On Convergence Rates of Convex Regression in Multiple Dimensions

References

• Allon G, Beenstock M, Hackman S, Passy U, Shapiro A (2007) Nonparametric estimation of concave production technologies by entropy. J. Appl. Econometrics 22:795–816.Crossref, Google Scholar
• Asmussen S, Glynn PW (2007) Stochastic Simulation: Algorithms and Analysis (Springer, New York).Crossref, Google Scholar
• Bronshtein EM (1976) ϵ–entropy of convex sets and functions. Siberian Math. J. 17:393–398.Crossref, Google Scholar
• Cleveland WS (1979) Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74:829–836.Crossref, Google Scholar
• Cule ML, Samworth RJ (2010) Theoretical properties of the log–concave maximum likelihood estimator of a multidimensional density. Electron. J. Statist. 4:254–270.Crossref, Google Scholar
• Dümbgen L, Samworth RJ, Schuhmacher D (2010) Approximation by log–concave distributions with applications to regression. Ann. Statist. 39:702–730.Crossref, Google Scholar
• Groeneboom P, Jongbloed G, Wellner JA (2001) Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29:1653–1698.Crossref, Google Scholar
• Hall P, Huang L (2001) Nonparametric kernel regression subject to monotonicity constraints. Ann. Statist. 29:624–647.Crossref, Google Scholar
• Hanson DL, Pledger G (1976) Consistency in concave regression. Ann. Statist. 4:1038–1050.Crossref, Google Scholar
• Hildreth C (1954) Point estimates of ordinates of concave functions. J. Amer. Statist. Assoc. 49:598–619.Crossref, Google Scholar
• Kuosmanen T (2008) Representation theorem for convex nonparametric least squares. Econometrics J. 11:308–325.Crossref, Google Scholar
• Kurdila AJ, Zabarankin M (2005) Convex Functional Analysis (Systems and Control: Foundations and Applications) (Birkhäuser, Switzerland).Google Scholar
• Lim E, Glynn PW (2012) Consistency of multidimensional convex regression. Oper. Res. 60:196–208.Link, Google Scholar
• Mammen E (1991) Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19:741–759.Crossref, Google Scholar
• Nadaraya EA (1964) On estimating regression. Theoret. Probab. Appl. 9:141–142.Crossref, Google Scholar
• Nesterov Y (2004) Introductory Lectures on Convex Optimization (Kluwer Academic Publishers, Boston).Crossref, Google Scholar
• Reinsch CH (1967) Smoothing by spline functions. Numerische Mathematik 10:177–183.Crossref, Google Scholar
• Rockafellar RT (1970) Convex Analysis (Princeton University Press, Princeton, NJ).Crossref, Google Scholar
• Seijo E, Sen B (2011) Nonparametric least squares estimation of a multivariate convex regression function. Ann. Statist. 39:1633–1657.Crossref, Google Scholar
• Shanthikumar JG, Yao DD (1991) Strong stochastic convexity: Closure properties and applications. J. Appl. Probab. 28:131–145.Crossref, Google Scholar
• Shorack GR, Wellner JA (1986) Empirical Processes with Applications to Statistics (John Wiley & Sons, New York).Google Scholar
• Stone CJ (1977) Consistent nanparametric regression. Ann. Statist. 5:595–645.Crossref, Google Scholar
• Stone CJ (1982) Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10:1040–1053.Crossref, Google Scholar
• Varian HR (1984) The nonparametric approach to production analysis. Econometrica 52:579–597.Crossref, Google Scholar
• Varian HR (1985) Nonparametric analysis of optimizing behavior with measurement error. J. Econometrics 30:445–458.Crossref, Google Scholar
• Watson GS (1964) Smooth regression analysis. Sankhyā Ser. A 26:359–372.Google Scholar
• Wegge LL (1974) Mean value theorem for convex functions. J. Math. Econom. 1:207–208.Crossref, Google Scholar
• Wolff RW, Wang C-L (2002) On the convexity of loss probabilities. J. Appl. Probab. 39:402–406.Crossref, Google Scholar
• Yatchew AJ, Bos L (1997) Nonparametric least squares regression and testing in economic models. J. Quant. Econom. 13:81–131.Google Scholar