A Computational Framework for Multivariate Convex Regression and its Variants

Mazumder, Rahul; Choudhury, Arkopal; Iyengar, Garud; Sen, Bodhisattva

doi:10.48550/arxiv.1509.08165

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…In this last subsection, we return to the least-squares (LS) piecewise affine regression problem that has motivated this study. Being a generalization of the classic LS linear regression problem, the least-squares piecewise affine regression problem is to find, given data points {x , y } N =1 ∈ R d+1 , a LS estimator of a continuous piecewise affine function [9,23,3,16]. Since every piecewise affine function can be written in the form (1), we may formulate this regression problem as…”

Section: Least-squares Piecewise Affine Regressionmentioning

confidence: 99%

On the Finite Number of Directional Stationary Values of Piecewise Programs

Cui¹,

Pang²

2018

Preprint

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Extending a fundamental result for (indefinite) quadratic programs, this paper shows that certain non-convex piecewise programs have only a finite number of directional stationary values, and thus, possess only finitely many locally minimum values. We present various special cases of our main results, in particular, an application to a least-squares piecewise affine regression problem for which every directional stationary point is locally minimizing.

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Section: Least-squares Piecewise Affine Regressionmentioning

confidence: 99%

On the Finite Number of Directional Stationary Values of Piecewise Programs

Cui¹,

Pang²

2018

Preprint

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“…As the density of the evaluation points increases, the estimated function potentially has more hyperplane components and is more flexible; however, the computation time typically increases. If a smooth functional estimate is preferred, see Nesterov (2005) and Mazumder et al (2015), where methods for smoothing are provided. In practice, we propose to select the bandwidth vector h via the leave-one-out cross-validation based on the unconstrained estimator.…”

Section: Shape Constrained Kernel-weighted Least Squares (Sckls) With...mentioning

confidence: 99%

Shape-Constrained Kernel-Weighted Least Squares: Estimating Production Functions for Chilean Manufacturing Industries

Yagi

Chen

Johnson

et al. 2018

Journal of Business & Economic Statistics

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In this paper we examine a novel way of imposing shape constraints on a local polynomial kernel estimator. The proposed approach is referred to as Shape Constrained Kernel-weighted Least Squares (SCKLS). We prove uniform consistency of the SCKLS estimator with monotonicity and convexity/concavity constraints and establish its convergence rate. In addition, we propose a test to validate whether shape constraints are correctly specified. The competitiveness of SCKLS is shown in a comprehensive simulation study. Finally, we analyze Chilean manufacturing data using the SCKLS estimator and quantify production in the plastics and wood industries. The results show that exporting firms have significantly higher productivity.Nonparametric regression methods, such as the local linear (LL) estimator, avoid functional form misspecification. To model production with a production or a cost function, the flexible nature of nonparametric methods can cause difficulties in interpreting the results.Fortunately, microeconomic theory provides additional structure in the form of shape constraints. Recently several nonparametric shape constrained estimators have been proposed that combine the advantage of avoiding parametric functional specification with improved small sample performance relative to unconstrained nonparametric estimators. Nevertheless, the existing methods have limitations regarding either estimation performance or computational feasibility. In this paper, we propose a new estimator that imposes shape restrictions on local kernel weighting methods. By combining local averaging with shape constrained estimation, we improve finite sample performance by avoiding overfitting.Work on shape-constrained regression first started in the 1950s with Hildreth (1954), who studied the univariate regressor case with a least squares objective subject to monotonicity and concavity/convexity constraints. See also Brunk (1955) and Grenander (1956) for alternative shape constrained estimators. Under the concavity/convexity constraint, properties such as consistency, rate of convergence, and asymptotic distribution have been shown by Hanson and Pledger (1976), Mammen (1991), and Groeneboom et al. (2001, respectively. In the multivariate case, Kuosmanen (2008) developed the characterization of the least squares estimator subject to concavity/convexity and monotonicity constraints, which we will refer to as Convex Nonparametric Least Squares (CNLS) throughout this paper. Furthermore, consistency of the least squares estimator was shown independently by Seijo andSen (2011) andLim andGlynn (2012).Regarding the nonparametric estimation implemented using kernel based methods, Birke and Dette (2007), Carroll et al. (2011), and Hall and Huang (2001) investigated the univariate case and proposed smooth estimators that can impose derivative-based constraints including monotonicity and concavity/convexity. Du et al. (2013) proposed Constrained Weighted Bootstrap (CWB) by generalizing Hall and Huang's method to the multivariate regression setting. Ber...

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A Computational Framework for Multivariate Convex Regression and its Variants

Cited by 2 publications

References 26 publications

On the Finite Number of Directional Stationary Values of Piecewise Programs

On the Finite Number of Directional Stationary Values of Piecewise Programs

Shape-Constrained Kernel-Weighted Least Squares: Estimating Production Functions for Chilean Manufacturing Industries

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