Electronic Journal of Statistics

Distribution-free properties of isotonic regression

Jake A. Soloff, Adityanand Guntuboyina, and Jim Pitman

Full-text: Open access

Abstract

It is well known that the isotonic least squares estimator is characterized as the derivative of the greatest convex minorant of a random walk. Provided the walk has exchangeable increments, we prove that the slopes of the greatest convex minorant are distributed as order statistics of the running averages. This result implies an exact non-asymptotic formula for the squared error risk of least squares in homoscedastic isotonic regression when the true sequence is constant that holds for every exchangeable error distribution.

Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 3243-3253.

Dates
Received: February 2019
First available in Project Euclid: 24 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1569290688

Digital Object Identifier
doi:10.1214/19-EJS1594

Keywords
Shape-constrained regression statistical dimension convex minorant fluctuation theory quantiles of stochastic processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Soloff, Jake A.; Guntuboyina, Adityanand; Pitman, Jim. Distribution-free properties of isotonic regression. Electron. J. Statist. 13 (2019), no. 2, 3243--3253. doi:10.1214/19-EJS1594. https://projecteuclid.org/euclid.ejs/1569290688


Export citation

References

  • [1] J. Abramson, J. Pitman, N. Ross, and G. U. Bravo. Convex minorants of random walks and Lévy processes., Electronic Communications in Probability, 16:423–434, 2011.
  • [2] D. Amelunxen, M. Lotz, M. B. McCoy, and J. A. Tropp. Living on the edge: Phase transitions in convex programs with random data., Information and Inference: A Journal of the IMA, 3(3):224–294, 2014.
  • [3] P. C. Bellec. Sharp oracle inequalities for least squares estimators in shape restricted regression., The Annals of Statistics, 46(2):745–780, 2018.
  • [4] H. D. Brunk, R. E. Barlow, D. J. Bartholomew, and J. M. Bremner., Statistical inference under order restrictions; the theory and application of isotonic regression. Wiley, New York, 1972.
  • [5] C. Carolan and R. Dykstra. Asymptotic behavior of the grenander estimator at density flat regions., Canadian Journal of Statistics, 27(3):557–566, 1999.
  • [6] C. Carolan and R. Dykstra. Marginal densities of the least concave majorant of Brownian motion., Ann. Statist., 29(6) :1732–1750, 2001.
  • [7] H. S. M. Coxeter and W. O. J. Moser., Generators and relations for discrete groups, volume 14. Springer Science & Business Media, 2013.
  • [8] A. Dassios. On the quantiles of Brownian motion and their hitting times., Bernoulli, 11(1):29–36, 2005.
  • [9] B. Fang and A. Guntuboyina. On the risk of convex-constrained least squares estimators under misspecification., Bernoulli, 25(3) :2206–2244, 2019.
  • [10] P. Groeneboom, G. Jongbloed, and J. A. Wellner. Estimation of a convex function: characterizations and asymptotic theory., The Annals of Statistics, 29(6) :1653–1698, 2001.
  • [11] S. J. Grotzinger and C. Witzgall. Projections onto order simplexes., Applied mathematics and Optimization, 12(1):247–270, 1984.
  • [12] A. Guntuboyina and B. Sen. Nonparametric shape-restricted regression., Statistical Science, 33(4):568–594, 2018.
  • [13] O. Kallenberg., Foundations of modern probability. Springer Science & Business Media, 2006.
  • [14] F. B. Knight. The uniform law for exchangeable and Lévy process bridges., Astérisque, (236):171–188, 1996.
  • [15] A. B. Németh and S. Z. Németh. How to project onto the monotone nonnegative cone using pool adjacent violators type algorithms., arXiv preprint arXiv :1201.2343, 2012.
  • [16] S. Oymak and B. Hassibi. Sharp mse bounds for proximal denoising., Foundations of Computational Mathematics, 16(4):965 –1029, 2016.
  • [17] T. Robertson, F. T. Wright, and R. L. Dysktra., Order restricted statistical inference. 1988.
  • [18] F. Spitzer and H. Widom. The circumference of a convex polygon., Proceedings of the American Mathematical Society, 12(3):506–509, 1961.
  • [19] E. H. Zarantonello. Projections on convex sets in Hilbert space and spectral theory. In, Contributions to Nonlinear Functional Analysis, 237–424, 1971.
  • [20] C.-H. Zhang. Risk bounds in isotonic regression., The Annals of Statistics, 30(2):528–555, 2002.