Variational Problems Arising in Statistics

  • B. T. Polyak
Part of the Progress in Systems and Control Theory book series (PSCT, volume 2)

Abstract

Mathematical statistics is a nice source of nonstandard variational problems. As an example we can mention the famous Neyman—Pearson lemma on hypothesis testing: in optimization language it is a variational problem with integral type functional (not including derivatives) subject to specific constraints. In this paper we deal with variational problems of another kind arising in such areas of statistics as parameter estimation and nonparametric regression. Among them there are such nonstandard problems as minimization of a functional which is a ratio of two integrals, minimization of a matrix-valued criteria, finding a saddle point of a functional over some classes of functions etc. Some of these problems can be solved in explicit form by use of a technique which is untypical for the classical calculus of variations.

Keywords

Variational Problem Maximum Likelihood Estimator Nonparametric Regression Polynomial Spline Minimax Estimator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. J. Huber, Robust estimation of a location parameter, Ann. Math. Stat., 35 (1964), pp. 13–101.CrossRefGoogle Scholar
  2. [2]
    P. J. Huber, Robust statistics, Wiley, New York, 1981.CrossRefGoogle Scholar
  3. [3]
    B. T. Polyak, Ya. Z. Tsypkin, Optimal pseudogradient adaptation algorithms, Autom. and Remote Contr., 41 (1981), pp. 1101–1110.Google Scholar
  4. [4]
    B. T. Polyak, Ya. Z. Tsypkin, Robust pseudogradient adaptation algorithms, Autom. and Remote Contr., 41 (1981), pp. 1404–1409.Google Scholar
  5. [5]
    B. T. Polyak, Ya. Z. Tsypkin, Robust identification, Automatica, 16 (1980), pp. 53–69.CrossRefGoogle Scholar
  6. [6]
    A. S. Nemirovskii, B. T. Polyak, A. B. Tsybakov, Estimators of maximum likelihood type for nonparametric regression, Soviet Math. Dokl., 28 (1983), pp. 788–792.Google Scholar
  7. [7]
    A. S. Nemirovskii, B. T. Polyak, A. B. Tsybakov, Signal processing by the nonparametric maximum likelihood method, Probl. Inform. Transmiss. 20 (1984), pp. 177–191.Google Scholar
  8. [8]
    C. H. Reinsch, Smoothing by spline functions I, II, Numer. Math. 10 (1967), pp. 177–183 and 16 (1971), pp. 451–454.CrossRefGoogle Scholar
  9. [9]
    B. T. Polyak, Ya, Z. Tsypkin,Optimal and robust estimation of autoregression coefficients, Engrg. Cybern., 21 (1983), No. 1.Google Scholar
  10. [10]
    L. Devroye, L. Gyorgi, Nonparametric density estimation: L1-view, Wiley, New York, 1985.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • B. T. Polyak
    • 1
  1. 1.Institute of Control ProblemsMoscow, B 279USSR

Personalised recommendations