Statistical Science

Nonparametric Shape-Restricted Regression

Adityanand Guntuboyina and Bodhisattva Sen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression and constrained single index model. We review some of the theoretical properties of the least squares estimator (LSE) in these problems, emphasizing on the adaptive nature of the LSE. In particular, we study the behavior of the risk of the LSE, and its pointwise limiting distribution theory, with special emphasis to isotonic regression. We survey various methods for constructing pointwise confidence intervals around these shape-restricted functions. We also briefly discuss the computation of the LSE and indicate some open research problems and future directions.

Article information

Statist. Sci., Volume 33, Number 4 (2018), 568-594.

First available in Project Euclid: 29 November 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Adaptive risk bounds bootstrap Chernoff’s distribution convex regression isotonic regression likelihood ratio test monotone function order preserving function estimation projection on a closed convex set tangent cone


Guntuboyina, Adityanand; Sen, Bodhisattva. Nonparametric Shape-Restricted Regression. Statist. Sci. 33 (2018), no. 4, 568--594. doi:10.1214/18-STS665.

Export citation


  • [1] Abrevaya, J. and Huang, J. (2005). On the bootstrap of the maximum score estimator. Econometrica 73 1175–1204.
  • [2] Aït-Sahalia, Y. and Duarte, J. (2003). Nonparametric option pricing under shape restrictions. frontiers of financial econometrics and financial engineering. J. Econometrics 116 9–47.
  • [3] Amelunxen, D., Lotz, M., McCoy, M. B. and Tropp, J. A. (2014). Living on the edge: Phase transitions in convex programs with random data. Inf. Inference 3 224–294.
  • [4] Anevski, D. and Hössjer, O. (2006). A general asymptotic scheme for inference under order restrictions. Ann. Statist. 34 1874–1930.
  • [5] Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information. Ann. Math. Stat. 26 641–647.
  • [6] Bacchetti, P. (1989). Additive isotonic models. J. Amer. Statist. Assoc. 84 289–294.
  • [7] Bagchi, P., Banerjee, M. and Stoev, S. A. (2016). Inference for monotone functions under short- and long-range dependence: Confidence intervals and new universal limits. J. Amer. Statist. Assoc. 111 1634–1647.
  • [8] Balabdaoui, F. (2007). Consistent estimation of a convex density at the origin. Math. Methods Statist. 16 77–95.
  • [9] Balabdaoui, F., Durot, C. and Jankowski, H. (2016). Least squares estimation in the monotone single index model. Preprint. Available at arXiv:1610.06026.
  • [10] Balabdaoui, F., Jankowski, H., Pavlides, M., Seregin, A. and Wellner, J. (2011). On the Grenander estimator at zero. Statist. Sinica 21 873–899.
  • [11] Balabdaoui, F., Rufibach, K. and Wellner, J. A. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist. 37 1299–1331.
  • [12] Balázs, G. (2016). Convex regression: Theory, practice, and applications. Ph.D. thesis, Univ. Alberta.
  • [13] Banerjee, M. (2007). Likelihood based inference for monotone response models. Ann. Statist. 35 931–956.
  • [14] Banerjee, M. (2009). Inference in exponential family regression models under certain shape constraints using inversion based techniques. In Advances in Multivariate Statistical Methods. Stat. Sci. Interdiscip. Res. 4 249–271. World Sci. Publ., Hackensack, NJ.
  • [15] Banerjee, M. and Wellner, J. A. (2001). Likelihood ratio tests for monotone functions. Ann. Statist. 29 1699–1731.
  • [16] Banerjee, M. and Wellner, J. A. (2005). Confidence intervals for current status data. Scand. J. Stat. 32 405–424.
  • [17] Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference Under Order Restrictions. The Theory and Application of Isotonic Regression. Wiley, London.
  • [18] Bellec, P. C. (2016). Adaptive confidence sets in shape restricted regression. Preprint. Available at arXiv:1601.05766.
  • [19] Bellec, P. C. (2018). Sharp oracle inequalities for least squares estimators in shape restricted regression. Ann. Statist. 46 745–780.
  • [20] Bertsekas, D. P. (2003). Convex Analysis and Optimization. Athena Scientific, Belmont, MA.
  • [21] Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge Univ. Press, Cambridge.
  • [22] Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation. J. Amer. Statist. Assoc. 80 580–619.
  • [23] Brunk, H. D. (1955). Maximum likelihood estimates of monotone parameters. Ann. Math. Stat. 26 607–616.
  • [24] Brunk, H. D. (1970). Estimation of isotonic regression. In Nonparametric Techniques in Statistical Inference (Proc. Sympos., Indiana Univ., Bloomington, Ind., 1969) 177–197. Cambridge Univ. Press, London.
  • [25] Carolan, C. and Dykstra, R. (1999). Asymptotic behavior of the Grenander estimator at density flat regions. Canad. J. Statist. 27 557–566.
  • [26] Chatterjee, S. (2014). A new perspective on least squares under convex constraint. Ann. Statist. 42 2340–2381.
  • [27] Chatterjee, S. (2016). An improved global risk bound in concave regression. Electron. J. Stat. 10 1608–1629.
  • [28] Chatterjee, S., Guntuboyina, A. and Sen, B. (2015). On risk bounds in isotonic and other shape restricted regression problems. Ann. Statist. 43 1774–1800.
  • [29] Chatterjee, S., Guntuboyina, A. and Sen, B. (2018). On matrix estimation under monotonicity constraints. Bernoulli 24 1072–1100. Preprint. Available at arXiv:1506.03430.
  • [30] Chatterjee, S. and Lafferty, J. (2015). Adaptive risk bounds in unimodal regression. Preprint. Available at arXiv:1512.02956.
  • [31] Chatterjee, S. and Lafferty, J. (2018). Denoising flows on trees. IEEE Trans. Inform. Theory 64 1767–1783. Preprint. Available at arXiv:1602.08048.
  • [32] Chatterjee, S. and Mukherjee, S. (2017). On estimation in tournaments and graphs under monotonicity constraints. Preprint. Available at arXiv:1603.04556.
  • [33] Chen, X., Guntuboyina, A. and Zhang, Y. (2017). A note on the approximate admissibility of regularized estimators in the Gaussian sequence model. Electron. J. Stat. 11 4746–4768. Preprint. Available at arXiv:1703.00542.
  • [34] Chen, X., Lin, Q. and Sen, B. (2015). On degrees of freedom of projection estimators with applications to multivariate shape restricted regression. Preprint. Available at arXiv:1509.01877.
  • [35] Chen, Y. and Samworth, R. J. (2016). Generalized additive and index models with shape constraints. J. R. Stat. Soc. Ser. B. Stat. Methodol. 78 729–754.
  • [36] Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31–41.
  • [37] Cui, X., Härdle, W. K. and Zhu, L. (2011). The EFM approach for single-index models. Ann. Statist. 39 1658–1688.
  • [38] Demetriou, I. and Tzitziris, P. (2017). Infant mortality and economic growth: Modeling by increasing returns and least squares. In Proceedings of the World Congress on Engineering 2.
  • [39] Donoho, D. L. (1990). Gelfand $n$-widths and the method of least squares. Technical Report 282, Dept. Statistics, Univ. California, Berkeley.
  • [40] Doss, C. R. and Wellner, J. A. (2016). Inference for the mode of a log-concave density. Preprint. Available at arXiv:1611.10348.
  • [41] Doss, C. R. and Wellner, J. A. (2016). Mode-constrained estimation of a log-concave density. Preprint. Available at arXiv:1611.10335.
  • [42] Dümbgen, L. (2003). Optimal confidence bands for shape-restricted curves. Bernoulli 9 423–449.
  • [43] Dümbgen, L., Freitag, S. and Jongbloed, G. (2004). Consistency of concave regression with an application to current-status data. Math. Methods Statist. 13 69–81.
  • [44] Dümbgen, L. and Spokoiny, V. G. (2001). Multiscale testing of qualitative hypotheses. Ann. Statist. 29 124–152.
  • [45] Dykstra, R. L. (1983). An algorithm for restricted least squares regression. J. Amer. Statist. Assoc. 78 837–842.
  • [46] Fang, B. and Guntuboyina, A. (2017). On the risk of convex-constrained least squares estimators under misspecification. Preprint. Available at arXiv:1706.04276.
  • [47] Flammarion, N., Mao, C. and Rigollet, P. (2016). Optimal rates of statistical seriation. Preprint. Available at arXiv:1607.02435.
  • [48] Fraser, D. A. S. and Massam, H. (1989). A mixed primal-dual bases algorithm for regression under inequality constraints. Application to concave regression. Scand. J. Stat. 16 65–74.
  • [49] Frisén, M. (1986). Unimodal regression. Statistician 479–485.
  • [50] Gao, C., Han, F. and Zhang, C.-H. (2017). Minimax risk bounds for piecewise constant models. Preprint. Available at arXiv:1705.06386.
  • [51] Gebhardt, F. (1970). An algorithm for monotone regression with one or more independent variables. Biometrika 57 263–271.
  • [52] Ghosal, P. and Sen, B. (2017). On univariate convex regression. Sankhyā A 79 215–253.
  • [53] Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125–153 (1957).
  • [54] Groeneboom, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016–1027.
  • [55] Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II (Berkeley, Calif., 1983). 539–555. Wadsworth, Belmont, CA.
  • [56] Groeneboom, P. and Hendrickx, K. (2018). Confidence intervals for the current status model. Scand. J. Stat. 45 135–163. Preprint. Available at arXiv:1611.08299.
  • [57] Groeneboom, P. and Hendrickx, K. (2018). Current status linear regression. Ann. Statist. 46 1415–1444. Available at arXiv:1601.00202.
  • [58] Groeneboom, P. and Jongbloed, G. (1995). Isotonic estimation and rates of convergence in Wicksell’s problem. Ann. Statist. 23 1518–1542.
  • [59] Groeneboom, P. and Jongbloed, G. (2014). Nonparametric Estimation Under Shape Constraints: Estimators, Algorithms and Asymptotics. Cambridge Series in Statistical and Probabilistic Mathematics 38. Cambridge Univ. Press, New York.
  • [60] Groeneboom, P. and Jongbloed, G. (2015). Nonparametric confidence intervals for monotone functions. Ann. Statist. 43 2019–2054.
  • [61] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653–1698.
  • [62] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2008). The support reduction algorithm for computing non-parametric function estimates in mixture models. Scand. J. Stat. 35 385–399.
  • [63] Groeneboom, P., Jongbloed, G. and Witte, B. I. (2010). Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model. Ann. Statist. 38 352–387.
  • [64] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. DMV Seminar 19. Birkhäuser, Basel.
  • [65] Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388–400.
  • [66] Guntuboyina, A., Lieu, D., Chatterjee, S. and Sen, B. (2017). Spatial adaptation in trend filtering. Preprint. Available at arXiv:1702.05113.
  • [67] Guntuboyina, A. and Sen, B. (2015). Global risk bounds and adaptation in univariate convex regression. Probab. Theory Related Fields 163 379–411.
  • [68] Guntuboyina, A. and Sen, B. (2018). Supplement to “Nonparametric Shape-Restricted Regression.” DOI:10.1214/18-STS665SUPP.
  • [69] Han, Q., Wang, T., Chatterjee, S. and Samworth, R. J. (2017). Isotonic regression in general dimensions. Preprint. Available at arXiv:1708.09468.
  • [70] Hanson, D. L. and Pledger, G. (1976). Consistency in concave regression. Ann. Statist. 4 1038–1050.
  • [71] Hanson, D. L., Pledger, G. and Wright, F. T. (1973). On consistency in monotonic regression. Ann. Statist. 1 401–421.
  • [72] Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Monographs on Statistics and Applied Probability 43. CRC Press, London.
  • [73] Hildreth, C. (1954). Point estimates of ordinates of concave functions. J. Amer. Statist. Assoc. 49 598–619.
  • [74] Hu, J., Kapoor, M., Zhang, W., Hamilton, S. R. and Coombes, K. R. (2005). Analysis of dose–response effects on gene expression data with comparison of two microarray platforms. Bioinformatics 21 3524–3529.
  • [75] Huang, J. and Wellner, J. A. (1995). Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Stat. 22 3–33.
  • [76] Huang, J. and Wellner, J. A. (1997). Interval censored survival data: A review of recent progress. In Proceedings of the First Seattle Symposium in Biostatistics: Survival Analysis (D. Y. Lin and T. R. Fleming, eds.) 123–169.
  • [77] Jongbloed, G. and van der Meulen, F. H. (2009). Estimating a concave distribution function from data corrupted with additive noise. Ann. Statist. 37 782–815.
  • [78] Kakade, S. M., Kanade, V., Shamir, O. and Kalai, A. (2011). Efficient learning of generalized linear and single index models with isotonic regression. In Advances in Neural Information Processing Systems 927–935.
  • [79] Keshavarz, A., Wang, Y. and Boyd, S. (2011). Imputing a convex objective function. In Intelligent Control (ISIC), 2011 IEEE International Symposium on 613–619. IEEE.
  • [80] Kosorok, M. R. (2008). Bootstrapping in Grenander estimator. In Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen. Inst. Math. Stat. (IMS) Collect. 1 282–292. IMS, Beachwood, OH.
  • [81] Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29 1–27.
  • [82] Kuchibhotla, A. K., Patra, R. K. and Sen, B. (2017). Efficient estimation in convex single index models. Preprint. Available at arXiv:1708.00145.
  • [83] Kulikov, V. N. and Lopuhaä, H. P. (2006). The behavior of the NPMLE of a decreasing density near the boundaries of the support. Ann. Statist. 34 742–768.
  • [84] Kuosmanen, T. (2008). Representation theorem for convex nonparametric least squares. Econom. J. 11 308–325.
  • [85] Kyng, R., Rao, A. and Sachdeva, S. (2015). Fast, provable algorithms for isotonic regression in all $l$-$p$-norms. In Advances in Neural Information Processing Systems 2719–2727.
  • [86] Leurgans, S. (1982). Asymptotic distributions of slope-of-greatest-convex-minorant estimators. Ann. Statist. 10 287–296.
  • [87] Li, K.-C. and Duan, N. (1989). Regression analysis under link violation. Ann. Statist. 17 1009–1052.
  • [88] Li, Q. and Racine, J. S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton Univ. Press, Princeton, NJ.
  • [89] Luss, R., Rosset, S. and Shahar, M. (2012). Efficient regularized isotonic regression with application to gene–gene interaction search. Ann. Appl. Stat. 6 253–283.
  • [90] Magnani, A. and Boyd, S. P. (2009). Convex piecewise-linear fitting. Optim. Eng. 10 1–17.
  • [91] Mammen, E. (1991). Estimating a smooth monotone regression function. Ann. Statist. 19 724–740.
  • [92] Mammen, E. (1991). Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19 741–759.
  • [93] Mammen, E., Linton, O. and Nielsen, J. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist. 27 1443–1490.
  • [94] Mammen, E. and Thomas-Agnan, C. (1999). Smoothing splines and shape restrictions. Scand. J. Stat. 26 239–252.
  • [95] Mammen, E. and Yu, K. (2007). Additive isotone regression. In Asymptotics: Particles, Processes and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 55 179–195. IMS, Beachwood, OH.
  • [96] Mammen, E. and Yu, K. (2007). Additive isotone regression. In Asymptotics: Particles, Processes and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 55 179–195. IMS, Beachwood, OH.
  • [97] Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications, 2nd ed. Springer, New York.
  • [98] Matzkin, R. L. (1991). Semiparametric estimation of monotone and concave utility functions for polychotomous choice models. Econometrica 59 1315–1327.
  • [99] Mazumder, R., Choudhury, A., Iyengar, G. and Sen, B. (2015). A computational framework for multivariate convex regression and its variants. J. Amer. Statist. Assoc. To appear. Preprint. Available at arXiv:1509.08165.
  • [100] Meyer, M. and Woodroofe, M. (2000). On the degrees of freedom in shape-restricted regression. Ann. Statist. 28 1083–1104.
  • [101] Meyer, M. C. (1999). An extension of the mixed primal-dual bases algorithm to the case of more constraints than dimensions. J. Statist. Plann. Inference 81 13–31.
  • [102] Meyer, M. C. (2008). Inference using shape-restricted regression splines. Ann. Appl. Stat. 2 1013–1033.
  • [103] Meyer, M. C. (2013). Semi-parametric additive constrained regression. J. Nonparametr. Stat. 25 715–730.
  • [104] Meyer, M. C. (2013). A simple new algorithm for quadratic programming with applications in statistics. Comm. Statist. Simulation Comput. 42 1126–1139.
  • [105] Mukerjee, H. (1988). Monotone nonparameteric regression. Ann. Statist. 16 741–750.
  • [106] Murphy, S. A., van der Vaart, A. W. and Wellner, J. A. (1999). Current status regression. Math. Methods Statist. 8 407–425.
  • [107] Nemirovskiĭ, A. S., Polyak, B. T. and Tsybakov, A. B. (1985). The rate of convergence of nonparametric estimates of maximum likelihood type. Problemy Peredachi Informatsii 21 17–33.
  • [108] Newey, W. K. (1990). Semiparametric efficiency bounds. J. Appl. Econometrics 5 99–135.
  • [109] Obozinski, G., Lanckriet, G., Grant, C., Jordan, M. I. and Noble, W. S. (2008). Consistent probabilistic outputs for protein function prediction. Genome Biol. 9 S6.
  • [110] Oymak, S. and Hassibi, B. (2016). Sharp MSE bounds for proximal denoising. Found. Comput. Math. 16 965–1029.
  • [111] Powell, J. L., Stock, J. H. and Stoker, T. M. (1989). Semiparametric estimation of index coefficients. Econometrica 57 1403–1430.
  • [112] Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhyā Ser. A 31 23–36.
  • [113] Pya, N. and Wood, S. N. (2015). Shape constrained additive models. Stat. Comput. 25 543–559.
  • [114] Robertson, T. and Wright, F. T. (1975). Consistency in generalized isotonic regression. Ann. Statist. 3 350–362.
  • [115] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, Chichester.
  • [116] Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton, NJ.
  • [117] Schrijver, A. (1986). Theory of Linear and Integer Programming. Wiley, Chichester.
  • [118] Seijo, E. and Sen, B. (2011). Nonparametric least squares estimation of a multivariate convex regression function. Ann. Statist. 39 1633–1657.
  • [119] Sen, B., Banerjee, M. and Woodroofe, M. (2010). Inconsistency of bootstrap: The Grenander estimator. Ann. Statist. 38 1953–1977.
  • [120] Sen, B. and Meyer, M. (2017). Testing against a linear regression model using ideas from shape-restricted estimation. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 423–448.
  • [121] Sen, B. and Woodroofe, M. (2012). Bootstrap confidence intervals for isotonic estimators in a stereological problem. Bernoulli 18 1249–1266.
  • [122] Sen, B. and Xu, G. (2015). Model based bootstrap methods for interval censored data. Comput. Statist. Data Anal. 81 121–129.
  • [123] Shah, N. B., Balakrishnan, S., Guntuboyina, A. and Wainwright, M. J. (2017). Stochastically transitive models for pairwise comparisons: Statistical and computational issues. IEEE Trans. Inform. Theory 63 934–959.
  • [124] Shapiro, A., Dentcheva, D. and Ruszczyński, A. (2009). Lectures on Stochastic Programming: Modeling and Theory. MPS/SIAM Series on Optimization 9. SIAM, Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA.
  • [125] Simchi-Levi, D., Chen, X. and Bramel, J. (2005). The Logic of Logistics: Theory, Algorithms, and Applications for Logistics and Supply Chain Management, 2nd ed. Springer, New York.
  • [126] Slawski, M. and Hein, M. (2013). Non-negative least squares for high-dimensional linear models: Consistency and sparse recovery without regularization. Electron. J. Stat. 7 3004–3056.
  • [127] Stout, Q. F. (2008). Unimodal regression via prefix isotonic regression. Comput. Statist. Data Anal. 53 289–297.
  • [128] Stout, Q. F. (2014). Fastest isotonic regression algorithms.
  • [129] Stout, Q. F. (2015). Isotonic regression for multiple independent variables. Algorithmica 71 450–470.
  • [130] Talagrand, M. (2014). Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems. 60. Springer, Heidelberg.
  • [131] Tukey, J. W. (1961). Curves as parameters, and touch estimation. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 681–694. Univ. California Press, Berkeley, CA.
  • [132] van Eeden, C. (1956). Maximum likelihood estimation of ordered probabilities. Indag. Math. (N.S.) 18 444–455.
  • [133] van de Geer, S. (1990). Estimating a regression function. Ann. Statist. 18 907–924.
  • [134] van de Geer, S. and Wainwright, M. J. (2017). On concentration for (regularized) empirical risk minimization. Sankhya A 79 159–200.
  • [135] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • [136] Varian, H. R. (1984). The nonparametric approach to production analysis. Econometrica 52 579–597.
  • [137] Wei, Y., Wainwright, M. J. and Guntuboyina, A. (2017). The geometry of hypothesis testing over convex cones: Generalized likelihood tests and minimax radii. Preprint. Available at arXiv:1703.06810.
  • [138] Woodroofe, M. and Sun, J. (1993). A penalized maximum likelihood estimate of $f(0+)$ when $f$ is nonincreasing. Statist. Sinica 3 501–515.
  • [139] Wright, F. T. (1981). The asymptotic behavior of monotone regression estimates. Ann. Statist. 9 443–448.
  • [140] Wu, J., Meyer, M. C. and Opsomer, J. D. (2015). Penalized isotonic regression. J. Statist. Plann. Inference 161 12–24.
  • [141] Yang, F. and Barber, R. F. (2017). Contraction and uniform convergence of isotonic regression. Preprint. Available at arXiv:1706.01852.
  • [142] Yatchew, A. (2003). Semiparametric Regression for the Applied Econometrician. Cambridge Univ. Press, Cambridge.
  • [143] Zhang, C.-H. (2002). Risk bounds in isotonic regression. Ann. Statist. 30 528–555.

Supplemental materials

  • Supplement to “Nonparametric Shape-Restricted Regression”. The supplement contains some of the detailed proofs of results in the paper.