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Minimax Estimation in Regression under Sample Conformity Constraints

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  • Andrey Borisov

    (Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44/2 Vavilova Str., 119333 Moscow, Russia
    Moscow Aviation Institute, 4, Volokolamskoe Shosse, 125993 Moscow, Russia
    Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, 1-52 Leninskiye Gory, 119991 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University, GSP-1, Leninskie Gory, 119991 Moscow, Russia)

Abstract

The paper is devoted to the guaranteeing estimation of parameters in the uncertain stochastic nonlinear regression. The loss function is the conditional mean square of the estimation error given the available observations. The distribution of regression parameters is partially unknown, and the uncertainty is described by a subset of probability distributions with a known compact domain. The essential feature is the usage of some additional constraints describing the conformity of the uncertain distribution to the realized observation sample. The paper contains various examples of the conformity indices. The estimation task is formulated as the minimax optimization problem, which, in turn, is solved in terms of saddle points. The paper presents the characterization of both the optimal estimator and the set of least favorable distributions. The saddle points are found via the solution to a dual finite-dimensional optimization problem, which is simpler than the initial minimax problem. The paper proposes a numerical mesh procedure of the solution to the dual optimization problem. The interconnection between the least favorable distributions under the conformity constraint, and their Pareto efficiency in the sense of a vector criterion is also indicated. The influence of various conformity constraints on the estimation performance is demonstrated by the illustrative numerical examples.

Suggested Citation

  • Andrey Borisov, 2021. "Minimax Estimation in Regression under Sample Conformity Constraints," Mathematics, MDPI, vol. 9(10), pages 1-21, May.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:10:p:1080-:d:552072
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    References listed on IDEAS

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    1. Epstein, Larry G. & Ji, Shaolin, 2014. "Ambiguous volatility, possibility and utility in continuous time," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 269-282.
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    Cited by:

    1. Mikhail Posypkin & Andrey Gorshenin & Vladimir Titarev, 2022. "Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”," Mathematics, MDPI, vol. 10(13), pages 1-8, June.

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