142
Views
7
CrossRef citations to date
0
Altmetric
Original Articles

On ℓ-overlapping Runs of Ones of Length k in Sequences of Independent Binary Random Variables

&
Pages 3865-3884 | Received 08 Nov 2012, Accepted 19 Mar 2013, Published online: 15 Sep 2015

References

  • Agarwal, M., Mohan, P. (2008). GERT analysis of m-consecutive-k-out-of-n:F system with overlapping runs and k-1-step Markov dependence. Int. J. Operat. Res. 3:36–51.
  • Aki, S. (2012). Statistical modeling for discrete patterns in a sequence of exchangeable trials. Ann. Instit. Statist. Math. 64:633–655.
  • Aki, S., Hirano, K. (1988). Some characteristics of the binomial distribution of order k and related distributions. In: Matusita, K., Ed. Statistical Theory and Data Analysis II. Amsterdam: Elsevier, pp.211–222.
  • Aki, S., Hirano, K. (2000). Numbers of success-runs of specified length until certain stopping time rules and generalized binomial distributions of order k. Ann. Instit. Statist. Math. 52:767–777.
  • Alevizos, P.D., Papastavridis, S.G., Sypsas, P. (1993). Reliability of cyclic m-consecutive-k-out-of-n:F systems. Proc. 2nd IASTED Int. Conf. Reliab. Qual. Control, and Risk Assess., pp.140–143, Anaheim: IASTED-ACTA Press.
  • Antzoulakos, D.L. (2003). Waiting times and number of appearances of runs: A unified approach. Commun. Statist. Theor. Meth. 32:1289–1315.
  • Antzoulakos, D.L., Chadjiconstantinidis, S. (2001). Distributions of numbers of success runs of fixed length in Markov dependent trials. Ann. Instit. Statist. Math. 53:599–619.
  • Atkinson, K. (1985). Elementary Numerical Analysis. New York: John Wiley and Sons, Inc.
  • Balakrishnan, N., Koutras, M.V. (2002). Runs and Scans with Applications. New York: Wiley.
  • Chao, M.T., Fu, J.C., Koutras, M.V. (1995). A survey of the reliability studies of consecutive-k-out-of-n:F systems and its related systems. IEEE Trans. Reliab. 44:120–127.
  • Charalambides, C.A. (1994). Success runs in a circular sequence of independent Bernoulli trials. In: Godbole, A.P., Papastavridis, S.G., Eds. Runs and Patterns in Probability. Amsterdam: Kluwer, pp.15–30.
  • Chern, H-H, Hwang, H-K (2005). Limit distribution of the number of consecutive records. Random Struct. Algorithms 26:404–417.
  • Chern, H-H, Hwang, H-K, Yeh, Y-N (2000). Distribution of the number of consecutive records. Random Struct. Algorithms 17:167–196.
  • Csörgö, S., Wu, W.B. (2000). On sums of overlapping products of independent Bernoulli random variables. Ukr. Math. J. 52:1496–1503.
  • Dafnis, S. D., Makri, F. S., Psillakis, Z. M. (2010). On the reliability of consecutive systems. Proc. World Congress on Eng. 2010, London, III:1817–1823.
  • Dafnis, S.D., Philippou, A.N., Antzoulakos, D.L. (2012). Distributions of patterns of two successes separated by a string of k-2 failures. Statist. Pap. 53:323–344.
  • Demir, S., Eryilmaz, S. (2010). Run statistics in a sequence of arbitrarily dependent binary trials. Statist. Pap. 51:959–973.
  • Derman, C., Lieberman, G.J., Ross, S.M. (1982). On the consecutive-k-out-of-n:F system. IEEE Trans. Reliab. 31:57–63.
  • Eryilmaz, S. (2010a). Review of recent advances in reliability of consecutive k-out-of-n and related systems. Proc. Instit. Mech. Eng. Part O: J. Risk Reliab. 224:225–237.
  • Eryilmaz, S. (2010b). Mixture representations for the reliability of consecutive-k systems. Mathemat. Comput. Model. 51:405–412.
  • Eryilmaz, S. (2011a). On the mean and extreme distances between failures in Markovian binary sequences. J. Computat. Appl. Math. 236:1502–1510.
  • Eryilmaz, S. (2011b). Joint distribution of run statistics in partially exchangeable processes. Statist. Probab. Lett. 81:163–168.
  • Eryilmaz, S. (2012a). On distributions of runs in the compound binomial risk model. Methodol. Comput. Appl. Math. DOI 10.1007/s11009-012-9303-x, pp.1–11.
  • Eryilmaz, S. (2012b). m-consecutive-k-out-of-n:F system with overlapping runs: signature based reliability analysis. Int. J. Operat. Res. 15:64–73.
  • Eryilmaz, S. (2012c). The number of failed components in a coherent system with exchangeable components. IEEE Trans. Reliab. 61:203–207.
  • Eryilmaz, S. (2013). On the lifetime behavior of a discrete time shock model. J. Computat. Appl. Math. 237:384–388.
  • Eryilmaz, S., Demir, S. (2007). Success runs in a sequence of exchangeable binary trials. J. Statist. Plann. Infer., 137:2954–2963.
  • Eryilmaz, S., Koutras, M.V., Triantafyllou, I.S. (2011). Signature based analysis of m-consecutive-k-out-of-n:F systems with exchangeable components. Nav. Res. Logist., 58:344–354.
  • Eryilmaz, S., Mahmoud, B. (2012). Linear m-consecutive-k, ℓ-out–of-n:F System. IEEE Trans. Reliab., 61:787–791.
  • Eryilmaz, S., Yalcin, F. (2011). Distribution of run statistics in partially exchangeable processes. Metrika 73:293–304.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I, 3rd edn. New York: John Wiley.
  • Fu, J.C., Johnson, B.C., Chang, Y-M (2012). Approximating the extreme right-hand tail probability for the distribution of the number of patterns in a sequence of multi-state trials. J. Statist. Plann. Infer. 142:473–480.
  • Fu, J.C., Koutras, M.V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89:1050–1058.
  • Fu, J.C., Lou, W. Y.W. (2003). Distribution Theory of Runs and Patterns and its Applications: A Finite Markov Chain Imbedding Approach. River Edge, NJ: World Scientific.
  • Griffith, W.S. (1986). On consecutive-k-out-of-n failure systems and their generalizations. In: Basu, A.P., Ed., Reliability and Quality Control. North Holland: Elsevier, pp.157–165.
  • Han, S., Aki, S. (2000). A unified approach to binomial-type distributions of order k. Commun. Statist. Theor. Meth. 29:1929–1943.
  • Hirano, K. (1986). Some properties of the distributions of order k. In: Philippou, A.N., Horadam, A.F., Bergum, G.E., Eds., Fibonacci Numbers and Their Applications. Dordrecht: Reidel, pp.43–53.
  • Hirano, K., Aki, S., Kashiwagi, N., Kuboki, H. (1991). On Ling’s binomial and negative binomial distributions of order k. Statist. Probab. Lett. 11:503–509.
  • Holst, L. (2007). Counts of failure strings in certain Bernoulli sequences. J. Appl. Probab. 44:824–830.
  • Holst, L. (2008a). The number of two consecutive successes in a Hope-Pólya urn. J. Appl. Probab. 45:901–906.
  • Holst, L. (2008b). A note on embedding certain Bernoulli sequences in marked Poisson processes. J. Appl. Probab. 45:1181–1185.
  • Holst, L. (2009). On consecutive records in certain Bernoulli sequences. J. Appl. Probab. 46:1201–1208.
  • Holst, L. (2011). A note on records in a binary sequence. Arkiv för Matematic 49:351–356.
  • Hsiau, S-R (2010). Selecting the last consecutive record in a record process. Adv. Appl. Probab. 42:739–760.
  • Huffer, F.W., Sethuraman, J., Sethuraman, S. (2008). A study of counts of Bernoulli strings via conditional Poisson processes. Proc. Amer. Mathemat. Soc., 137:2125–2134.
  • Inoue, K., Aki, S. (2003). Generalized binomial and negative binomial distributions of order k by the ℓ-overlapping enumeration scheme. Ann. Instit. Statist. Math. 55:153–167.
  • Inoue, K., Aki, S. (2010). On the conditional and unconditional distributions of the number of success runs on a circle with applications. Statist. Probab. Lett. 80:874–885.
  • Inoue, K., Aki, S., Hirano, K. (2011). Distributions of simple patterns in some kinds of exchangeable sequences. J. Statist. Plann. Infer. 141:2532–2544.
  • Joffe, A., Marchand, E., Perron, F., Popadiuk, P. (2004). On sums of products of Bernoulli variables and random permutations. J. Theoret. Probab. 17:285–292.
  • Kontoleon, J.M. (1980). Reliability determination of r-consecutive-out-of-n:F system. IEEE Trans. Reliab. 29:437–438.
  • Koutras, M.V. (1997). Waiting times and number of appearances of events in a sequence of discrete random variables. In: Balakrishnan, N., Ed., Advances in Combinatorial Methods and Applications to Probability and Statistics. Boston: Birkhauser, pp.363–384.
  • Koutras, M.V. (2003). Applications of Markov chains to the distribution theory of runs and patterns. In: Shanbhag, D.N., Rao, C.R., Eds., Stochastic Processes: Modelling and Simulation. Amsterdam: North-Holland, Handbook of Statistics. 44:431–472.
  • Koutras, M.V., Milienos, F.S. (2012). Exact and asymptotic results for pattern waiting times. J. Statist. Plann. Infer., 142:1464–1479.
  • Koutras, M.V., Papadopoulos, G.K., Papastavridis, S.G. (1995). Runs on a circle. J. Appl. Probab. 32:396–404.
  • Kuo, W., Zuo, M.J. (2003). Optimal Reliability Modeling: Principles and Applications. Hoboken, NJ: Wiley.
  • Levitin, G. (2010). The Universal Generating Function in Reliability Analysis and Optimization. London: Springer-Verlag Limited.
  • Ling, K.D. (1988). On binomial distributions of order k. Statist. Probab. Lett. 6:247–250.
  • Makri, F.S. (2010). On occurrences of F-S strings in linearly and circularly ordered binary sequences. J. Appl. Probab. 47:157–178.
  • Makri, F.S. (2011). Minimum and maximum distances between failures in binary sequences. Statist. Probab. Lett. 81:402–410.
  • Makri, F.S., Philippou, A.N. (1994). Binomial distributions of order k on the circle. In: Godbole, A.P., Papastavridis, S.G., Eds., Runs and Patterns in Probability. Amsterdam: Kluwer, pp.65–81.
  • Makri, F.S., Philippou, A.N. (2005). On binomial and circular binomial distributions of order k for ℓ-overlapping success runs of length k. Statist. Pap. 46:411–432.
  • Makri, F.S., Philippou, A.N., Psillakis, Z.M. (2007). Polya, inverse Polya, and circular Polya distributions of order k for ℓ-overlapping success runs. Commun. Statist. Theor. Meth. 36:657–668.
  • Makri, F.S., Psillakis, Z.M. (2011a). On success runs of a fixed length in Bernoulli sequences: exact and asymptotic results. Comput. Math. Applic. 61:761–772.
  • Makri, F.S., Psillakis, Z.M. (2011b). On success runs of length exceeded a threshold. Methodol. Comput. Appl. Math. 13:269–305.
  • Makri, F.S., Psillakis, Z.M. (2011c). On runs of length exceeding a threshold: normal approximation. Statist. Pap. 52:531–551.
  • Makri, F.S., Psillakis, Z.M. (2012a). Counting certain binary strings. J. Statist. Plann. Infer. 142:908–924.
  • Makri, F.S., Psillakis, Z.M. (2012b). Exact distributions of constrained k,ℓ strings of failures between subsequent successes. Statist. Pap., DOI 10.1007/s00362-012-0462-1, pp.1–24.
  • Mori, T. (2001). On the distribution of sums of overlapping products. Acta Scientiarum Mathematicarum (Szeged) 67:833–841.
  • Mytalas, G.C., Zazanis, M.A. (2013). Central limit theorem approximations for the number of runs in Markov-dependent binary sequences. J. Statist. Plann. Infer. 143:321–333.
  • Nevzorov, V.B. (2001). Records: Mathematical Theory. Providence.RI: American Mathematical Society.
  • Philippou, A.N., Makri, F.S. (1986). Successes, runs and longest runs. Statist. Probab. Lett. 4:211–215.
  • Roussas, G.G. (1997). A Course in Mathematical Statistics. 2nd ed. San Diego: Academic Press.
  • Shao, J. (2003). Mathematical Statistics. New York: Springer.
  • Sinha, K., Sinha, B.P. (2009). On the distribution of runs of ones in binary strings. Comput. Math. Applic. 58:1816–1829.
  • Triantafyllou, I.S., Koutras, M.V. (2008). On the signature of coherent systems and applications. Probab. Eng. Inform. Sci. 22:19–35.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.