Abstract
In this paper we study exact distributions of runs on directed trees. On the assumption that the collection of random variables indexed by the vertices of a directed tree has a directed Markov distribution, the exact distribution theory of runs is extended from based on random sequences to based on directed trees. The distribution of the number of success runs of a specified length on a directed tree along the direction is derived. A consecutive-k-out-of-n:F system on a directed tree is introduced and investigated. By assuming that the lifetimes of the components are independent and identically distributed, we give the exact distribution of the lifetime of the consecutive system. The results are not only theoretical but also suitable for computation.
Similar content being viewed by others
REFERENCES
Aki, S. (1985). Discrete distributions of order k on a binary sequence, Ann. Inst. Statist. Math., 37, 205-224.
Aki, S. (1992). Waiting time problems for a sequence of discrete random variables, Ann. Inst. Statist. Math., 44, 363-378.
Aki, S., Balakrishnan, N. and Mohanty, S. G. (1996). Sooner and later waiting time problems and failure runs in higher order Markov dependent trials, Ann. Inst. Statist. Math., 48, 773-787.
Aki, S. and Hirano, K. (1988). Some characteristics of the binomial distribution of order k and related distributions, Statistical Theory and Data Analysis II, Proceedings of the 2nd Pacific Area Statistical Conference (ed. K. Matusita), 211-222, North-Holland, Amsterdam.
Aki, S. and Hirano, K. (1993). Discrete distributions related to succession events in a two-state Markov chain, Statistical Science & Data Analysis (eds. K. Matusita, M. L. Puri and T. Hayakawa), 467-474, VSP Publishers, Amsterdam.
Aki, S. and Hirano, K. (1995). Joint distributions of numbers of success-runs and failures until the first consecutive k successes, Ann. Inst. Statist. Math., 47, 225-235.
Aki, S. and Hirano, K. (1996). Lifetime distribution and estimation problems of consecutive-k-out-of-n:F systems, Ann. Inst. Statist. Math., 48, 185-199.
Balakrishnan, N., Mohanty, S. G. and Aki, S. (1997). Start-up demonstration tests under Markov dependence model with corrective actions, Ann. Inst. Statist. Math., 49, 155-169.
Balasubramanian, K., Viveros, R. and Balakrishnan, N. (1993). Sooner and later waiting time problem for Markovian Bernoulli trials, Statist. Probab. Lett., 18, 153-161.
Chao, M. T., Fu, J. C. and Koutras, M. V. (1995). Survey of reliability studies of consecutive-k-out-of-n:F & related systems, IEEE Transactions on Reliability, 40, 120-127.
Ebneshahrashoob, M. and Sobel, M. (1990). Sooner and later problems for Bernoulli trials: frequency and run quotas, Statist. Probab. Lett., 9, 5-11.
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multistate trials, Statistica Sinica, 6, 957-974.
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach, J. Amer. Statist. Assoc., 89, 1050-1058.
Godbole, A. P. and Papastavridis, S. G. (1994). Runs and Patterns in Probability: Selected Papers, Kluwer, Dordrecht.
Hirano, K. (1986). Some properties of the distributions of order k, Fibonacci Numbers and Their Applications (eds. A. N. Philippou, G. E. Bergum and A. F. Horadam), 43-53, Reidel, Dordrecht.
Hirano, K. (1994). Consecutive-k-out-of-n:F Systems, Proc. Inst. Statist. Math., 42, 45-61 (in Japanese).
Hirano, K. and Aki, S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain, Statistica Sinica, 3, 313-320.
Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete distributions, Wiley, New York.
Koutras, M. V. and Alexandrou, V. A. (1995). Runs, scans and urn model distributions: a unified Markov chain approach, Ann. Inst. Statist. Math., 47, 743-766.
Lauritzen, S. L. (1996). Graphical Models, Clarendon Press, Oxford.
Lou, W. Y. W. (1996). On runs and longest run tests: a method of finite Markov chain imbedding, J. Amer. Statist. Assoc., 91, 1595-1601.
Mohanty, S. G. (1994). Success runs of length k in Markov dependent trials, Ann. Inst. Statist. Math., 46, 777-796.
Philippou, A. N. (1986). Distributions and Fibonacci polynomials of order k, longest runs, and reliability of consecutive-k-out-of-n:F system, Fibonacci Numbers and Their Applications (eds. A. N. Philippou, G. E. Bergum and A. F. Horadam), 203-227, Reidel, Dordrecht.
Philippou, A. N., Georghiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties, Statist. Probab. Lett., 1, 171-175.
Philippou, A. N. and Makri, F. S. (1986). Successes, runs, and longest runs, Statist. Probab. Lett., 4, 101-105.
Philippou, A. N. and Muwafi, A. A. (1982). Waiting for the k-th consecutive success and the Fibonacci sequence of order k, The Fibonacci Quart., 20, 28-32.
Rajarshi, M. B. (1974). Success runs in a two-state Markov chain, J. Appl. Probab., 11, 190-192.
Riordan, J. (1958). An Introduction to Combinatorial Analysis, Wiley, New York.
Ripley, B. D. (1996). Pattern Recognition and Neutral Networks, Cambridge University Press, Cambridge.
Schwager, S. J. (1983). Run probabilities in sequences of Markov-dependent trials, J. Amer. Statist. Assoc., 78, 168-175.
Stanley, R. T. (1986). Enumerative Combinatorics, Wadsworth Publishers, Kentucky.
Uchida, M. and Aki, S. (1995). Sooner and later waiting time problems in two-state Markov chain, Ann. Inst. Statist. Math., 47, 415-433.
Author information
Authors and Affiliations
About this article
Cite this article
Aki, S. Distributions of Runs and Consecutive Systems on Directed Trees. Annals of the Institute of Statistical Mathematics 51, 1–15 (1999). https://doi.org/10.1023/A:1003877016437
Issue Date:
DOI: https://doi.org/10.1023/A:1003877016437