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On Multivariate Distributions of Various Orders Obtained by Waiting for the r-th Success Run of Length k in Trials With Multiple Outcomes

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Advances in Combinatorial Methods and Applications to Probability and Statistics

Part of the book series: Statistics for Industry and Technology ((SIT))

Abstract

A sequence of independent trials with m + 1 mutually exclusive outcomes S, F 1 , F 2,…, F m is considered until the occurrence of the r-th nonoverlapping success run of length k, and the distributions of related random and the distributions of related random vectors are derived. First a new genesis scheme is established for the multivariate negative binomial distribution of order k type I, of Philippou, Antzoulakos and Tripsiannis (1988). It is shown that it is the distribution of the sum of two random vectors: the i-th component of the first one is the number of occurrences of F i and the i-th component of the second one is the total number of S’s which precede directly the occurrences of F i but do not belong to any success run of length k (1 ≤ i ≤ m). Furthermore, we obtain exact distributions of random vectors whose components are numbers of failures, non-overlapping runs of failures, successes, overlapping success runs of length l and success runs of length at least l. The majority of the above problems are also treated in the case of the generalized sequence of order k and corresponding results are established regarding the k and corresponding results are established regarding the multivariate extended negative binomial distribution of order k of Philippou and Antzoulakos (1990). The present paper generalizes several results of Aki and Hirano (1994, 1995).

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Author information

Authors and Affiliations

  1. University of Crete, Rethymno, Greece

    Demetrios L. Antzoulakos

  2. University of Patras, Patras, Greece

    Andreas N. Philippou

Authors
  1. Demetrios L. Antzoulakos
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  2. Andreas N. Philippou
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Editor information

Editors and Affiliations

  1. Department of Mathematics, McMaster University, L8S 4Kl, Hamilton, Ontario, Canada

    N. Balakrishnan

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© 1997 Birkhäuser Boston

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Antzoulakos, D.L., Philippou, A.N. (1997). On Multivariate Distributions of Various Orders Obtained by Waiting for the r-th Success Run of Length k in Trials With Multiple Outcomes. In: Balakrishnan, N. (eds) Advances in Combinatorial Methods and Applications to Probability and Statistics. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4140-9_24

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  • DOI: https://doi.org/10.1007/978-1-4612-4140-9_24

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-8671-4

  • Online ISBN: 978-1-4612-4140-9

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