Representation theorem for convex nonparametric least squares

By Timo Kuosmanen. Econometrics Journal, 2008

246 citations recorded by the source corpus.

Summary We examine a nonparametric least-squares regression model that endogenously selects the functional form of the regression function from the family of continuous, monotonic increasing and globally concave functions that can be nondifferentiable. We show that this family of functions can be characterized without a loss of generality by a subset of continuous, piece-wise linear functions whose intercept and slope coefficients are constrained to satisfy the required monotonicity and concavity conditions. This representation theorem is useful at least in three respects. First, it enables us to derive an explicit representation for the regression function, which can be used for assessing marginal properties and for the purposes of forecasting and ex post economic modelling. Second, it enables us to transform the infinite dimensional regression problem into a tractable quadratic programming (QP) form, which can be solved by standard QP algorithms and solver software. Importantly, the QP formulation applies to the general multiple regression setting. Third, an operational computational procedure enables us to apply bootstrap techniques to draw statistical inference.

Subfields: Economics and Econometrics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/465192b6318313843dffe517235af509b0ec49e2

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Citations in the corpus

Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints (2012, methodology): “As Kuosmanen (2008) emphasizes, the Afriat inequalities are the key to modeling the concavity axiom in the general multiple regression setting where there is no unambiguous way of sorting input vectors x.”
On estimating shadow prices of undesirable outputs with efficiency models: A literature review (2014, methodology): “Mekaroonreung and Johnson [51] applied the convex nonparametric least squares (CNLS) [60,61] and the stochastic nonparametric envelopment of data (StoNED) method [62], to compute the shadow prices of SO2 and NOx for U.S. power plants.”
Near-optimal max-affine estimators for convex regression (2015, background): “Recently, Lecué and Mendelson (2013); Mendelson (2014) also developed a new technique to deal with subgaussian noise (or with even weaker assumptions on the noise) on the price of making stronger assumptions on the function class F (e.”
What is the best practice for benchmark regulation of electricity distribution? Comparison of DEA, SFA and StoNED methods (2013, methodology): “The cost frontier model (1) can be estimated with convex nonparametric least squares (CNLS: Johnson and Kuosmanen, 2011, 2012; Kuosmanen, 2008).”
Semiconvex Regression for Metamodeling-Based Optimization (2014, methodology): “Traditional methods rely on fitting a least squares objective function subject to first or second order convexity constraints [40, 42, 2, 3, 62, 45]; these methods only scale to a few thousand observations due to the number of constraints.”
Nonparametric Shape-Restricted Regression (2017, background): “ing regression function f : Rd!R (d1) is known to be convex, i.e., for every u;v2Rd, f(u+ (1 )v) f(u) + (1 )f(v); for every 2(0;1): (5) Convexity appears naturally in many applications; see e.g., [73,86,38] and the references therein. The convexity of fconstrains to lie in a (polyhedral) convex set”
Nonparametric Least Squares Estimation of a Multivariate Convex Regression Function (2010, methodology): “The method of least squares has been applied to multidimensional concave regression in Kuosmanen (2008).”
Efficiency Analysis: A Primer on Recent Advances (2014, background): “11See Kuosmanen & Fosgerau (2009) for a test on the sign of the skewness of the OLS residuals.”