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On a random number of disorders

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  • Szajowski, Krzysztof

Abstract

We register a random sequence which has the following properties: it has three segments being the homogeneous Markov processes. Each segment has his own one step transition probability law and the length of the segment is unknown and random. It means that at two random successive moments (they can be equal also and equal zero too) the source of observations is changed and the first observation in new segment is chosen according to new transition probability starting from the last state of the previous segment. In effect the number of homogeneous segments is random. The transition probabilities of each process are known and a priori distribution of the disorder moments is given. The former research on such problem has been devoted to various questions concerning the distribution changes. The random number of distributional segments creates new problems in solutions with relation to analysis of the model with deterministic number of segments. Two cases are presented in details. In the first one the objectives is to stop on or between the disorder moments while in the second one our objective is to find the strategy which immediately detects the distribution changes. Both problems are reformulated to optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of optimal decision function.

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File URL: http://mpra.ub.uni-muenchen.de/20256/
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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 20256.

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Date of creation: 23 Nov 2008
Date of revision: 02 Jan 2010
Handle: RePEc:pra:mprapa:20256

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Keywords: disorder problem; sequential detection; optimal stopping; Markov process; change point; double optimal stopping;

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  1. Bojdecki, Tomasz & Hosza, Jerzy, 1984. "On a generalized disorder problem," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 18(2), pages 349-359, November.
  2. Sarnowski, Wojciech & Szajowski, Krzysztof, 2008. "On-line detection of a part of a sequence with unspecified distribution," Statistics & Probability Letters, Elsevier, Elsevier, vol. 78(15), pages 2511-2516, October.
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