COBS: qualitatively constrained smoothing via linear programming. Popular smoothing techniques generally have a difficult time accommodating qualitative constraints like monotonicity, convexity or boundary conditions on the fitted function. In this paper, we attempt to bring the problem of constrained spline smoothing to the foreground and describe the details of a constrained B-spline smoothing (COBS) algorithm that is being made available to S-plus-users. Recent work of He & Shi (1998) considered a special case and showed that the L 1 projection of a smooth function into the space of B-splines provides a monotone smoother that is flexible, efficient and achieves the optimal rate of convergence. Several options and generalizations are included in COBS: it can handle small or large data sets either with user interaction or full automation. Three examples are provided to show how COBS works in a variety of real-world applications.

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  1. Wu, Yan-ke; Hu, Ya-nan; Zhou, Jian; Tian, Mao-zai: Simultaneous estimation of multiple conditional regression quantiles (2020)
  2. Jhong, Jae-Hwan; Koo, Ja-Yong: Simultaneous estimation of quantile regression functions using B-splines and total variation penalty (2019)
  3. Rodrigues, T.; Dortet-Bernadet, J.-L.; Fan, Y.: Simultaneous Fitting of Bayesian penalised quantile splines (2019)
  4. Stasinopoulos, Mikis D.; Rigby, Robert A.; De Bastiani, Fernanda: GAMLSS: a distributional regression approach (2018)
  5. Mhalla, Linda; Chavez-Demoulin, Valérie; Naveau, Philippe: Non-linear models for extremal dependence (2017)
  6. Fengler, Matthias R.; Hin, Lin-Yee: Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints (2015)
  7. Cormier, Eric; Genest, Christian; Nešlehová, Johanna G.: Using B-splines for nonparametric inference on bivariate extreme-value copulas (2014)
  8. Baraldo, Stefano; Ieva, Francesca; Paganoni, Anna Maria; Vitelli, Valeria: Outcome prediction for heart failure telemonitoring via generalized linear models with functional covariates (2013)
  9. Qu, Long; Nettleton, Dan; Dekkers, Jack C. M.: Improved estimation of the noncentrality parameter distribution from a large number of (t)-statistics, with applications to false discovery rate estimation in microarray data analysis (2012)
  10. Xu, Ganggang; Xiang, Yanbiao; Wang, Suojin; Lin, Zhengyan: Regularization and variable selection for infinite variance autoregressive models (2012)
  11. Laurini, Márcio Poletti: Imposing no-arbitrage conditions in implied volatilities using constrained smoothing splines (2011)
  12. von Davier, Alina A. (ed.): Statistical models for test equating, scaling, and linking. With a foreword by Paul W. Holland. (2011)
  13. Laurini, Márcio Poletti; Moura, Marcelo: Constrained smoothing (B)-splines for the term structure of interest rates (2010)
  14. Lee, Jong Soo; Cox, Dennis D.: Robust smoothing: smoothing parameter selection and applications to fluorescence spectroscopy (2010)
  15. Shows, Justin Hall; Lu, Wenbin; Zhang, Hao Helen: Sparse estimation and inference for censored median regression (2010)
  16. Hyndman, Rob J.; Ullah, Md. Shahid: Robust forecasting of mortality and fertility rates: a functional data approach (2007)
  17. Leitenstorfer, Florian; Tutz, Gerhard: Knot selection by boosting techniques (2007)
  18. Ng, Pin; Maechler, Martin: A fast and efficient implementation of qualitatively constrained quantile smoothing splines (2007)
  19. Bollaerts, Kaatje; Eilers, Paul H. C.; Aerts, Marc: Quantile regression with monotonicity restrictions using (P)-splines and the (L_1)-norm (2006)
  20. Kim, Mi-Ok: Quantile regression with shape-constrained varying coefficients (2006)

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