Student or educator? Sign up with your school email and get 6 months of Pro free.
CoChat
Sign Up
Cited by 3,666

The DOI matches its Crossref record; 3 of the 3 references it cites were found in a registry or an index.

Flexible smoothing with B-splines and penalties

Open access
Eilers, Paul H. C., Marx, Brian D.·1996·Statistical Science
DOI 10.1214/ss/1038425655
Open this paper in CoChat — free account

Abstract

B-splines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent B-splines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of B-splines, of their construction and of penalized likelihood is presented. We discuss properties of penalized B-splines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.

Overview

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1214/ss/1038425655

Cites

3 of 3 references verified0 citation errors · 0 not checkable

Statistics

Verification

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Citations

Cites

Show all 3 references

Research works printed but not linked: 58

  1. Silverman, B. W. (1986). Density Estimation for Statistics and Data Analy sis. Chapman and Hall, London. · doi:10.1007/978-1-4899-3324-9 · Paper page
  2. Scott, D. W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, New York. · DOI · Paper page
  3. Rice, J. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12 1215-1230. · DOI · Paper page
  4. Fan, J. and Gijbels, I. (1995). Data-driven bandwidth selection in local poly nomial fitting: variable bandwidth and spatial adaptation. J. Roy. Statist. Soc. Ser. B 57 371-394. · DOI · Paper page
  5. Fan, J. and Marron, J. S. (1994). Fast implementations of nonparametric curve estimators. J. Comput. Graph. Statist. 3 35-56. · doi:10.1080/10618600.1994.10474629
  6. Fan, J. (1993). Local linear regression smoothers and their minimax efficiency. Ann. Statist. 21 196-216. · DOI · Paper page
  7. Mallows, C. (1973). Some comments on cp. Technometrics 15 661-675. · DOI · Paper page
  8. Kneip, A. (1994). Ordered linear smoothers. Ann. Statist. 22 835-866. · doi:10.1214/aos/1176325498
  9. Hjort, N. L. and Jones, M. C. (1996). Locally parametric nonparametric density estimation. Ann. Statist. 24 1619-1647. · doi:10.1214/aos/1032298288
  10. Ruppert, D., Sheather, S. J. and Wand, M. P. (1995). An effective bandwidth selector for local least squares regression. J. Amer. Statist. Assoc. 90 1257-1270. · doi:10.2307/2291516
  11. Scott, D. W. and Terrell, G. R. (1987). Biased and unbiased cross-validation in density estimation. J. Amer. Statist. Assoc. 82 1131-1146. · DOI
  12. Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models. Chapman and Hall, London. · DOI · Paper page
  13. Simonoff, J. S. (1996). Smoothing Methods in Statistics. Springer, New York. · DOI · Paper page
  14. Wahba, G. (1978). Improper priors, spline smoothing and the problem of guarding against model errors in regression. J. Roy. Statist. Soc. Ser. B 40 364-372. · DOI · Paper page
  15. Friedman, J. H. (1991). Multivariate adaptive regression splines (with discussion). Ann. Statist. 19 1-141. · DOI · Paper page
  16. KOOPERBERG, C. and STONE, C. J. (1991). A study of logspline density estimation. Comput. Statist. Data Anal. 12 327-347. · DOI · Paper page
  17. Green, P. J. and Yandell, B. S. (1985). Semi-parametric generalized linear models. In Generalized Linear Models (B. Gilchrist et al., eds.). Springer, New York. Hand, D. J., Daly, F., Lunn, A. D., McConway, K. J. and Ostrowski, E. (1994). A Handbook of Small Data Sets. Chapman and Hall, London. · DOI
  18. Whittaker, E. T. (1923). On a new method of graduation. Proc. Edinburgh Math. Soc. 41 63-75. · DOI · Paper page
  19. Ashford, R. and Walker, P. J. (1972). Quantal response analysis for a mixture of populations. Biometrics 28 981-988. · doi:10.2307/2528634
  20. Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975). Discrete Multivariate Analy sis: Theory and Practice. MIT Press.
  21. Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatter plots. J. Amer. Statist. Assoc. 74 829-836. · DOI
  22. Cox, M. G. (1981). Practical spline approximation. In Topics in Numerical Analy sis (P. R. Turner, ed.). Springer, Berlin. · doi:10.1007/bfb0063201
  23. de Boor, C. (1977). Package for calculating with B-splines. SIAM J. Numer. Anal. 14 441-472. · doi:10.1137/0714026
  24. de Boor, C. (1978). A Practical Guide to Splines. Springer, Berlin. · DOI · Paper page
  25. Dierckx, P. (1993). Curve and Surface Fitting with Splines. Clarendon, Oxford. · DOI · Paper page
  26. Diggle P. and Marron J. S. (1988). Equivalence of smoothing parameter selectors in density and intensity estimation. J. Amer. Statist. Assoc. 83 793-800. · doi:10.2307/2289308
  27. Eilers, P. H. C. (1990). Smoothing and interpolation with generalized linear models. Quaderni di Statistica e Matematica Applicata alle Scienze Economico-Sociali 12 21-32. Eilers, P. H. C. (1991a). Penalized regression in action: estimating pollution roses from daily averages. Environmetrics 2 25-48. Eilers, P. H. C. (1991b). Nonparametric density estimation with grouped observations. Statist. Neerlandica 45 255-270. · DOI
  28. Eilers, P. H. C. (1995). Indirect observations, composite link models and penalized likelihood. In Statistical Modelling (G. U. H. Seeber et al., eds.). Springer, New York. · doi:10.1007/978-1-4612-0789-4_12
  29. Eilers, P. H. C. and Marx, B. D. (1992). Generalized linear models with P-splines. In Advances in GLIM and Statistical Modelling (L. Fahrmeir et al., eds.). Springer, New York. · doi:10.1007/978-1-4612-2952-0_12
  30. Friedman, J. and Silverman, B. W. (1989). Flexible parsimonious smoothing and additive modeling (with discussion). Technometrics 31 3-39. · doi:10.2307/1270359
  31. Hardle, W. (1990). Applied Nonparametric Regression. Cambridge Univ. Press. · DOI · Paper page
  32. Kooperberg, C. and Stone, C. J. (1992). Logspline density estimation for censored data. J. Comput. Graph. Statist. 1 301- 328. · DOI
  33. Marron, J. S. and Ruppert, D. (1994). Transformations to reduce boundary bias in kernel density estimation. J. Roy. Statist. Soc. Ser. B 56 653-671. · DOI · Paper page
  34. Marx, B. D. and Eilers, P. H. C. (1994). Direct generalized additive modelling with penalized likelihood. Paper presented at the 9th Workshop on Statistical Modelling, Exeter, 1994.
  35. O'Sullivan, F. (1986). A statistical perspective on ill-posed inverse problems (with discussion). Statist. Sci. 1 505-527. · DOI · Paper page
  36. O'Sullivan, F. (1988). Fast computation of fully automated logdensity and log-hazard estimators. SIAM J. Sci. Statist. Comput. 9 363-379. · DOI · Paper page
  37. Reinsch, C. (1967). Smoothing by spline functions. Numer. Math. 10 177-183. · DOI · Paper page
  38. Wand, M. P. and Jones, M. C. (1993). Kernel Smoothing. Chapman and Hall, London. · DOI · Paper page
  39. Chiu, S.-T. (1996). A comparative review of bandwidth selection for kernel density estimation. Statist. Sinica 6 129-145.
  40. Hall, P. and Johnstone, I. (1992). Empirical functionals and efficient smoothing parameter selection. J. Roy. Statist. Soc. Ser. B 54 519-521. · DOI
  41. Hall, P., Marron, J. S. and Park, B. U. (1992). Smoothed crossvalidation. Probab. Theory Related Fields 92 1-20. · DOI · Paper page
  42. Barry, D. (1993). Testing for additivity of a regression function. Ann. Statist. 21 235-254. · doi:10.1214/aos/1176349024
  43. Cox, D. D., Koh, E., Wahba, G. and Yandell, B. S. (1988). Testing the (parametric) null model hy pothesis in (semiparametric) partial and generalized spline models. Ann. Statist. 16 113-119. · doi:10.1214/aos/1176350693
  44. Gu, C. (1992). Cross validating non Gaussian data. Journal of Computational and Graphical Statistics 1 169-179. · DOI
  45. Wahba, G. (1983). Bayesian "confidence intervals" for the crossvalidated smoothing spline. J. Roy. Statist. Soc. Ser. B 45 133-150. · DOI · Paper page
  46. Ansley, C. F., Kohn, R. and Wong, C. M. (1993). Nonparametric spline regression with prior information. Biometrika 80 75- 88. · doi:10.1093/biomet/80.1.75
  47. Efron, B. and Tibshirani, R. (1996). Using specially designed exponential families for density estimation. Ann. Statist. 24 000-000. · DOI
  48. Jones, M. C. (1991). On correcting for variance inflation in kernel density estimation. Comput. Statist. Data Anal. 11 3-15. · doi:10.1016/0167-9473(91)90049-8
  49. Jones, M. C. (1996). On close relations of local likelihood density estimation. Unpublished manuscript. · doi:10.1007/bf02562622
  50. Loader, C. R. (1996). Local likelihood density estimation. Ann. Statist. 24 1602-1618 · DOI · Paper page
  51. Marron, J. S. (1996). A personal view of smoothing and statistics (with discussion). Comput. Statist. To appear. · doi:10.1007/978-3-642-48425-4_1
  52. Cook, R. D. and Weisberg, S. (1994). Regression Graphics. Wiley, New York.
  53. Eilers, P. H. C. (1988). Autoregressive models with latent variables. In COMPSTAT 1988 Proceedings (D. Edwards and N. E. Raun, eds.). physica-Verlag. · doi:10.1007/978-3-642-46900-8_50
  54. Engel, J. and Gasser, T. (1995). A minimax result for a class of nonparametric density estimators. Nonparametric Statistics 4 327-334. · DOI
  55. Fan, J., Hall, P., Martin, M. A. and Patil, P. (1996). On local smoothing of nonparametric curve estimators. J. Amer. Statist. Assoc. 91 258-266. Foley, J. D., van Dam, A., Feiner, S. K. and Hughes, J. F. · doi:10.2307/2291403
  56. Jones, M. C., Marron, J. S. and Sheather, S. J. (1996). A brief survey of bandwidth selection for density estimation. J. Amer. Statist. Assoc. To appear. · DOI
  57. Kooperberg, C., Bose, S. and Stone, C. J. (1997). Poly chotomous regression. J. Amer. Statist. Assoc. To appear. Kooperberg, C., Stone, C. J. and Truong, Y. K. (1995a). Hazard regression. J. Amer. Statist. Assoc. 90 78-94. Kooperberg, C., Stone, C. J. and Truong, Y. K. (1995b). Logspline estimation of a possibly mixed spectral distribution. J. Time Ser. Anal. 16 359-388. · doi:10.1111/j.1467-9892.1995.tb00240.x
  58. Speckman, P. L. (1983). Spline smoothing and optimal rates of convergence in nonparametric regression models. Ann. Statist. 13 970-983. Stone, C. J., Hansen, M., Kooperberg, C. and Truong, Y. K. · DOI · Paper page

Cited by

See all 3,666

Metadata

Authors
Eilers, P. H. C., & Marx, B. D.
Venue
Statistical Science
Publisher
Institute of Mathematical Statistics
Published
1996-05-01
Volume
11
Issue
2
Pages
89-121
ISSN
0883-4237 (ISSN-L)
DOI
10.1214/ss/1038425655
OpenAlex
W1990420052
Cited by
3,666
References
3
FWCI
8.31
Type
journal article
Language
en
Open access
bronze

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review

How to cite

Cite CoChat for the checks on this page, and the paper for its own findings.

This page
CoChat. (2026). Citation check: Flexible smoothing with B-splines and penalties. https://app.cochat.ai/papers/10.1214/ss/1038425655
The paper
Eilers, P. H. C., & Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science. https://doi.org/10.1214/ss/1038425655

Data sources

Bibliographic data: Semantic Scholar, OpenAlex, Crossref.

Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).

Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.