Abstract
The paper deals with three generalized dependent setups arising from a sequence of Bernoulli trials. Various distributional properties, such as probability generating function, probability mass function and moments are discussed for these setups and their waiting time. Also, explicit forms of probability generating function and probability mass function are obtained. Finally, two applications to demonstrate the relevance of the results are given.
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References
Aki S (1997) On sooner and later problems between success and failure runs. In: Balakrishnan N (ed) Advances in combinatorial methods and applications to probability and statistics. Borkhäuser, Boston, pp 385–400
Aki S, Kuboki H, Hirano K (1984) On discrete distributions of order \(k\). Ann Inst Stat Math 36:431–440
Antzoulakos DL, Chadjiconstantinidis S (2001) Distributions of numbers of success runs of fixed length in Markov dependent trials. Ann Inst Stat Math 53:599–619
Antzoulakos DL, Bersimis S, Koutras MV (2003) Waiting times associated with the sum of success run lengths. In: Lindqvist B, Doksum K (eds) Mathematical and statistical methods in reliability. World Scientific, Singapore, pp 141–157
Balakrishnan N, Koutras MV (2002) Runs and scans with applications. Wiley, New York
Balakrishnan N, Mohanty SG, Aki S (1997) Start-up demonstration tests under Markov dependence model with corrective actions. Ann Inst Stat Math 49:155–169
Balakrishnan N, Koutras MV, Milienos FS (2014a) Some binary start-up demonstration tests and associated inferential methods. Ann Inst Stat Math 66:759–787
Balakrishnan N, Koutras MV, Milienos FS (2014b) Start-up demonstration tests: models, methods and applications, with some unifications. Appl Stoch Models Bus Ind 30:373–413
Berstel J (1986) Fibonacci words—a survey. In: Rozenberg G, Salomaa A (eds) The book of L. Springer, Berlin
Dafnis SD, Antzoulakos DL, Philippou AN (2010) Distribution related to \((k_1, k_2)\) events. J Stat Plan Inference 140:1691–1700
Fu JC (1986) Reliability of consecutive-\(k\)-out-of-\(n\): F system with (\(k-1\))-step Markov dependence. IEEE Trans Reliab 35:602–606
Fu JC, Hu B (1987) On reliability of a large consecutive-\(k\)-out-of-\(n\): F system with (\(k-1\))-step Markov dependence. IEEE Trans Reliab 36:75–77
Fu JC, Koutras MV (1994) Distribution theory of runs: a Markov chain approach. J Am Stat Assoc 89:1050–1058
Fu JC, Lou WYW, Chen SC (1999) On the probability of pattern matching in nonaligned DNA sequences: a finite Markov chain imbedding approach. In: Glaz J, Balakrishnan N (eds) Scan statistics and applications. Birkhäuser, Boston, pp 287–302
Greenberg I (1970) The first occurrence of n successes in N trials. Technometrics 12:627–634
Huang WT, Tsai CS (1991) On a modified binomial distribution of order \(k\). Stat Prob Lett 11:125–131
Koutras MV (1996) On a waiting time distribution in a sequence of Bernoulli trials. Ann Inst Stat Math 48:789–806
Koutras MV (1997) Waiting time distributions associated with runs of fixed length in two-state Markov chains. Ann Inst Stat Math 49:123–139
Kumar AN, Upadhye NS (2017) On discrete Gibbs measure approximation to runs. Preprint. arXiv:1701.03294
Makri FS, Philippou AN, Psillakis ZM (2007) Shortest and longest length of success runs in binary sequences. J Stat Plan Inference 137:2226–2239
Moore PT (1958) Some properties of runs in quality control procedures. Biometrika 45:89–95
Philippou AN, Makri A (1986) Success, runs and longest runs. Stat Prob Lett 4:211–215
Philippou AN, Georghiou C, Philippou GN (1983) A generalized distribution and some of its properties. Stat Prob Lett 1:171–175
Sinha K, Sinha BP, Datta D (2010) CNS: a new energy efficient transmission scheme for wireless sensor networks. Wirel Netw J 16:2087–2104
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The authors are grateful to the associate editor and reviewers for many valuable suggestions, critical comments which improved the presentation of the paper.
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Kumar, A.N., Upadhye, N.S. Generalizations of distributions related to (\(k_1,k_2\))-runs. Metrika 82, 249–268 (2019). https://doi.org/10.1007/s00184-018-0668-x
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DOI: https://doi.org/10.1007/s00184-018-0668-x
Keywords
- (\(k_1, k_2\))-runs
- Waiting time
- Probability generating function
- Probability mass function
- Moments
- Markov dependent trials