Computational Statistics

, Volume 33, Issue 2, pp 639–658 | Cite as

New designs to consistently estimate the isotonic regression

  • Ana Colubi
  • J. Santos Dominguez-Menchero
  • Gil Gonzalez-Rodriguez
Original Paper
  • 12 Downloads

Abstract

The usual estimators of the regression under isotonicity assumptions are too sensitive at the tails. In order to avoid this problem, some new strategies for fixed designs are analyzed. The uniform consistency of certain estimators on a closed and bounded working interval are obtained. It is shown that the usual isotonic regression can be employed when the number of observations at the edges of the interval is suitably controlled. Moreover, two modifications are proposed which substantially improve the results. One modification is based on the reallocation of part of the edge observations, and the other one forces the isotonic regression to take values within some horizontal bands. The theoretical results are complemented with some examples and simulation studies that illustrate the performance of the proposed estimators in practice.

Keywords

Isotonic regression Estimation Uniform consistency Designs 

Notes

Acknowledgements

The research in this paper has been partially supported by MTM2013-44212-P, GRUPIN14-005 and the COST Action IC1408.

References

  1. Alvarez E, Yohai V (2012) M-estimators for isotonic regression. J Stat Plan Inference 142:2351–2368MathSciNetCrossRefMATHGoogle Scholar
  2. Ayer M, Brunk HD, Ewing WT, Reid WT, Silverman E (1955) An empirical distribution function for sampling with incomplete information. Ann Math Stat 26:641–647MathSciNetCrossRefMATHGoogle Scholar
  3. Barlow RE, Bartholomew D, Bremner JM, Brunk HD (1972) Statistical inference under order restrictions: the theory and applications of isotonic regression. Wiley, New YorkMATHGoogle Scholar
  4. Brunk HD (1970) Nonparametric techniques in statistical inference, chap. estimation of isotonic regression. Cambridge University Press, London, pp 177–195Google Scholar
  5. Colubi A, Domínguez-Menchero JS, González-Rodríguez G (2006) Testing constancy for isotonic regressions. Scand J Stat 33:463–475MathSciNetCrossRefMATHGoogle Scholar
  6. Colubi A, Domínguez-Menchero JS, González-Rodríguez G (2007) A test for constancy of isotonic regressions using the \({L}_2\)-norm. Stat Sin 17:713–724MATHGoogle Scholar
  7. Cuesta-Albertos JA, Domínguez-Menchero JS, Matrán C (1995) Consistency of \({L}_p\)-best monotone approximations. J Stat Plan Inference 47:295–318CrossRefMATHGoogle Scholar
  8. de Leeuw J, Hornik K, Mair P (2009) Isotone optimization in R: pool-adjacent-violators algorithm (PAVA) and active set methods. J Stat Softw 32:1–24CrossRefGoogle Scholar
  9. Domínguez-Menchero JS, González-Rodríguez G (2007) Analyzing an extension of the isotonic regression problem. Metrika 66:19–30MathSciNetCrossRefMATHGoogle Scholar
  10. Domínguez-Menchero JS, González-Rodríguez G, López-Palomo MJ (2005) An \({L}_2\) point of view in testing monotone regression. Nonparametric Stat 17:135–153MathSciNetCrossRefMATHGoogle Scholar
  11. Domínguez-Menchero JS, López-Palomo MJ (1995) On the estimation of monotone uniform approximations. Stat Probab Lett 35:355–362MathSciNetCrossRefMATHGoogle Scholar
  12. Groeneboom P, Jongbloed G (2010) Generalized continuous isotonic regression. Stat Probab Lett 80:248–253MathSciNetCrossRefMATHGoogle Scholar
  13. Hanson DL, Pledger G, Wright FT (1973) On consistency in monotonic regression. Ann Stat 1:401–421MathSciNetCrossRefMATHGoogle Scholar
  14. Mammen E (1991) Estimating a smooth monotone regression function. Ann Stat 19:724–740MathSciNetCrossRefMATHGoogle Scholar
  15. Mammen E, Thomas-Agnan C (1999) Smoothing splines and shape restrictions. Scand J Stat 26:239–252MathSciNetCrossRefMATHGoogle Scholar
  16. Meyer M, Woodroofe M (2000) On the degrees of freedom in shape-restricted regression. Ann Stat 28:1083–1104MathSciNetCrossRefMATHGoogle Scholar
  17. Mukerjee H (1988) Monotone nonparametric regression. Ann Stat 16:741–75CrossRefMATHGoogle Scholar
  18. Pal JK (2008) Spiking problem in monotone regression: penalized residual sum of squares. Stat Probab Lett 78:1548–1556MathSciNetCrossRefMATHGoogle Scholar
  19. Pal JK, Woodroofe M (2007) Large sample properties of shape restricted regression estimators with smoothness adjustments. Stat Sin 17:1601–1616MathSciNetMATHGoogle Scholar
  20. Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New YorkMATHGoogle Scholar
  21. Sampson AR, Singh H, Whitaker L (2009) Simultaneous confidence bands for isotonic functions. J Stat Plan Inference 139:828–842MathSciNetCrossRefMATHGoogle Scholar
  22. Tibshirani RJ, Hoefling H, Tibshirani R (2011) Nearly-isotonic regression. Technometrics 53:54–61MathSciNetCrossRefGoogle Scholar
  23. Wang X, Li F (2008) Isotonic smoothing spline regression. J Comput Graph Stat 17:281–287MathSciNetGoogle Scholar
  24. Wang Y, Huang J (2002) Limiting distribution for monotone median regression. J Stat Plan Inference 107:281–287MathSciNetCrossRefMATHGoogle Scholar
  25. Wright FT (1981) The asymptotic behavior of monotone regression estimates. Ann Stat 9:443–448MathSciNetCrossRefMATHGoogle Scholar
  26. Wu WB, Woodroofe M, Mentz G (2001) Isotonic regression: another look at the changepoint problem. Biometrika 88:793–804MathSciNetCrossRefMATHGoogle Scholar
  27. Yeganova L, Wilbur WJ (2009) Isotonic regression under Lipschitz constraint. J Optim Theory Appl 141:429–443MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dpto. de Estadistica e I.O. y D.M.Universidad de OviedoOviedoSpain

Personalised recommendations