Nonparametric least squares estimation of a multivariate convex regression function

Seijo, Emilio; Sen, Bodhisattva

doi:10.1214/10-aos852

Cited by 143 publications

(125 citation statements)

References 34 publications

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“…In [3] consistency and further asymptotic results for the sample analogue of SCR are shown. Since the estimation of SR is very similar to the estimation of a convex function via the least squares approach, we think that the results in [52,34,25] can possibly be used to show consistency and maybe also further asymptotic results for the sample analogue estimator of SR under certain assumptions. The situation for the sharp marrow region is more difficult.…”

Section: Sample Analogues Of Identification Regionsmentioning

confidence: 98%

Statistical modeling under partial identification: Distinguishing three types of identification regions in regression analysis with interval data

Schollmeyer

Augustin

2015

International Journal of Approximate Reasoning

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One of the most promising applications of the methodology of imprecise probabilities in statistics is the reliable analysis of interval data (or more generally coarsened data). As soon as one refrains from making strong, often unjustified assumptions on the coarsening process, statistical models are naturally only partially identified and set-valued parameter estimators (identification regions) have to be derived. In this paper we consider linear regression analysis under interval data in the dependent variable. While in the traditional case of neglected imprecision different understandings of regression modeling lead to the same parameter estimators, we now have to distinguish between two different types of identification regions, called (Sharp) Marrow Region (SMR) and (Sharp) Collection Region (SCR) here. In addition, we propose the Set-loss Region (SR) as a compromise between SMR and SCR based on a set-domained loss function. We elaborate and discuss some fundamental properties of these regions and then illustrate the methodology in detail by an example, where the influence of different covariates on wine quality, measured by a coarse rating scale, is investigated. We also compare the different identification regions to classical estimates from a naive analysis and from common interval censorship modeling.

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Section: Sample Analogues Of Identification Regionsmentioning

confidence: 98%

Statistical modeling under partial identification: Distinguishing three types of identification regions in regression analysis with interval data

Schollmeyer

Augustin

2015

International Journal of Approximate Reasoning

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“…. , 0) ∈ G n ), closed and convex by Lemma 2.3 of Seijo and Sen (2011). Note that ϕ n is continuous and coercive (i.e., |ϕ n (g 1 , .…”

Section: Mathematical Frameworkmentioning

confidence: 99%

“…While statistical properties of the convex regression estimator are well-established (Hildreth 1954, Hanson and Pledger 1976, Wu 1982, Fraser and Massam 1989, Mammen 1991, Groeneboom, Jongbloed, and Wellner 2001, Turlach 2005, Birke and Dette 2007, Chang, Chien, Hsiung, C.-C.Wen, and Wu 2007, Meyer 2008, Kuosmanen 2008, Shively, Walker, and Damien 2011, Seijo and Sen 2011, Lim and Glynn 2012, Lim 2014, the convex regression estimator suffers from computational inefficiency. Minimization of ψ n over C can be formulated as a quadratic program (QP) with (d + 1)n decision variables and n 2 constraints (Kuosmanen 2008).…”

Section: Introductionmentioning

confidence: 99%

On a least absolute deviations estimator of a multivariate convex function

Lim¹,

Luo²

2014

Proceedings of the Winter Simulation Conference 2014

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When estimating a performance measure f * of a complex system from noisy data, the underlying function f * is often known to be convex. In this case, one often uses convexity to better estimate f * by fitting a convex function to data. The traditional way of fitting a convex function to data, which is done by computing a convex function minimizing the sum of squares, takes too long to compute. It also runs into an "out of memory" issue for large-scale datasets. In this paper, we propose a computationally efficient way of fitting a convex function by computing the best fit minimizing the sum of absolute deviations. The proposed least absolute deviations estimator can be computed more efficiently via a linear program than the traditional least squares estimator. We illustrate the efficiency of the proposed estimator through several examples.

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“…We begin by proving the existence and the uniqueness off c ,f m , andf p . The existence and the uniqueness off c is proven in Lemma 2.3 of Seijo & Sen (2011). To prove the existence off m , we note that Problem (4) is a minimization problem of a coercive function over a non-empty closed subset of R n .…”

Section: The Asymptotic Behavior Of the Test Statistics And The Propomentioning

confidence: 99%

Nonparametric Tests for Convexity/Monotonicity/Positivity of Multivariate Functions with Noisy Observations

Lim¹,

Tavarez²

2017

IJSP

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We propose a new method of testing for a function's convexity, monotonicity, or positivity, based on some noisy observations of the function made over a finite set T of points in the domain, where the observations can be made multiple times at each point in T . One of the traditional approaches to the test of a function's shape characteristic is to fit a convex, a monotone, or a positive function, depending on the shape characteristic we wish to test for, to the data set minimizing the sum of squared errors, and to compute the sum of squared differences (SSD) between the fit and the data set. While the traditional approach proceeds by observing the SSD as the number of points in T increases to infinity, we propose observing the SSD as r, the number of observations taken at each point in T , increases to infinity. This new way of observing the asymptotic behavior of the SSD leads to a test procedure that does not require the estimation of any additional parameters, and hence, is easy to implement. The proposed test procedure is proven to achieve a prescribed power as r → ∞. Numerical examples illustrate that the proposed test successfully detects the convexity/monotonicity/positivity of a function, as well as the non-convexity/non-monotonicity/non-positivity of a function.

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Nonparametric least squares estimation of a multivariate convex regression function

Cited by 143 publications

References 34 publications

Statistical modeling under partial identification: Distinguishing three types of identification regions in regression analysis with interval data

Statistical modeling under partial identification: Distinguishing three types of identification regions in regression analysis with interval data

On a least absolute deviations estimator of a multivariate convex function

Nonparametric Tests for Convexity/Monotonicity/Positivity of Multivariate Functions with Noisy Observations

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