1932

Abstract

Statistical models defined by shape constraints are a valuable alternative to parametric models or nonparametric models defined in terms of quantitative smoothness constraints. While the latter two classes of models are typically difficult to justify a priori, many applications involve natural shape constraints, for instance, monotonicity of a density or regression function. We review some of the history of this subject and recent developments, with special emphasis on algorithmic aspects, adaptivity, honest confidence bands for shape-constrained curves, and distributional regression, i.e., inference about the conditional distribution of a real-valued response given certain covariates.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-statistics-033021-014937
2024-04-22
2024-05-07
Loading full text...

Full text loading...

/deliver/fulltext/statistics/11/1/annurev-statistics-033021-014937.html?itemId=/content/journals/10.1146/annurev-statistics-033021-014937&mimeType=html&fmt=ahah

Literature Cited

  1. Bacchetti P. 1989.. Additive isotonic models. . J. Am. Stat. Assoc. 84:(405):28994
    [Google Scholar]
  2. Balabdaoui F, Groeneboom P, Hendrickx K. 2019.. Score estimation in the monotone single-index model. . Scand. J. Stat. 46:(2):51744
    [Crossref] [Google Scholar]
  3. Balabdaoui F, Rufibach K, Wellner JA. 2009.. Limit distribution theory for maximum likelihood estimation of a log-concave density. . Ann. Stat. 37:(3):1299331
    [Crossref] [Google Scholar]
  4. Balabdaoui F, Wellner JA. 2010.. Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds. . Stat. Neerl. 64:(1):4570
    [Crossref] [Google Scholar]
  5. Banerjee M, Wellner JA. 2001.. Likelihood ratio tests for monotone functions. . Ann. Stat. 29:(6):1699731
    [Crossref] [Google Scholar]
  6. Borell C. 1975.. Convex set functions in d-space. . Period. Math. Hungar. 6:(2):11136
    [Crossref] [Google Scholar]
  7. Brunk HD. 1955.. Maximum likelihood estimates of monotone parameters. . Ann. Math. Stat. 26::60716
    [Crossref] [Google Scholar]
  8. Chatterjee S, Sen S. 2021.. Regret minimization in isotonic, heavy-tailed contextual bandits via adaptive confidence bands. . arXiv:2110.10245 [math.ST]
  9. Chen Y, Samworth RJ. 2016.. Generalized additive and index models with shape constraints. . J. R. Stat. Soc. Ser. B 78:(4):72954
    [Crossref] [Google Scholar]
  10. Clopper CJ, Pearson ES. 1934.. The use of confidence or fiducial limits illustrated in the case of the binomial. . Biometrika 26:(4):40413
    [Crossref] [Google Scholar]
  11. Colangelo A, Müller A, Scarsini M. 2006.. Positive dependence and weak convergence. . J. Appl. Probab. 43:(1):4859
    [Crossref] [Google Scholar]
  12. Cule M, Gramacy RB, Samworth R. 2009.. LogConcDEAD: An R package for maximum likelihood estimation of a multivariate log-concave density. . J. Stat. Softw. 29:(2):120
    [Crossref] [Google Scholar]
  13. Cule M, Samworth R. 2010.. Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. . Electron. J. Stat. 4::25470
    [Crossref] [Google Scholar]
  14. Cule M, Samworth R, Stewart M. 2010.. Maximum likelihood estimation of a multi-dimensional log-concave density. . J. R. Stat. Soc. Ser. B 72:(5):545607
    [Crossref] [Google Scholar]
  15. Davies PL. 1995.. Data features. . Stat. Neerl. 49:(2):185245
    [Crossref] [Google Scholar]
  16. de Leeuw J, Hornik K, Mair P. 2009.. Isotone optimization in R: Pool-adjacent-violators algorithm (PAVA) and active set methods. . J. Stat. Softw. 32:(5):124
    [Crossref] [Google Scholar]
  17. Dette H, Volgushev S. 2008.. Non-crossing non-parametric estimates of quantile curves. . J. R. Stat. Soc. Ser. B Stat. Methodol. 70:(3):60927
    [Crossref] [Google Scholar]
  18. Dimitriadis T, Dümbgen L, Henzi A, Puke M, Ziegel J. 2023.. Honest calibration assessment for binary outcome predictions. . Biometrika 110:(3):66380
    [Crossref] [Google Scholar]
  19. Donoho DL. 1988.. One-sided inference about functionals of a density. . Ann. Stat. 16:(4):1390420
    [Crossref] [Google Scholar]
  20. Doss CR, Wellner JA. 2019.. Inference for the mode of a log-concave density. . Ann. Stat. 47:(5):295076
    [Crossref] [Google Scholar]
  21. Dümbgen L. 1998.. New goodness-of-fit tests and their application to nonparametric confidence sets. . Ann. Stat. 26:(1):288314
    [Crossref] [Google Scholar]
  22. Dümbgen L. 2007.. Confidence bands for convex median curves using sign-tests. . In Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom, ed. E Cator, G Jongbloed, C Kraaikamp, R Lopuhaä, JA Wellner , pp. 85100 Beachwood, OH:: Inst. Math. Stat.
    [Google Scholar]
  23. Dümbgen L, Johns RB. 2004.. Confidence bands for isotonic median curves using sign tests. . J. Comput. Graph. Stat. 13:(2):51933
    [Crossref] [Google Scholar]
  24. Dümbgen L, Kolesnyk P, Wilke RA. 2017.. Bi-log-concave distribution functions. . J. Stat. Plan. Inference 184::117
    [Crossref] [Google Scholar]
  25. Dümbgen L, Lüthi L. 2022.. Honest confidence bands for isotonic quantile curves. . arXiv:2206.13069 [math.ST]
  26. Dümbgen L, Mösching A. 2023.. On stochastic orders and total positivity. . ESAIM Probab. Stat. 27:(1):46181
    [Crossref] [Google Scholar]
  27. Dümbgen L, Mösching A, Strähl C. 2021.. Active set algorithms for estimating shape-constrained density ratios. . Comput. Stat. Data Anal. 163::107300
    [Crossref] [Google Scholar]
  28. Dümbgen L, Rufibach K. 2009.. Maximum likelihood estimation of a log-concave density and its distribution function: basic properties and uniform consistency. . Bernoulli 15:(1):4068
    [Crossref] [Google Scholar]
  29. Dümbgen L, Rufibach K. 2011.. logcondens: Computations related to univariate log-concave density estimation. . J. Stat. Softw. 39:(6):128
    [Crossref] [Google Scholar]
  30. Dümbgen L, Samworth R, Schuhmacher D. 2011.. Approximation by log-concave distributions, with applications to regression. . Ann. Stat. 39:(2):70230
    [Crossref] [Google Scholar]
  31. Dykstra RL, Robertson T. 1982.. An algorithm for isotonic regression for two or more independent variables. . Ann. Stat. 10:(3):70816
    [Crossref] [Google Scholar]
  32. Fan J, Gijbels I. 1996.. Local Polynomial Modelling and Its Applications. London:: Chapman & Hall
  33. Feng OY, Guntuboyina A, Kim AKH, Samworth RJ. 2021.. Adaptation in multivariate log-concave density estimation. . Ann. Stat. 49:(1):12953
    [Crossref] [Google Scholar]
  34. Green PJ, Silverman BW. 1994.. Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. London:: Chapman & Hall
  35. Grenander U. 1956.. On the theory of mortality measurement. Part II. . Skand. Aktuarietidskr. 39::12553
    [Google Scholar]
  36. Groeneboom P. 1985.. Estimating a monotone density. . In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer II, pp. 53955 Belmont, CA:: Wadsworth
    [Google Scholar]
  37. Groeneboom P, Hendrickx K. 2018.. Current status linear regression. . Ann. Stat. 46:(4):141544
    [Crossref] [Google Scholar]
  38. Groeneboom P, Jongbloed G. 2014.. Nonparametric Estimation Under Shape Constraints—Estimators, Algorithms and Asymptotics. Cambridge, UK:: Cambridge Univ. Press
  39. Groeneboom P, Jongbloed G, Wellner JA. 2001.. Estimation of a convex function: characterizations and asymptotic theory. . Ann. Stat. 29:(6):165398
    [Google Scholar]
  40. Guntuboyina A, Sen B. 2018.. Nonparametric shape-restricted regression. . Stat. Sci. 33:(4):56894
    [Crossref] [Google Scholar]
  41. Han Q, Wellner JA. 2016.. Approximation and estimation of s-concave densities via Rényi divergences. . Ann. Stat. 44:(3):133259
    [Crossref] [Google Scholar]
  42. Hengartner NW, Stark PB. 1995.. Finite-sample confidence envelopes for shape-restricted densities. . Ann. Stat. 23:(2):52550
    [Crossref] [Google Scholar]
  43. Henzi A, Kleger GR, Ziegel JF. 2023.. Distributional (single) index models. . J. Am. Stat. Assoc. 118:(541):489503
    [Crossref] [Google Scholar]
  44. Henzi A, Mösching A, Dümbgen L. 2022.. Accelerating the pool-adjacent-violators algorithm for isotonic distributional regression. . Methodol. Comput. Appl. Probab. 24::263345
    [Crossref] [Google Scholar]
  45. Henzi A, Ziegel JF, Gneiting T. 2021.. Isotonic distributional regression. . J. R. Stat. Soc. Ser. B 83:(5):96393
    [Crossref] [Google Scholar]
  46. Hildreth C. 1954.. Point estimates of ordinates of concave functions. . J. Am. Stat. Assoc. 49::598619
    [Crossref] [Google Scholar]
  47. Hothorn T, Kneib T, Bühlmann P. 2014.. Conditional transformation models. . J. R. Stat. Soc. Ser. B 76:(1):327
    [Crossref] [Google Scholar]
  48. Kim AKH, Samworth RJ. 2016.. Global rates of convergence in log-concave density estimation. . Ann. Stat. 44:(6):275679
    [Google Scholar]
  49. Kneib T. 2013.. Beyond mean regression. . Stat. Model. 13:(4):275303
    [Crossref] [Google Scholar]
  50. Koenker R, Bassett G Jr. 1978.. Regression quantiles. . Econometrica 46:(1):3350
    [Crossref] [Google Scholar]
  51. Kuczmarski RJ, Ogden CL, Guo SS, Grummer-Strawn LM, Flegal KM, et al. 2002.. 2000 CDC Growth Charts for the United States: methods and development. . Vital Health Stat. 11:(246):1190
    [Google Scholar]
  52. Laha N, Miao Z, Wellner JA. 2021.. Bi-s*-concave distributions. . J. Stat. Plan. Inference 215::12757
    [Crossref] [Google Scholar]
  53. Mammen E. 1991.. Nonparametric regression under qualitative smoothness assumptions. . Ann. Stat. 19:(2):74159
    [Google Scholar]
  54. Mammen E, Yu K. 2007.. Additive isotone regression. . In Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom, ed. E Cator, G Jongbloed, C Kraaikamp, R Lopuhaä, JA Wellner , pp. 17995 Beachwood, OH:: Inst. Math. Stat.
    [Google Scholar]
  55. Mazumder R, Choudhury A, Iyengar G, Sen B. 2019.. A computational framework for multivariate convex regression and its variants. . J. Am. Stat. Assoc. 114:(525):31831
    [Crossref] [Google Scholar]
  56. Meyer MC. 2013.. Semi-parametric additive constrained regression. . J. Nonparametr. Stat. 25:(3):71530
    [Crossref] [Google Scholar]
  57. Mösching A, Dümbgen L. 2020.. Monotone least squares and isotonic quantiles. . Electron. J. Stat. 14:(1):2449
    [Crossref] [Google Scholar]
  58. Mösching A, Dümbgen L. 2022.. Estimation of a likelihood ratio ordered family of distributions. . arXiv:2007.11521 [math.ST]
  59. Müller S, Rufibach K. 2009.. Smooth tail-index estimation. . J. Stat. Comput. Simul. 79:(9–10):115567
    [Crossref] [Google Scholar]
  60. Prakasa Rao BLS. 1969.. Estimation of a unimodal density. . Sankhyā Ser. A 31::2336
    [Google Scholar]
  61. Rathke F, Schnörr C. 2019.. Fast multivariate log-concave density estimation. . Comput. Stat. Data Anal. 140::4158
    [Crossref] [Google Scholar]
  62. Robertson T, Wright FT, Dykstra RL. 1988.. Order Restricted Statistical Inference. New York:: Wiley
  63. Robeva E, Sturmfels B, Tran N, Uhler C. 2021.. Maximum likelihood estimation for totally positive log-concave densities. Scand. . J. Stat. 48:(3):81744
    [Google Scholar]
  64. Ryan TP. 2009.. Modern Regression Methods. New York:: Wiley, 2nd ed.
  65. Samworth RJ. 2018.. Recent progress in log-concave density estimation. . Stat. Sci. 33:(4):493509
    [Google Scholar]
  66. Saumard A. 2019.. Bi-log-concavity: some properties and some remarks towards a multi-dimensional extension. . Electron. Commun. Probab. 24:(61):18
    [Google Scholar]
  67. Shaked M, Shanthikumar JG. 2007.. Stochastic Orders. New York:: Springer
  68. Stout QF. 2015.. Isotonic regression for multiple independent variables. . Algorithmica 71:(2):45070
    [Crossref] [Google Scholar]
  69. Walther G. 2002.. Detecting the presence of mixing with multiscale maximum likelihood. . J. Am. Stat. Assoc. 97:(458):50813
    [Crossref] [Google Scholar]
  70. Walther G. 2009.. Inference and modeling with log-concave distributions. . Stat. Sci. 24:(3):31927
    [Crossref] [Google Scholar]
  71. Walther G, Ali A, Shen X, Boyd S. 2022.. Confidence bands for a log-concave density. . J. Comput. Graph. Stat. 31:(4):142638
    [Crossref] [Google Scholar]
  72. Woolridge J. 2002.. Ceosal2: instructional Stata data sets for econometrics. Database , Dep. Econ., Boston Coll., Boston, MA:
  73. Yang F, Barber RF. 2019.. Contraction and uniform convergence of isotonic regression. . Electron. J. Stat. 13:(1):64677
    [Crossref] [Google Scholar]
/content/journals/10.1146/annurev-statistics-033021-014937
Loading
/content/journals/10.1146/annurev-statistics-033021-014937
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error