Consistency of Multidimensional Convex Regression

Eunji Lim () and Peter W. Glynn ()
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Eunji Lim: Department of Industrial Engineering, University of Miami, Coral Gables, Florida 33124
Peter W. Glynn: Department of Management Science and Engineering, Stanford University, Stanford, California 94305

Operations Research, 2012, vol. 60, issue 1, 196-208

Abstract: Convex regression is concerned with computing the best fit of a convex function to a data set of n observations in which the independent variable is (possibly) multidimensional. Such regression problems arise in operations research, economics, and other disciplines in which imposing a convexity constraint on the regression function is natural. This paper studies a least-squares estimator that is computable as the solution of a quadratic program and establishes that it converges almost surely to the “true” function as n (rightarrow) (infinity) under modest technical assumptions. In addition to this multidimensional consistency result, we identify the behavior of the estimator when the model is misspecified (so that the “true” function is nonconvex), and we extend the consistency result to settings in which the function must be both convex and nondecreasing (as is needed for consumer preference utility functions).

Keywords: nonparametric regression; multidimensional convex functions; asymptotic properties; consistency (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (17)

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