Binomial distribution of order \(k\) in a modified binary sequence. (English) Zbl 1492.60035
Summary: Let us consider a sequence of \(n\) binary trials (signals). A counter registers successes, but once a success is registered the mechanism is locked for a number of trials following each registration. Under this framework the observed sequence of outcomes turns to a dependent sequence with non-identical success probabilities even if the original trials were independent and identically distributed. In the present paper, we study the distribution of the number of success runs registered by the counter after the completion of the \(n\) signals. Our study covers the general case where the original trials are independent but not necessarily identically distributed. The special case of identically distributed trials gives birth to the modified binomial distribution of order \(k\), which generalizes binomial distributions extensively studied in the literature. In this case, we derive neat recursive relations for the probability mass function, the probability generating function and the moments. The applicability of the modified binomial distribution of order \(k\) in several research areas is highlighted and after developing theoretical results we discuss how they can be exploited to study a biomedical engineering problem.
MSC:
| 60E05 | Probability distributions: general theory |
| 92C10 | Biomechanics |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
| 62E10 | Characterization and structure theory of statistical distributions |
References:
| [1] | Aliverti, A., Wearable technology: role in respiratory health and disease, Breathe, 13, e27-e36 (2017) · doi:10.1183/20734735.008417 |
| [2] | Antzoulakos, DL; Bersimis, S.; Koutras, MV, On the distribution of the total number of run lengths, Ann Inst Stat Math, 55, 865-884 (2003) · Zbl 1047.62013 · doi:10.1007/BF02523398 |
| [3] | Balakrishnan, N.; Koutras, MV, Runs and scans with applications (2002), New York: Wiley, New York · Zbl 0991.62087 |
| [4] | Balakrishnan, N.; Koutras, MV; Milienos, FS, Start-up demonstration tests: models, methods and applications, with some unifications, Appl Stoch Model Bus Ind, 30, 373-413 (2014) · Zbl 07880602 · doi:10.1002/asmb.2040 |
| [5] | Balakrishnan, N.; Koutras, M.; Milienos, F., Reliability analysis and plans for successive testing: start-up demonstration tests and applications (2021), Cambridge: Academic Press, Cambridge · Zbl 1473.62004 |
| [6] | Bersimis, S.; Koutras, MV; Papadopoulos, GK, Waiting time for an almost perfect run and applications in statistical process control, Methodol Comput Appl Probab, 16, 207-222 (2014) · Zbl 1296.60025 · doi:10.1007/s11009-012-9307-6 |
| [7] | Chadjiconstantinidis, S.; Antzoulakos, DL; Koutras, MV, Joint distributions of successes, failures and patterns in enumeration problems, Adv Appl Probab, 32, 866-884 (2000) · Zbl 0963.60003 · doi:10.1239/aap/1013540248 |
| [8] | Dafnis, SD; Makri, FS, Weak runs in sequences of binary trials, Metrika (2021) · Zbl 1493.62061 · doi:10.1007/s00184-021-00842-1 |
| [9] | Dafnis, SD; Makri, FS; Koutras, MV, Generalizations of runs and patterns distributions for sequences of binary trials, Methodol Comput Appl Probab, 23, 165-185 (2021) · Zbl 1473.62054 · doi:10.1007/s11009-020-09810-0 |
| [10] | Dandekar, V. M. (1955). Certain modified forms of binomial and Poisson distributions. Sankhyā Indian J Stat 15:237-250 · Zbl 0067.10502 |
| [11] | De Moivre, A., The doctrine of chance (1738), New York: Chelsea Publishing Co, New York |
| [12] | Eryilmaz, S.; Demir, S., Run statistics in a sequence of arbitarily dependent binary trials, Stat Pap, 45, 1007-1023 (2007) |
| [13] | Feller, W., An introduction to probability theory and its applications (1968), New York: Wiley, New York · Zbl 0155.23101 |
| [14] | Fu, JC; Koutras, MV, Distribution theory of runs: a Markov chain approach, J Am Stat Assoc, 89, 1050-1058 (1994) · Zbl 0806.60011 · doi:10.1080/01621459.1994.10476841 |
| [15] | Fu, JC; Lou, WYW, Distribution theory of runs and patterns and its applications: a finite Markov imbedding approach (2003), New Jersey: World Scientific Publishing, New Jersey · Zbl 1030.60063 · doi:10.1142/4669 |
| [16] | Godbole, AP, Specific formulae for some success run distributions, Stat Probab Lett, 10, 119-124 (1990) · Zbl 0706.60008 · doi:10.1016/0167-7152(90)90006-S |
| [17] | Han, Q.; Aki, S., Joint distributions of runs in a sequence of multi-state trials, Ann Inst Stat Math, 51, 419-447 (1999) · Zbl 0961.60029 · doi:10.1023/A:1003941920316 |
| [18] | Hirano, K.; Philippou, AN, Some properties of the distributions of order \(k\), Fibonacci numbers and their applications, 43-53 (1986), Dordrecht: Reidel, Dordrecht · Zbl 0601.62023 · doi:10.1007/978-94-009-4311-7_4 |
| [19] | Johnson, NL; Kotz, S.; Kemp, AW, Univariate discrete distributions (1992), New York: Wiley, New York · Zbl 0773.62007 |
| [20] | Kamps, AW; Roorda, RJ; Brand, PL, Peak flow diaries in childhood asthma are unreliable, Thorax, 56, 180-182 (2001) · doi:10.1136/thorax.56.3.180 |
| [21] | Koutras, MV; Alexandrou, VA, Runs, scans and urn model distributions: a unified Markov chain approach, Ann Inst Stat Math, 47, 743-766 (1995) · Zbl 0848.60021 · doi:10.1007/BF01856545 |
| [22] | Koutras, MV; Bersimis, S.; Antzoulakos, DL, Improving the performance of the chi-square control chart via runs rules, Methodol Comput Appl Probab, 8, 409-426 (2006) · Zbl 1103.62119 · doi:10.1007/s11009-006-9754-z |
| [23] | Koutras, MV; Eryilmaz, S., Compound geometric distribution of order \(k\), Methodol Comput Appl Probab, 19, 377-393 (2017) · Zbl 1380.62064 · doi:10.1007/s11009-016-9482-y |
| [24] | Kumar, CS, A class of discrete distributions of order \(k\), J Stat Theory Pract, 3, 4, 795-803 (2009) · Zbl 1211.62027 · doi:10.1080/15598608.2009.10411960 |
| [25] | Philippou, AN; Georghiou, C.; Philippou, GN, A generalized geometric distribution and some of its properties, Statist Probab Lett, 1, 171-175 (1983) · Zbl 0517.60010 · doi:10.1016/0167-7152(83)90025-1 |
| [26] | Philippou, AN; Makri, FS, Successes, runs and longest runs, Statist Probab Lett, 4, 101-105 (1986) · Zbl 0584.60024 · doi:10.1016/0167-7152(86)90025-8 |
| [27] | Sen, K.; Agarwal, ML; Bhattacharya, S., Geiger counter-type Pólya-Eggenberger distributions, Commun Stat Theory Methods, 44, 4912-4926 (2015) · Zbl 1336.60014 · doi:10.1080/03610926.2012.742109 |
| [28] | Thamrin, C.; Stern, G.; Strippoli, MP; Kuehni, CE; Suki, B.; Taylor, DR; Frey, U., Fluctuation analysis of lung function as a predictor of long-term response to beta2-agonists, Eur Respir J, 33, 486-493 (2009) · doi:10.1183/09031936.00106208 |
| [29] | Todhunter, I., A history of the mathematical theory of probability, Cambridge, Macmillan, reprinted 1965 (1865), New York: Chelsea, New York |
| [30] | Yalcin, F.; Eryilmaz, S., \(q\)-geometric and \(q\)-binomial distributions of order \(k\), J Comput Appl Math, 71, 31-38 (2014) · Zbl 1327.60043 · doi:10.1016/j.cam.2014.03.025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
