Variational Problems Arising in Statistics
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.
KeywordsVariational Problem Maximum Likelihood Estimator Nonparametric Regression Polynomial Spline Minimax Estimator
Unable to display preview. Download preview PDF.
- B. T. Polyak, Ya. Z. Tsypkin, Optimal pseudogradient adaptation algorithms, Autom. and Remote Contr., 41 (1981), pp. 1101–1110.Google Scholar
- B. T. Polyak, Ya. Z. Tsypkin, Robust pseudogradient adaptation algorithms, Autom. and Remote Contr., 41 (1981), pp. 1404–1409.Google Scholar
- 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
- 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
- B. T. Polyak, Ya, Z. Tsypkin,Optimal and robust estimation of autoregression coefficients, Engrg. Cybern., 21 (1983), No. 1.Google Scholar
- L. Devroye, L. Gyorgi, Nonparametric density estimation: L1-view, Wiley, New York, 1985.Google Scholar