# On the Exact Distributions of Pattern Statistics for a Sequence of Binary Trials: A Combinatorial Approach

Living reference work entry

## Abstract

Consider a sequence of exchangeable or Markov-dependent binary (zero-one) trials. A sequence of independent and identically distributed binary trials is covered as a particular case of both the prementioned ones. For counting/waiting time pattern statistics defined on such model sequences, we point out how their exact probability distributions can be established using enumerative combinatorics. The expressions for the distributions contain probabilities depending on the internal structure of the model sequence and combinatorial numbers denoting set cardinalities. The latter numbers depend on the considered pattern statistics and the number of ones, for an exchangeable sequence, as well as the number of runs of ones, for a Markov-dependent sequence. These numbers become concrete when certain patterns and enumerative schemes are studied on the model sequences. Exact distributions for statistics connected to patterns of limited length, as well as to certain runs and scans, are provided using proper combinatorial numbers and exemplify the approach.

## Keywords

Exact distributions Enumerative combinatorics Runs, scans, and patterns Binary trials

## References

1. Balakrishnan N, Koutras MV (2002) Runs and scans with applications. Wiley, New York
2. Boutsikas MV, Koutras MV, Milienos FS (2017) Asymptotic results for the multiple scan statistic. J Appl Prob 54:320–330
3. Charalambides CA (2002) Enumerative combinatorics. Chapman & Hall/CRC, Boca RatonGoogle Scholar
4. Chern HH, Hwang HK, Yeh YN (2000) Distribution of the number of consecutive records. Random Struct Algorithms 17:169–196
5. Dafnis SD, Philippou AN (2011) Distributions of patterns with applications in engineering. IAENG Int J Appl Math 41:68–75
6. Dafnis SD, Philippou AN, Antzoulakos DL (2012) Distributions of patterns of two successes separated by a string of k − 2 failures. Stat Papers 53:323–344
7. Eryilmaz S (2010) Discrete scan statistics generated by exchangeable binary trials. J Appl Prob 47:1084–1092
8. Eryilmaz S (2011) Joint distribution of run statistics in partially exchangeable processes. Statist Probab Lett 81:163–168
9. Eryilmaz S (2016) A new class of life time distributions. Statist Probab Lett 112:63–71
10. Eryilmaz S (2017) The concept of weak exchangeability and its applications. Metrika 80:259–271
11. Eryilmaz S, Zuo M (2010) Constrained (k, d)-out-of-n systems. Inst J Syst Sci 41:679–685
12. Eryilmaz S, Yalcin F (2011) On the mean and extreme distances between failures in Markovian binary sequences. J Comput Appl Math 236:1502–1510
13. Feller W (1968) An introduction to probability theory and its applications, vol I, 3rd edn. Wiley, New York
14. Fu JC, Lou WYW (2003) Distribution theory of runs and patterns and its applications: a finite Markov chain imbedding approach. World Scientific Publishing Co. Inc., River Edge
15. George EO, Bowman D (1995) A full likelihood procedure for analyzing exchangeable binary data. Biometrics 51:512–523
16. Glaz J, Naus J, Wallenstein S (2001) Scan statistics. Springer, New York
17. Holst L (2007) Counts of failure strings in certain Bernoulli sequences. J Appl Prob 44:824–830
18. Holst L (2008) The number of two consecutive successes in a Hope-Polya urn. J Appl Prob 45:901–906
19. Holst L (2009) On consecutive records in certain Bernoulli sequences. J Appl Prob 46:1201–1208
20. Huffer FW, Sethuraman J, Sethuraman S (2009) A study of counts of Bernoulli strings via conditional Poisson processes. Proc Am Math Soc 137:2125–2134
21. Jacquet P, Szpankowski W (2006) On (d, k) sequences not containing a given word. In: International symposium on information theory, ISIT 2006, Seatle, pp 1486–1489Google Scholar
22. Joffe A, Marchand E, Perron F, Popadiuk P (2004) On sums of products of Bernoulli variables and random permutations. J Theor Probab 17:285–292
23. Johnson BC, Fu JC (2014) Approximating the distributions of runs and patterns. J Stat Distrib Appl 1:5
24. Koutras MV, Eryilmaz S (2017) Compound geometric distribution of order k. Methodol Comput Appl Probab 19:377–393
25. Koutras MV, Lyberopoulos DP (2018) Asymptotic results for jump probabilities associated to the multiple scan statistic. Ann Inst Stat Math 70:951–968
26. Koutras VM, Koutras MV, Yalcin F (2016) A simple compound scan statistic useful for modeling insurance and risk management problems. Insur Math Econ 69:202–209
27. Kumar AN, Upadhye NS (2018) Generalizations of distributions related to (k 1, k 2)-runs. Metrika. https://doi.org/10.1007/s00184-018-0668-x Google Scholar
28. Ling KD (1988) On binomial distributions of order k. Statist Probab Lett 6:247–250
29. Makri FS (2010) On occurrences of F − S strings in linearly and circularly ordered binary sequences. J Appl Prob 47:157–178
30. Makri FS (2011) Minimum and maximum distances between failures in binary sequences. Statist Probab Lett 81:402–410
31. Makri FS, Psillakis ZM (2011) On success runs of a fixed length in Bernoulli sequences: exact and asymtotic results. Comput Math Appl 61:761–772
32. Makri FS, Psillakis ZM (2012) Counting certain binary strings. J Statist Plan Inference 142:908–924
33. Makri FS, Psillakis ZM (2013) Exact distributions of constrained (k, ) strings of failures between subsequent successes. Stat Papers 54:783–806
34. Makri FS, Psillakis ZM (2014) On the expected number of limited length binary strings derived by certain urn models. J Probab. https://doi.org/10.1155/2014/646140
35. Makri FS, Psillakis ZM (2016) On runs of ones defined on a q-sequence of binary trials. Metrika 79:579–602
36. Makri FS, Psillakis ZM (2017) On limited length binary strings with an application in statistical control. Open Stat Probab J 8:1–6
37. Makri FS, Philippou AN, Psillakis ZM (2007) Success run statistics defined on an urn model. Adv Appl Prob 39:991–1019
38. Makri FS, Psillakis ZM, Arapis AN (2019) On the concentration of runs of ones of length exceeding a threshold in a Markov chain. J Appl Statist 46:85–100
39. Mood AM (1940) The distribution theory of runs. Ann Math Statist 11:367–392
40. Mori TF (2001) On the distribution of sums of overlapping products. Acta Sci Math (Szeged) 67:833–841
41. Mytalas GC, Zazanis MA (2013) Cental limit theorem approximations for the number of runs in Markov-dependent binary sequences. J Statist Plann Inference 143:321–333
42. Riordan J (1964) An introduction to combinatorial analysis, 2nd edn. Wiley, New York
43. Sarkar A, Sen K, Anuradha (2004) Waiting time distributions of runs in higher order Markov chains. Ann Inst Stat Math 56:317–349Google Scholar
44. 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
45. Stefanov VT, Szpankowski W (2007) Waiting time distributions for pattern occurrence in a constrained sequence. Discret Math Theor Comput Sci 9:305–320
46. Zehavi E, Wolf JK (1988) On runlength codes. IEEE Trans Inf Theory 34:45–54

© Springer Science+Business Media, LLC, part of Springer Nature 2019

## Authors and Affiliations

1. 1.Department of MathematicsUniversity of PatrasPatrasGreece
2. 2.Department of PhysicsUniversity of PatrasPatrasGreece

## Section editors and affiliations

• Joseph Glaz
• 1
• Markos V. Koutras
• 2
1. 1.Department of StatisticsUniversity of ConnecticutStorrsUSA
2. 2.Department of Statistics and Insurance ScienceUniversity of PiraeusPiraeusGreece