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Strong consistency of a modified maximum likelihood estimator for controlled Markov chains

  • Bharat Doshi (a1) and Steven E. Shreve (a2)
Abstract

A controlled Markov chain with finite state space has transition probabilities which depend on an unknown parameter α lying in a known finite set A. For each α, a stationary control law ϕ α is given. This paper develops a control scheme whereby at each stage t a parameter α t is chosen at random from among those parameters which nearly maximize the log likelihood function, and the control ut is chosen according to the control law ϕ αt. It is proved that this algorithm leads to identification of the true α under conditions weaker than any previously considered.

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Corresponding author
Postal address: HP1B323, Holmdel, NJ 07733, U.S.A. Research carried out when the author was at Rutgers University.
∗∗ Postal address: Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.
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Research sponsored in part by the Air Force Office of Scientific Research (AFSC), United States Air Force, under Contract F-49620–79-C-0165.

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[1] Borkar, V. and Varaiya, P. (1979) Adaptive control of Markov chains, I: Finite parameter set. IEEE Trans. Auto. Control 24, 953957.
[2] Loève, M. (1960) Probability Theory Van Nostrand, Princeton, NJ.
[3] Mandl, P. (1974) Estimation and control in Markov chains. Adv. Appl. Prob. 6, 4060.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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