bodhisattva Sen. 2010

175 citations recorded by the source corpus.

This paper deals with the consistency of the least squares estimator of a convex regression function when the predictor is multidimensional. We characterize and discuss the computation of such an estimator via the solution of certain quadratic and linear programs. Mild sufficient conditions for the consistency of this estimator and its subdifferentials in fixed and stochastic design regression settings are provided. We also consider a regression function which is known to be convex and componentwise nonincreasing and discuss the characterization, computation and consistency of its least squares estimator.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ebc8b5cffdbba02837863f0d1a86355c693f706d

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In Statistics and Probability

Citations in the corpus

Plugin estimation of smooth optimal transport maps (2021, background): “…optimization problem can be reduced to that of solving a finite-dimensional quadratic program, by a direct extension of well-known solvers for shape-constrained nonparametric regression with Lipschitz and convex constraints (cf. Seijo and Sen (2011), Mazumder et al. (2019), and references therein).”
Near-optimal max-affine estimators for convex regression (2015, background): “While early papers on convex regression explored the singlevariable case, recently attention shifted towards multivariate problems (see, e.g., Seijo and Sen, 2011; Hannah and Dunson, 2012).”
Nonparametric Shape-Restricted Regression (2017, background): “Thus the computation of the convex LSE θ̂ involves solving a quadratic program with n(d + 1) variables and n(n − 1) linear constraints; see [118] for the characterization, computation and consistency of the convex LSE where offthe-shelf interior point solvers (e.”
Covering Numbers for Convex Functions (2012, methodology): “We consequently use Dryanov’s ideas from [2, Theorem 3.1] to solve this quasi one-dimensional problem which allows us to complete the induction step.”
Multivariate convex regression with adaptive partitioning (2011, methodology): “The characterization (Kuosmanen 2008) and consistency (Seijo & Sen 2011, Lim & Glynn 2011) of the least squares problem have only recently been studied.”
Faithful Variable Screening for High-Dimensional Convex Regression (2014, background): “Guntuboyina and Sen (2013) analyze univariate convex regression and show surprisingly that the risk of the MLE is adaptive to the complexity of the true function.”
Generalized additive and index models with shape constraints (2014, methodology): “The SCMLE method takes roughly 1–2 s with a sample size of n = 1000 and 4–8 additive components, which is unsurprisingly slower than Tree or MARS, but it is typically faster than other shape-constrained methods such as SCAM and MCR. Note that MCR is particularly slow compared with the other methods ”