Statistical Science

Some Developments in the Theory of Shape Constrained Inference

Piet Groeneboom and Geurt Jongbloed

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

Abstract

Shape constraints enter in many statistical models. Sometimes these constraints emerge naturally from the origin of the data. In other situations, they are used to replace parametric models by more versatile models retaining qualitative shape properties of the parametric model. In this paper, we sketch a part of the history of shape constrained statistical inference in a nutshell, using landmark results obtained in this area. For this, we mainly use the prototypical problems of estimating a decreasing probability density on $[0,\infty )$ and the estimation of a distribution function based on current status data as illustrations.

Article information

Source
Statist. Sci., Volume 33, Number 4 (2018), 473-492.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ss/1543482054

Digital Object Identifier
doi:10.1214/18-STS657

Mathematical Reviews number (MathSciNet)
MR3881204

Zentralblatt MATH identifier
07032825

Keywords
Isotonic regression Grenander estimator inverse problem monotonicity interval censoring current status regression single index model bootstrap Chernoff’s distribution Airy functions

Citation

Groeneboom, Piet; Jongbloed, Geurt. Some Developments in the Theory of Shape Constrained Inference. Statist. Sci. 33 (2018), no. 4, 473--492. doi:10.1214/18-STS657. https://projecteuclid.org/euclid.ss/1543482054


Export citation

References

  • 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.
  • Azadbakhsh, M., Jankowski, H. and Gao, X. (2014). Computing confidence intervals for log-concave densities. Comput. Statist. Data Anal. 75 248–264.
  • Balabdaoui, F. and Durot, C. (2015). Marshall lemma in discrete convex estimation. Statist. Probab. Lett. 99 143–148.
  • Balabdaoui, F., Groeneboom, P. and Hendrickx, K. (2017). Score estimation in the monotone single index model. Submitted.
  • Balabdaoui, F. and Pitman, J. (2011). The distribution of the maximal difference between a Brownian bridge and its concave majorant. Bernoulli 17 466–483.
  • 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.
  • Balabdaoui, F., Jankowski, H., Rufibach, K. and Pavlides, M. (2013). Asymptotics of the discrete log-concave maximum likelihood estimator and related applications. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 769–790.
  • Banerjee, M. and Wellner, J. A. (2001). Likelihood ratio tests for monotone functions. Ann. Statist. 29 1699–1731.
  • 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 Series in Probability and Mathematical Statistics. Wiley, London.
  • Birgé, L. (1999). Interval censoring: A nonasymptotic point of view. Math. Methods Statist. 8 285–298.
  • Böhning, D. (1986). A vertex-exchange-method in $D$-optimal design theory. Metrika 33 337–347.
  • Carolan, C. and Dykstra, R. (2001). Marginal densities of the least concave majorant of Brownian motion. Ann. Statist. 29 1732–1750.
  • Cator, E. (2011). Adaptivity and optimality of the monotone least-squares estimator. Bernoulli 17 714–735.
  • Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31–41.
  • Çinlar, E. (1992). Sunset over Brownistan. Stochastic Process. Appl. 40 45–53.
  • Cosslett, S. R. (2007). Efficient estimation of semiparametric models by smoothed maximum likelihood. Internat. Econom. Rev. 48 1245–1272.
  • Cule, M. and Samworth, R. (2010). Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Electron. J. Stat. 4 254–270.
  • Cule, M., Samworth, R. and Stewart, M. (2010). Maximum likelihood estimation of a multi-dimensional log-concave density. J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 545–607.
  • Daniels, H. E. and Skyrme, T. H. R. (1985). The maximum of a random walk whose mean path has a maximum. Adv. in Appl. Probab. 17 85–99.
  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1–38.
  • Dümbgen, L. and Rufibach, K. (2011). logcondens: Computations related to univariate log-concave density estimation. J. Stat. Softw. 39 1–28.
  • Dümbgen, L., Rufibach, K. and Wellner, J. A. (2007). Marshall’s lemma for convex density estimation. In Asymptotics: Particles, Processes and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 55 101–107. IMS, Beachwood, OH.
  • Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125–153 (1957).
  • Groeneboom, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016–1027.
  • Groeneboom, P. (1984). Brownian motion with a parabolic drift and Airy functions. CWI Technical Report, Dept. Mathematical Statistics-R 8413, CWI. Available at http://oai.cwi.nl/oai/asset/6435/6435A.pdf.
  • 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). Wadsworth Statist./Probab. Ser. 539–555. Wadsworth, Belmont, CA.
  • Groeneboom, P. (1987). Asymptotics for incomplete censored observations. Report 87-18, Mathematical Institute, Univ. Amsterdam.
  • Groeneboom, P. (1991). Nonparametric maximum likelihood estimators for interval censoring and deconvolution. Technical Report 378, Dept. Statistics, Stanford Univ. Available at https://statistics.stanford.edu/research/nonparametric-maximum-likelihood-estimators-interval-censoring-and-deconvolution.
  • Groeneboom, P. (1996). Lectures on inverse problems. In Lectures on Probability Theory and Statistics (Saint-Flour, 1994). Lecture Notes in Math. 1648 67–164. Springer, Berlin.
  • Groeneboom, P. (2010). The maximum of Brownian motion minus a parabola. Electron. J. Probab. 15 1930–1937.
  • Groeneboom, P. (2011). Vertices of the least concave majorant of Brownian motion with parabolic drift. Electron. J. Probab. 16 2234–2258.
  • Groeneboom, P. (2013). Nonparametric (smoothed) likelihood and integral equations. J. Statist. Plann. Inference 143 2039–2065.
  • Groeneboom, P. (2015). Rcpp scripts and C$+$$+$ code. Available at https://github.com/pietg/book.
  • Groeneboom, P. (2018). Chernoff’s distribution and differential equations of parabolic and Airy type. Talk at meeting Shape-Constrained Methods: Inference, Applications, and Practice. Banff, 1-28 to 2-2, 2018.
  • Groeneboom, P. and Hendrickx, K. (2017a) curstatCI. R package, 2017a. Available at https://cran.r-project.org/web/packages/curstatCI/index.html. Version 0.1.1.
  • Groeneboom, P. and Hendrickx, K. (2017b). The nonparametric bootstrap for the current status model. Electron. J. Stat. 11 3446–3484.
  • Groeneboom, P. and Hendrickx, K. (2018a). Confidence intervals for the current status model. Scand. J. Stat. 45 135–163.
  • Groeneboom, P. and Hendrickx, K. (2018b). Current status linear regression. Ann. Statist. 46 1415–1444.
  • Groeneboom, P., Hooghiemstra, G. and Lopuhaä, H. P. (1999). Asymptotic normality of the $L_{1}$ error of the Grenander estimator. Ann. Statist. 27 1316–1347.
  • 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.
  • Groeneboom, P. and Jongbloed, G. (2015). Nonparametric confidence intervals for monotone functions. Ann. Statist. 43 2019–2054.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653–1698.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). A canonical process for estimation of convex functions: The “invelope” of integrated Brownian motion $+t^{4}$. Ann. Statist. 29 1620–1652.
  • 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.
  • Groeneboom, P. and Ketelaars, T. (2011). Estimators for the interval censoring problem. Electron. J. Stat. 5 1797–1845.
  • Groeneboom, P., Lalley, S. and Temme, N. (2015). Chernoff’s distribution and differential equations of parabolic and Airy type. J. Math. Anal. Appl. 423 1804–1824.
  • Groeneboom, P., Maathuis, M. H. and Wellner, J. A. (2008a). Current status data with competing risks: Consistency and rates of convergence of the MLE. Ann. Statist. 36 1031–1063.
  • Groeneboom, P., Maathuis, M. H. and Wellner, J. A. (2008b). Current status data with competing risks: Limiting distribution of the MLE. Ann. Statist. 36 1064–1089.
  • Groeneboom, P. and Pyke, R. (1983). Asymptotic normality of statistics based on the convex minorants of empirical distribution functions. Ann. Probab. 11 328–345.
  • Groeneboom, P. and Temme, N. M. (2011). The tail of the maximum of Brownian motion minus a parabola. Electron. Commun. Probab. 16 458–466.
  • Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. DMV Seminar 19. Birkhäuser, Basel.
  • Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388–400.
  • Janson, S. (2013). Moments of the location of the maximum of Brownian motion with parabolic drift. Electron. Commun. Probab. 18 no. 15, 8.
  • Janson, S., Louchard, G. and Martin-Löf, A. (2010). The maximum of Brownian motion with parabolic drift. Electron. J. Probab. 15 1893–1929.
  • Jongbloed, G. (1998). The iterative convex minorant algorithm for nonparametric estimation. J. Comput. Graph. Statist. 7 310–321.
  • Jongbloed, G. (2001). Sieved maximum likelihood estimation in Wicksell’s problem and related deconvolution problems. Scand. J. Stat. 28 161–183.
  • Keiding, N., Hansen, O. K. H., Sørensen, D. N. and Slama, R. (2012). The current duration approach to estimating time to pregnancy. Scand. J. Stat. 39 185–204.
  • Kiefer, J. and Wolfowitz, J. (1976). Asymptotically minimax estimation of concave and convex distribution functions. Z. Wahrsch. Verw. Gebiete 34 73–85.
  • Kim, A. K. H. and Samworth, R. J. (2016). Global rates of convergence in log-concave density estimation. Ann. Statist. 44 2756–2779.
  • Klein, R. W. and Spady, R. H. (1993). An efficient semiparametric estimator for binary response models. Econometrica 61 387–421.
  • 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.
  • Kuchibhotla, A. K., Patra, R. K. and Sen, B. (2017). Score estimation in the convex single index models. Submitted. Available at https://arxiv.org/abs/1708.00145.
  • Li, G. and Zhang, C.-H. (1998). Linear regression with interval censored data. Ann. Statist. 26 1306–1327.
  • Marshall, A. W. (1969). Discussion on Barlow and van Zwet’s paper. In Nonparametric Techniques in Statistical Inference. Proceedings of the First International Symposium on Nonparametric Techniques Held at Indiana Univ. 174–176.
  • Murphy, S. A., van der Vaart, A. W. and Wellner, J. A. (1999). Current status regression. Math. Methods Statist. 8 407–425.
  • Peto, R. (1973). Experimental survival curves for interval-censored data. J. R. Stat. Soc. Ser. C. Appl. Stat. 22 86–91.
  • Pimentel, L. P. R. (2014). On the location of the maximum of a continuous stochastic process. J. Appl. Probab. 51 152–161.
  • Pitman, J. W. (1983). Remarks on the convex minorant of Brownian motion. In Seminar on Stochastic Processes, 1982 (Evanston, Ill., 1982). Progr. Probab. Statist. 5 219–227. Birkhäuser, Boston, MA.
  • Pitman, J. and Ross, N. (2012). The greatest convex minorant of Brownian motion, meander, and bridge. Probab. Theory Related Fields 153 771–807.
  • Pitman, J. and Uribe Bravo, G. (2012). The convex minorant of a Lévy process. Ann. Probab. 40 1636–1674.
  • Prakasa Rao, B. L. S. (1969). Estkmation of a unimodal density. Sankhyā Ser. A 31 23–36.
  • Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, Chichester.
  • Sen, B., Banerjee, M. and Woodroofe, M. (2010). Inconsistency of bootstrap: The Grenander estimator. Ann. Statist. 38 1953–1977.
  • Sen, B. and Xu, G. (2015). Model based bootstrap methods for interval censored data. Comput. Statist. Data Anal. 81 121–129.
  • Slama, R., Hansen, O. K. H., Ducot, B., Bohet, A., Sorensen, D., Allemand, L., Eijkemans, M. J., Rosetta, L., Thalabard, J. C., Keiding, N. et al. (2012). Estimation of the frequency of involuntary infertility on a nation-wide basis. Hum. Reprod. 27 1489–1498.
  • Tanaka, H. (2008). Semiparametric least squares estimation of monotone single index models and its application to the iterative least squares estimation of binary choice models.
  • van der Vaart, A. (1991). On differentiable functionals. Ann. Statist. 19 178–204.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York.
  • van Eeden, C. (1956). Maximum likelihood estimation of ordered probabilities. Nederl. Akad. Wetensch. Proc. Ser. A 18 444–455.
  • Vardi, Y. (1982). Nonparametric estimation in the presence of length bias. Ann. Statist. 10 616–620.
  • Vardi, Y. (1989). Multiplicative censoring, renewal processes, deconvolution and decreasing density: Nonparametric estimation. Biometrika 76 751–761.
  • Walther, G. (2001). Multiscale maximum likelihood analysis of a semiparametric model, with applications. Ann. Statist. 29 1297–1319.
  • Watson, G. S. (1971). Estimating functionals of particle size distributions. Biometrika 58 483–490.
  • Wellner, J. A. and Zhan, Y. (1997). A hybrid algorithm for computation of the nonparametric maximum likelihood estimator from censored data. J. Amer. Statist. Assoc. 92 945–959.
  • Wright, S. J. (1997). Primal-Dual Interior-Point Methods. SIAM, Philadelphia, PA.