Abstract
In the present paper we study the distributions of families of patterns which generalize runs and patterns distributions extensively examined in the literature during the last decades. In our analysis we assume that the sequence of outcomes under investigation includes independent, but not necessarily identically distributed trials. An illustration is also provided how our new results could be exploited to enrich a new system, still in research, related to patients’ weaning from mechanical ventilation.
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Antzoulakos DL, Bersimis S, Koutras MV (2003) On the distribution of the total number of run lengths. Ann Inst Statist Math 55:865–884
Balakrishnan N, Koutras MV (2002) Runs and scans with applications. Wiley, New York
Bersimis S, Koutras MV, Maravelakis PE (2014) A compound control chart for monitoring and controlling high quality processes. Eur J Oper Res 233:595–603
Buliev I, Pallikarakis N, Filos K (2006) Computer-based system for automated weaning of conscious patients from assisted mechanical ventilation: preliminary tests. Annuals of Technical University of Varna Bulgaria
Chadjiconstantinidis S, Antzoulakos DL, Koutras MV (2000) Joint distributions of successes, failures and patterns in enumeration problems. Adv Appl Probab 32:866–884
Dafnis SD, Makri FS, Philippou AN (2019) The reliability of a generalized consecutive system. Appl Math Comput 359:186–193
Dafnis SD, Philippou AN (2011) Distributions of patterns with applications in engineering. IAENG Int J Appl Math 41:68–75
Dafnis SD, Philippou AN, Antzoulakos DL (2012) Distributions of patterns of two successes separated by a string of k − 2 failures. Stat Pap 53:323–344
Demir S, Eryilmaz S (2010) Run statistics in a sequence of arbitarily dependent binary trials. Stat Pap 51:959–973
Dermitzakis A, Buliev I, Filos K (2008) Fuzzy logic controller for weaning of conscious patients from mechanical ventilation: a simulation study. In: 6th European Symposium on Biomedical Engineering, Chania, Greece, Code: s3.08
Dojat M, Harf A, Touchard D, Lemaire F, Brochard L (2000) Clinical evaluation of a computer controlled pressure support mode. Am J Respir Crit Care Med 161:1161–1166
Eryilmaz S (2005) On the distribution and expectation of success runs in nonhomogeneous Markov dependent trials. Stat Pap 46:117–128
Esteban A, Alia I, Ibanez J, Benito S, Tobin MJ (1994) Modes of mechanical ventilation. A national survey of Spanish hospitals. Chest 106:1188–1193
Fu JC, Koutras MV (1994a) Distribution theory of runs: a Markov chain approach. J Am Stat Assoc 89:1050–1058
Fu JC, Koutras MV (1994b) Poisson approximations for 2-dimensional patterns. Ann Inst Statist Math 45:179–192
Fu JC, Lou W Y W (2003) Distribution theory of runs and patterns and its applications: a finite Markov imbedding approach. World Scientific Publishing, New Jersey
Goldstein L (1990) Poisson approximations in DNA sequence matching. Commun Stat-Theor M 19:4167–4179
Hirano K (1986) Some properties of the distributions of order k. Fibonacci Numbers and Their Applications, Reidel Publishing Company, 43–53
Inoue K, Aki S (2003) Generalized binomial and negative binomial distributions of order k by the l-overlapping enumeration scheme. Ann Inst Statist Math 55:153–167
Koutras MV (2003) Applications of Markov chains to the distribution theory of runs and patterns. Handbook of Statistics, Elsevier, North Holland, 431–472
Koutras MV, Alexandrou VA (1995) Runs, scans and urn model distributions: a unified Markov chain approach. Ann Inst Statist Math 47:743–766
Koutras MV, Papadopoulos GK, Papastavridis SG (1995) Runs on a circle. J Appl Probab 32:396–404
Makri FS, Philippou AN (2005) On binomial and circular binomial distributions of order k for l-overlapping success runs of length k. Stat Pap 46:411–432
Makri FS, Philippou AN, Psillakis ZM (2007) Success run statistics defined on an urn model. Adv Appl Probab 39:991–1019
Makri FS, Psillakis ZM (2011) On success runs of length exceeded a threshold. Methodol Comput Appl 13:269–305
Makri FS, Psillakis ZM (2012) Counting certain binary strings. J Stat Plann Inference 142:908–924
Makri FS, Psillakis ZM (2013) Exact distributions of constrained (k,l) strings of failures between subsequent successes. Stat Pap 54:783–806
Makri FS, Psillakis ZM (2014) On the expected number of limited length binary strings derived by certain urn models. J Probab doi:10.11552014646140
Makri FS, Psillakis ZM (2015) On l-overlapping runs of ones of length k in sequences of independent binary random variables. Commun Stat-Theor M 44:3865–3884
Makri FS, Psillakis ZM (2017) On limited length binary strings with an application in statistical control. The Open Statistics and Probability Journal, 8. doi:10.21741876527001708010001
Mytalas GC, Zazanis MA (2013) Central limit theorem approximations for the number of runs in Markov-dependent binary sequences. J Stat Plann Inference 143:321–333
Philippou AN, Makri FS (1986) Successes, runs and longest runs. Stat Probabil Lett 1:171–175
Sen K, Goyal B (2004) Distributions of patterns of two failures separated by success runs of length k. J Korean Stat Soc 33:35–58
Yalcin F, Eryilmaz S (2014) q-geometric and q-binomial distributions of order k. J Comput Appl Math 371:31–38
Acknowledgments
The authors wish to thank the referees for the thorough reading, useful comments and suggestions which helped to improve the article.
Work funded by National Matching Funds 2016 − 2017 of the Greek Government, and more specifically by the General Secretariat for Research and Technology (GSRT), related to EU project ”ISMPH: Inference for a Semi-Markov Process” (GA No 329128).
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Work done while SDD was apostgraduate student at the Department of Statistics and Insurance Science, University of Piraeus, Greece.
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Dafnis, S.D., Makri, F.S. & Koutras, M.V. Generalizations of Runs and Patterns Distributions for Sequences of Binary Trials. Methodol Comput Appl Probab 23, 165–185 (2021). https://doi.org/10.1007/s11009-020-09810-0
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DOI: https://doi.org/10.1007/s11009-020-09810-0