Minimax estimation in systems of observation with Markovian chains by integral criterion

Abstract

A problem of estimation of states and parameters in stochastic dynamic systems of observation with discrete time containing a Markovian chain is studied. Matrices of transient probabilities and observation plans are random with unknown distribution with a given compact carrier. Observations, on the basis of which the estimation is made, are available at a fixed interval of time [0, T]. As a loss function, we have a conditional mathematical expectation with respect to the available observations of 2-norm of the estimation error of a signal process on [0, T]. The problem is in constructing an estimate minimizing losses correspondent to the worst distribution of the pair “a matrix of transient probabilities—a matrix of observation plan” form a set of allowable distributions. For a correspondent minimax problem is demonstrated the existence of a saddle point and is obtained a form of the wanted minimax estimation. The applicability of the obtained results is illustrated by a numerical example of the estimation of a state of TCP under the conditions of uncertainty of communication channel parameters.

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Correspondence to V. Borisov.

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Original Russian Text © A.V. Borisov, A.V. Bosov, A.I. Stefanovich, 2011, published in Avtomatika i Telemekhanika, 2011, No. 2, pp. 41–55.

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Borisov, V., Bosov, A.V. & Stefanovich, A.I. Minimax estimation in systems of observation with Markovian chains by integral criterion. Autom Remote Control 72, 255–268 (2011). https://doi.org/10.1134/S0005117911020056

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Keywords

  • Saddle Point
  • Remote Control
  • Hide Markov Model
  • Loss Function
  • Observation Plan