Skip to main content

A posteriori minimax estimation with likelihood constraints

Abstract

Consideration was given to the problem of guaranteed estimation of the parameters of an uncertain stochastic regression. The loss function is the conditional mean squared error relative to the available observations. The uncertainty set is a subset of the probabilistic distributions lumped on a certain compact with additional linear constraints generated by the likelihood function. Solution of this estimation problem comes to determining the saddle point defining both the minimax estimator and the set of the corresponding worst distributions. The saddle point is the solution of a simpler finite-dimensional dual optimization problem. A numerical algorithm to solve this problem was presented, and its precision was determined. Model examples demonstrated the impact of the additional likelihood constraints on the estimation performance.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Kurzhanskii, A.B., Upravlenie i otsenivanie v usloviyakh neopredelennosti (Control and Estimation under Uncertainty), Moscow: Nauka, 1977.

    Google Scholar 

  2. 2.

    Pankov, A.R. and Semenikhin, K.V., Minimax Identification of the Generalized Uncertain Stochastic Linear Model, Autom. Remote Control, 1998, vol. 59, no. 11, part 2, pp. 1632–1643.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Solov’ev, V.N., On the Theory of Minimax Bayes Estimation, Teor. Veroyat. Primen., 1999, vol. 44, no. 4, pp. 738–756.

    Google Scholar 

  4. 4.

    Anan’ev, B.I., Minimax Estimation of Statistically Uncertain Systems under the Choice of a Feedback Parameter, J. Math. Syst., 1995, vol. 5, no. 2, pp. 1–17.

    MathSciNet  Google Scholar 

  5. 5.

    Calafiore, G. and El Ghaoui, L., Worst-case Maximum Likelihood Estimation in the Linear Model, Automatica, 2001, vol. 37, no. 4, pp. 573–580.

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Matasov, A.I., Estimators for Uncertain Dynamic Systems, Dordrecht: Kluwer, 1999.

    Google Scholar 

  7. 7.

    Poor, V. and Looze, D.P., Minimax State Estimation for Linear Stochastic Systems with Noise Uncertainty, IEEE Trans. Automat. Control, 1981, vol. 26, no. 4, pp. 902–906.

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Borovkov, A.A., Matematicheskaya statistika (Mathematical Statistics), Moscow: Fizmatlit, 2006.

    Google Scholar 

  9. 9.

    Blackwell, D.H. and Girshick, M.A., Theory of Games and Statistical Decisions, New York: Wiley, 1954.

    MATH  Google Scholar 

  10. 10.

    Berger, J.O., Statistical Decision Theory and Bayesian Analysis, New York: Springer, 1985.

    MATH  Google Scholar 

  11. 11.

    Martin, C.J. and Mintz, M., Robust Filtering and Prediction for Linear Systems with Uncertain Dynamics: A Game-Theoretic Approach, IEEE Trans. Automat. Control, 1983, vol. 28, no. 9, pp. 888–896.

    MATH  Article  Google Scholar 

  12. 12.

    Germeier, Yu.B., Igry s neprotivopolozhnymi interesami (Games with Nonantagonistic Interests), Moscow: Fizmatlit, 1976.

    Google Scholar 

  13. 13.

    Billingsley, P., Convergence of Probability Measures, New York: Wiley, 1968. Translated under the title Skhodimost’ veroyatnostnykh mer, Moscow: Nauka, 1977.

    MATH  Google Scholar 

  14. 14.

    Bulinskii, A.V. and Shiryaev, A.N., Teoriya sluchainykh protsessov (Theory of Random Processes), Moscow: Fizmatlit, 2005.

    Google Scholar 

  15. 15.

    Kats, I.Ya. and Kurzhanskii, A.B., Minimax Multistep Filtering in Statically Uncertain Situations, Autom. Remote Control, 1978, vol. 39, no. 11, part 1, pp. 1643–1650.

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Elliott, R.J., Aggoun, L., and Moore, J.B., Hidden Markov Models: Estimation and Control, New York: Springer, 1994.

    Google Scholar 

  17. 17.

    Petersen, I.R., James, M.R., and Dupuis, P., Minimax Optimal Control of Stochastic Uncertain Systems with Relative Entropy Constraints, IEEE Trans. Automat. Control, 2000, vol. 45, pp. 398–412.

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Xie, L., Ugrinovskii, V.A., and Petersen, I.R., Finite Horizon Robust State Estimation for Uncertain Finite-alphabet Hidden MarkovModels with Conditional Relative Entropy Constraints, SIAM J. Control Optim., 2008, vol. 47, no. 1, pp. 476–508.

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    El Karoui, N. and Jeanblanc-Picquè, M., Contrôle de procesuss de Markov, Lect. Notes Math., Berlin: Springer, 1988, vol. 1321.

    Google Scholar 

  20. 20.

    Lee, E.B. and Marcus, L., Foundation of Optimal Control Theory, New York: Wiley, 1967.

    Google Scholar 

  21. 21.

    Low, S., Paganini, F., and Doyle, J., Internet Congestion Control, IEEE Control Syst. Mag., 2002, vol. 22, no. 1, pp. 28–43.

    Article  Google Scholar 

  22. 22.

    Floyd, S. and Jacobson, V., Random Early Detection Gateways for Congestion Avoidance, ACM/IEEE Trans. Networking, 1993, vol. 1(4), no. 9, pp. 397–413.

    Article  Google Scholar 

  23. 23.

    Al’tman, E., Avrachenkov, K.E., and Miller, G.B., Power Control of Signal in Wireless Networks: A Survey of Discrete States, Obozr. Prikl. Prom. Mat., 2007, vol. 14, no. 2, pp. 193–214.

    MATH  Google Scholar 

  24. 24.

    Sliding Mode Control in Engineering, Perruquetti, W. and Barbot, J.P., Eds., New York: Marcel Dekker, 2002.

    Google Scholar 

  25. 25.

    Kirillov, A.A. and Gvishiani, A.D., Teoremy i zadachi funktsional’nogo analiza (Theorems and Problems of Functional Analysis), Moscow: Nauka, 1988.

    MATH  Google Scholar 

  26. 26.

    Aivazyan, S.A., Bukhshtaber, V.M., Enyukov, I.S., and Meshalkin, L.D., Prikladnaya statistika: klassifikatsiya i snizhenie razmernosti (Applied Statistics: Classification and Reduction of Dimensionality), Moscow: Finansy i Statistika, 1989.

    Google Scholar 

  27. 27.

    Podinovskii, V.V. and Nogin, V.D., Pareto-optimal’nye resheniya mnogokriterial’nykh zadach (Paretooptimal Solutions of the Multicriteria Problems), Moscow: Fizmatlit, 1982.

    Google Scholar 

  28. 28.

    Balakrishnan, A.V., Applied Functional Analysis, New York: Springer, 1976. Translated under the title Prikladnoi funktsional’nyi analiz, Moscow: Nauka, 1980.

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. V. Borisov.

Additional information

Original Russian Text © A.V. Borisov, 2012, published in Avtomatika i Telemekhanika, 2012, No. 9, pp. 49–71.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Borisov, A.V. A posteriori minimax estimation with likelihood constraints. Autom Remote Control 73, 1481–1497 (2012). https://doi.org/10.1134/S0005117912090044

Download citation

Keywords

  • Saddle Point
  • Remote Control
  • Loss Function
  • Dual Problem
  • Conditional Expectation