Advertisement

Cumulative Distribution Estimation via Control Theoretic Smoothing Splines

  • Janelle K. Charles
  • Shan Sun
  • Clyde F. Martin

Summary

In this paper, we explore the relationship between control theory and statistics. Specifically, we consider the use of cubic monotone control theoretic smoothing splines in estimating the cumulative distribution function (CDF) defined on a finite interval [0,T]. The spline construction is obtained by imposing an infinite dimensional, non-negativity constraint on the derivative of the optimal curve. The main theorem of this paper states that the optimal curve y(t) is a piecewise polynomial of known degree with y(0) = 0 and y(T) = 1. The solution is determined through dynamic programming which takes advantage of a finite reparametrization of the problem.

References

  1. 1.
    Charles, J.K.: Probability Distribution Estimation using Control Theoretic Smoothing Splines. Dissertation, Texas Tech University (2009)Google Scholar
  2. 2.
    Eubank, R.L.: Nonparametric Regression and Spline Smoothing. Statistics: Textbooks and Monographs, vol. 157. Marcel Dekker, Inc., New York (1999)zbMATHGoogle Scholar
  3. 3.
    Egerstedt, M., Martin, C.F.: Monotone Smoothing Splines. In: Mathematical Theory of Networks and Systems, Perpignan, France (2000)Google Scholar
  4. 4.
    Egerstedt, M., Martin, C.F.: Control Theoretic Splines: Optimal Control, Statistics, and Path Planning. Princeton University Press (in press)Google Scholar
  5. 5.
    Luenbeger, D.G.: Optimization by Vector Space Methods. John Wiley & Sons, New York (1969)Google Scholar
  6. 6.
    Nagahara, M., Sato, K., Yamamoto, Y.: H  ∞  Optimal Nonparametric Density Estimation from Quantized Samples. Submitted to ISCIE SSSGoogle Scholar
  7. 7.
    Martin, C.F., Egerstedt, M.: Trajectory Planning for Linear Control Systems with Generalized Splines. In: Mathematical Theory of Networks and Systems, Padova, Italy (1998)Google Scholar
  8. 8.
    Meyer, M.C.: Inference Using Shape-Restricted Regression Splines. Annals of Applied Statistics 2(3), 1013–1033 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Silverman, B.W.: Density Esimation for Statistics and Data Analysis. Chapman and Hall, London (1986)Google Scholar
  10. 10.
    Silverman, B.W.: Spline Smoothing: The Equivalent Variable Kernel Method. Ann. Statist. 12, 898–916 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Silverman, B.W.: Some Aspects of the Spline Smoothing Approach Nonparametric Regression Curve Fitting. J. Royal Statist, Soc. B, 1–52 (1985)Google Scholar
  12. 12.
    Wahba, G.: Spline Models for Observational Data. In: CBMS-NSF Regional Conference Series in Applied Mathematics, 59, SIAM, SIAM, Philadelphia (1990)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2010

Authors and Affiliations

  • Janelle K. Charles
    • 1
  • Shan Sun
    • 2
  • Clyde F. Martin
    • 1
  1. 1.Department of Mathematics and StatisticsTexas Tech UniversityLubbockUSA
  2. 2.Department of MathematicsUniversity of Texas at ArlingtonArlingtonUSA

Personalised recommendations