Consistency of Multidimensional Convex Regression

References

• Allon G., Beenstock M., Hackman S., Passy U., Shapiro A. Nonparametric estimation of concave production technologies by entropy. J. Appl. Econometrics (2007) 22:795–816Crossref, Google Scholar
• Avriel M.Nonlinear Programming: Analysis and Methods (1976) (Prentice-Hall, Englewood Cliffs, NJ) Google Scholar
• Barlow R. E., Brunk H. D. The isotonic regression problem and its dual. J. Amer. Statist. Assoc. (1972) 67(337):140–147Crossref, Google Scholar
• Boyd S., Vandenberghe L.Convex Optimization (2004) (Cambridge University Press, Cambridge, UK) Crossref, Google Scholar
• Bronshtein E. M. ϵ-entropy of convex sets and functions. Siberian Math. J. (1976) 17(3):393–398Crossref, Google Scholar
• Brunk H. D. On the estimation of parameters restricted by inequalities. Ann. Math. Statist. (1958) 29(2):437–454Crossref, Google Scholar
• Brunk H. D. Conditional expectations given a σ-lattice and applications. Ann. Math. Statist. (1965) 36(5):1339–1350Crossref, Google Scholar
• Chen H., Yao D. D.Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization (2001) (Springer-Verlag, New York) Crossref, Google Scholar
• Copson E. T.Metric Spaces (1972) (Cambridge University Press, London) Google Scholar
• Cule M. L., Samworth R. J. Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Electron. J. Statist. (2010) 4:254–270Crossref, Google Scholar
• Dümbgen L., Samworth R. J., Schuhmacher D. Approximation by log-concave distributions with applications to regression. Ann. Statist. (2011) 39(2):702–730Crossref, Google Scholar
• Groeneboom P., Jongbloed G., Wellner J. A. Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. (2001) 29(6):1653–1698Crossref, Google Scholar
• Hall P., Huang L. Nonparametric kernel regression subject to monotonicity constraints. Ann. Statist. (2001) 29(3):624–647Crossref, Google Scholar
• Hanson D. L., Pledger G. Consistency in concave regression. Ann. Statist. (1976) 4(6):1038–1050Crossref, Google Scholar
• Kuosmanen T. Representation theorem for convex nonparametric least squares. Econometrics J. (2008) 11:308–325Crossref, Google Scholar
• Lim E., Glynn P. W. Simulation-based response surface computation in the presence of monotonicity. Proc. 2006 Winter Simulation Conf. (2006) Monterey, CA:264–271Crossref, Google Scholar
• Mammen E. Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. (1991) 19(2):741–759Crossref, Google Scholar
• Meyer R. F., Pratt J. W. The consistent assessment and fairing of preference functions. IEEE Trans. Systems Sci. Cybernetics (1968) 4(3):270–278Crossref, Google Scholar
• Munkres J. R.Topology (2000) 2nd ed.(Prentice Hall, Upper Saddle River, NJ) Google Scholar
• Roberts A. W., Varberg D. E. Another proof that convex functions are locally Lipschitz. Amer. Math. Monthly (1974) 81(9):1014–1016Crossref, Google Scholar
• Rockafellar R. T.Conjugate Duality and Optimization (1974) (Society for Industrial and Applied Mathematics, Philadelphia) Crossref, Google Scholar
• Rosenblatt F. Remarks on some nonparametric estimates of a density function. Ann. Statist. (1956) 27(3):832–837Crossref, Google Scholar
• Seijo E., Sen B. Nonparametric least squares estimation of a multivariate convex regression function. Ann. Statist. (2011) 39(3):1633–1657Crossref, Google Scholar
• Shanthikumar J. G., Yao D. D. Strong stochastic convexity: Closure properties and applications. J. Appl. Probab. (1991) 28:131–145Crossref, Google Scholar
• Van der Vaart A. W., Wellner J. A.Weak Convergence and Empirical Processes (1996) (Springer–Verlag, New York) Crossref, Google Scholar
• Varian H. R. The nonparametric approach to production analysis. Econometrica (1984) 52(3):579–597Crossref, Google Scholar
• Varian H. R. Nonparametric analysis of optimizing behavior with measurement error. J. Econometrics (1985) 30:445–458Crossref, Google Scholar
• Wright F. T. A strong law for variables indexed by a partially ordered set with applications to isotone regression. Ann. Probab. (1979) 7(1):109–127Crossref, Google Scholar
• Yatchew A. J., Bos L. Nonparametric least squares regression and testing in economic models. J. Quant. Econom. (1997) 13:81–131Google Scholar