Estimating a Function and Its Derivatives Under a Smoothness Condition

References

• [1] Adams RA, Fournier J (1977) Cone conditions and properties of Sobolev spaces. J. Math. Anal. Appl. 61(3):713–734.Crossref, Google Scholar
• [2] Adams RA, Fournier JJF (2003) Sobolev Spaces, 2nd ed. (Academic Press, Amsterdam).Google Scholar
• [3] Ankenman BE, Nelson BL, Staum J (2010) Stochastic kriging for simulation metamodeling. Oper. Res. 58(2):371–382.Link, Google Scholar
• [4] Bertsimas D, Mundru N (2020) Sparse convex regression. INFORMS J. Comput. 33(1):262–279.Link, Google Scholar
• [5] Birman MS, Solomyak MZ (1967) Piecewise-polynomial approximations of functions of the classes Wpα. Math. USSR-Sbornik 2(3):295–317.Crossref, Google Scholar
• [6] Boyd S, Vandenberghe L (2004) Convex Optimization (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [7] Brunk HD (1958) On the estimation of parameters restricted by inequalities. Ann. Math. Statist. 29(2):437–454.Crossref, Google Scholar
• [8] Brunk HD (1970) Estimation of isotonic regression. Puri ML, ed. Nonparametric Techniques in Statistical Inference (Cambridge University Press, London), 177–197.Google Scholar
• [9] Chen X, Ankenman BE, Nelson BL (2013) Enhancing stochastic kriging metamodels with gradient estimators. Oper. Res. 61(2):512–528.Link, Google Scholar
• [10] Chui CK (1988) Multivariate Splines (SIAM, Philadelphia).Crossref, Google Scholar
• [11] Chung KC, Yao TH (1977) On lattices admitting unique Lagrange interpolations. SIAM J. Numerical Anal. 14(4):735–743.Crossref, Google Scholar
• [12] Cleveland WS (1979) Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74(368):829–836.Crossref, Google Scholar
• [13] Cox DD (1983) Asymptotics for M-type smoothing splines. Ann. Statist. 11(2):530–551.Crossref, Google Scholar
• [14] Cox DD (1984) Multivariate smoothing spline functions. SIAM J. Numerical Anal. 21(4):789–813.Crossref, Google Scholar
• [15] Craven P, Wahba G (1979) Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross-validation. Numerische Mathematik 31:377–403.Crossref, Google Scholar
• [16] de Boor C (1978) A Practical Guide to Splines (Springer, New York).Crossref, Google Scholar
• [17] Dierckx P (1993) Curve and Surface Fitting with Splines (Oxford University Press, New York).Crossref, Google Scholar
• [18] Donoho DL, Johnstone IM (1994) Ideal spatial adaptation via wavelet shrinkage. Biometrika 81(3):425–455.Crossref, Google Scholar
• [19] Duchon J (1979) Splines minimizing rotation invariant semi-norms in Sobolev spaces. Schempp W, Zeller K, eds. Multivariate Approximation Theory (Birkhäuser-Verlag, Basel, Switzerland), 85–100.Google Scholar
• [20] Eubank RL (1999) Nonparametric Regression and Spline Smoothing (Marcel Dekker, New York).Crossref, Google Scholar
• [21] Franke R (1982) Scattered data interpolation: Tests of some methods. Math. Comput. 38(157):181–200.Google Scholar
• [22] Grant M, Boyd S (2014) CVX: MATLAB software for disciplined convex programming, version 2.1. Accessed April 17, 2024, http://cvxr.com/cvx.Google Scholar
• [23] Green PJ, Silverman BW (1994) Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach (Chapman & Hall, London).Crossref, Google Scholar
• [24] Green A, Balakrishnan S, Tibshirani R (2021) Minimax optimal regression over Sobolev spaces via Laplacian regularization on neighborhood graphs. Banerjee A, Fukumizu K, eds. Internat. Conf. Artificial Intelligence Statist. (PMLR, New York), 2602–2610.Google Scholar
• [25] Greville TNE (1969) Theory and Applications of Spline Functions (Academic Press, New York).Google Scholar
• [26] Groeneboom P, Jongbloed G, Wellner JA (2001) Estimation of a convex function: Characterization and asymptotic theory. Ann. Statist. 29(6):1653–1698.Crossref, Google Scholar
• [27] Györfi L, Kohler M, Krzyżak A, Walk H (2002) A Distribution-Free Theory of Nonparametric Regression (Springer, New York).Crossref, Google Scholar
• [28] Hanson DL, Pledger G (1976) Consistency in concave regression. Ann. Statist. 4(6):1038–1050.Crossref, Google Scholar
• [29] Härdle W (1990) Applied Nonparametric Regression (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [30] Hildreth C (1954) Point estimates of ordinates of concave functions. J. Amer. Statist. Assoc. 49(267):598–619.Crossref, Google Scholar
• [31] Hillier FS, Lieberman GJ (1967) Introduction to Operations Research, 9th ed. (Holden-Day, San Francisco).Google Scholar
• [32] Hull JC (2006) Options, Futures, and Other Derivatives (Prentice Hall, Hoboken, NJ).Google Scholar
• [33] Hutchinson MF, de Hoog FR (1985) Noisy data with spline functions. Numerische Mathematik 47:99–106.Crossref, Google Scholar
• [34] Johnson AL, Jiang DR (2018) Shape constraints in economics and operations research. Statist. Sci. 33(4):527–546.Crossref, Google Scholar
• [35] Kersey SN (2003) On the problems of smoothing and near-interpolation. Math. Comput. 72(244):1873–1885.Crossref, Google Scholar
• [36] Keshvari A (2018) Segmented concave least squares: A nonparametric piecewise linear regression. Eur. J. Oper. Res. 266(2):585–594.Crossref, Google Scholar
• [37] Kohler M, Krzyżak A, Walk H (2006) Rates of convergence for partitioning and nearest neighbor regression estimates with unbounded data. J. Multivariate Anal. 97(2):311–323.Crossref, Google Scholar
• [38] Kohler M, Krzyżak A, Walk H (2009) Optimal global rates of convergence for nonparametric regression with unbounded data. J. Statist. Planning Inference 139(4):1286–1296.Crossref, Google Scholar
• [39] Kolmogorov AN, Tihomirov VM (1961) ϵ-Entropy and ϵ-capacity of sets in functional spaces. Amer. Math. Soc. Translations 2(17):277–364.Crossref, Google Scholar
• [40] Kuosmanen T (2008) Representation theorem for convex nonparametric least squares. Econom. J. 11(2):308–325.Crossref, Google Scholar
• [41] Kuosmanen T, Johnson AL (2010) Data envelopment analysis as nonparametric least squares regression. Oper. Res. 58(1):149–160.Link, Google Scholar
• [42] Kuosmanen T, Johnson AL (2017) Modeling joint production of multiple outputs in StoNED: Directional distance function approach. Eur. J. Oper. Res. 262(2):792–801.Crossref, Google Scholar
• [43] Lee C, Johnson AL, Moreno-Centeno E, Kuosmanen T (2013) A more efficient algorithm for convex nonparametric least squares. Eur. J. Oper. Res. 227(2):391–400.Crossref, Google Scholar
• [44] Lim E (2020) The limiting behavior of isotonic and convex regression estimators when the model is misspecified. Electronic J. Statist. 14(1):2053–2097.Crossref, Google Scholar
• [45] Lim E (2021) Consistency of penalized convex regression. Internat. J. Statist. Probab. 10(1):69–78.Crossref, Google Scholar
• [46] Lim E, Glynn PW (2012) Consistency of multidimensional convex regression. Oper. Res. 60(1):196–208.Link, Google Scholar
• [47] Luo Z, Wahba G (1997) Hybrid adaptive splines. J. Amer. Statist. Assoc. 92(437):107–116.Crossref, Google Scholar
• [48] Mammen E (1991) Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19(2):741–759.Crossref, Google Scholar
• [49] Mazumder R, Choudhury A, Iyengar G, Sen B (2019) A computational framework for multivariate convex regression and its variants. J. Amer. Statist. Assoc. 114(525):318–331.Crossref, Google Scholar
• [50] Meinguet J (1979) Multivariate interpolation at arbitrary points made simple. Z. Angew. Math. Phys. 30:292–304.Crossref, Google Scholar
• [51] Meinguet J (1984) Surface spline interpolation: Basic theory and computational aspects. Singh SP, Burry JWH, Watson B, eds. Approximation Theory and Spline Functions, NATO ASI Series, vol. 136 (Springer, Dordrecht, Netherlands), 127–142.Crossref, Google Scholar
• [52] Myers RH, Montgomery DC (2002) Response Surface Methodology: Process and Product Optimization Using Designed Experiments (Wiley, New York).Google Scholar
• [53] Nadaraya EA (1964) On estimating regression. Theory Probab. Appl. 9(1):141–142.Crossref, Google Scholar
• [54] Oden JT, Reddy JN (1976) An Introduction to the Mathematical Theory of Finite Elements (Wiley, New York).Google Scholar
• [55] Reinsch CH (1967) Smoothing by spline functions. Numerische Mathematik 10:177–183.Crossref, Google Scholar
• [56] Reinsch CH (1971) Smoothing by spline functions II. Numerische Mathematik 16:451–454.Crossref, Google Scholar
• [57] Rice J, Rosenblatt M (1983) Smoothing splines: Regression, derivatives and deconvolution. Ann. Statist. 11(1):141–156.Crossref, Google Scholar
• [58] Ruppert D, Wand MP (1994) Multivariate locally weighted least squares regression. Ann. Statist. 22(3):1346–1370.Crossref, Google Scholar
• [59] Salemi P, Nelson BL, Staum J (2016) Moving least squares regression for high-dimensional stochastic simulation metamodeling. ACM Trans. Model. Comput. Simulation 26:16:1–16:25.Crossref, Google Scholar
• [60] Schoenberg IJ (1964) Spline functions and the problem of graduation. Proc. Natl. Acad. Sci. USA 52(4):947–950.Crossref, Google Scholar
• [61] Seijo E, Sen B (2011) Nonparametric least squares estimation of a multivariate convex regression function. Ann. Statist. 39(3):1633–1657.Crossref, Google Scholar
• [62] Shapiro A (2000) On the asymptotics of constrained local M-estimators. Ann. Statist. 28(3):948–960.Crossref, Google Scholar
• [63] Shapiro A, Dentcheva D, Ruszczyński AP (2009) Lectures on Stochastic Programming: Modeling and Theory (SIAM, Philadelphia).Crossref, Google Scholar
• [64] Stone CJ (1977) Consistent nonparametric regression. Ann. Statist. 5(4):595–645.Crossref, Google Scholar
• [65] Stone CJ (1980) Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8(6):1348–1360.Crossref, Google Scholar
• [66] Utreras FI (1981) Optimal smoothing of noisy data using spline functions. SIAM J. Sci. Statist. Comput. 2(3):349–362.Crossref, Google Scholar
• [67] Utreras FI (1988) Convergence rates for multivariate smoothing spline functions. J. Approx. Theory 52(1):1–27.Crossref, Google Scholar
• [68] van de Geer S (1990) Estimating a regression function. Ann. Statist. 18(2):907–924.Crossref, Google Scholar
• [69] van de Geer S (2000) Empirical Process in M-Estimation (Cambridge University Press, Cambridge, UK).Google Scholar
• [70] Varian HR (1985) Non-parametric analysis of optimizing behavior with measurement error. J. Econometrics 30(1):445–458.Crossref, Google Scholar
• [71] Wahba G (1990) Spline Models for Observational Data (SIAM, Philadelphia).Crossref, Google Scholar
• [72] Wand MP, Jones MC (1995) Kernel Smoothing (Chapman & Hall, London).Crossref, Google Scholar
• [73] Watson GS (1964) Smooth regression analysis. Sankhya Ser. A 26(4):359–372.Google Scholar
• [74] Wegman EJ, Wright IW (1983) Splines in statistics. J. Amer. Statist. Assoc. 78(382):351–365.Crossref, Google Scholar
• [75] Yagi D, Chen Y, Johnson AL, Kuosmanen T (2020) Shape-constrained kernel-weighted least squares: Estimating production functions for Chilean manufacturing industries. J. Bus. Econom. Statist. 38(1):43–54.Crossref, Google Scholar
• [76] Zangwill WI (1969) Nonlinear Programming: A Unified Approach (Prentice-Hall, Hoboken, NJ).Google Scholar