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On the Exact Distributions of Pattern Statistics for a Sequence of Binary Trials: A Combinatorial Approach

In: Handbook of Scan Statistics

Author

Listed:
  • Frosso S. Makri

    (University of Patras, Department of Mathematics)

  • Zaharias M. Psillakis

    (University of Patras, Department of Physics)

Abstract

Consider a sequence of exchangeable or Markov-dependent binary (zero-one) trials. A sequence of independent and identically distributed binary trials is covered as a particular case of both the prementioned ones. For counting/waiting time pattern statistics defined on such model sequences, we point out how their exact probability distributions can be established using enumerative combinatorics. The expressions for the distributions contain probabilities depending on the internal structure of the model sequence and combinatorial numbers denoting set cardinalities. The latter numbers depend on the considered pattern statistics and the number of ones, for an exchangeable sequence, as well as the number of runs of ones, for a Markov-dependent sequence. These numbers become concrete when certain patterns and enumerative schemes are studied on the model sequences. Exact distributions for statistics connected to patterns of limited length, as well as to certain runs and scans, are provided using proper combinatorial numbers and exemplify the approach.

Suggested Citation

  • Frosso S. Makri & Zaharias M. Psillakis, 2024. "On the Exact Distributions of Pattern Statistics for a Sequence of Binary Trials: A Combinatorial Approach," Springer Books, in: Joseph Glaz & Markos V. Koutras (ed.), Handbook of Scan Statistics, chapter 18, pages 339-358, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-8033-4_48
    DOI: 10.1007/978-1-4614-8033-4_48
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