Skip to main content
SpringerLink
Log in

Waiting for Scans Containing Two Successes

  • Living reference work entry
  • First Online:
Handbook of Scan Statistics

Abstract

In the present chapter, we present a review of results pertaining to the distribution of waiting times for the occurrence(s) of scans of type 2∕r in sequences of binary trials. Our review covers the geometric distribution of order 2∕r, the negative binomial distributions of order 2∕r, and their generalizations. Exact and asymptotic results are presented and illustrated through numerical examples. In the case of the geometric distribution of order 2∕r, a new closed, exact formula is established. Several applications of the reviewed waiting time distributions in various scientific areas are discussed in some detail.

Work done while SDD was a postgraduate student at the Department of Statistics and Insurance Science, Greece.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Similar content being viewed by others

References

  • Aparisi F, Champ CW, Garcia-Diaz JC (2004) A performance analysis of Hotelling’s χ2 control chart with supplementary runs rules. Qual Eng 16:359–368

    Article  Google Scholar 

  • Arnold BC (1972) The waiting time until first duplication. J Appl Probab 9:841–846

    Article  MathSciNet  MATH  Google Scholar 

  • Arratia R, Goldstein L, Gordon L (1989) Two moments suffice for Poisson approximations: the Chen-Stein method. Ann Probab 17:9–25

    Article  MathSciNet  MATH  Google Scholar 

  • Arratia R, Goldstein L, Gordon L (1990) Poisson approximation and the Chen-Stein method. Stat Sci 5:403–424

    MathSciNet  MATH  Google Scholar 

  • Balakrishnan N, Koutras MV (2002) Runs and scans with applications. Wiley, New York

    MATH  Google Scholar 

  • Barbour AD, Holst L, Janson S (1992) Poisson approximation. Oxford University Press, New York

    MATH  Google Scholar 

  • Barlow R, Proschan F (1981) Statistical theory of reliability and life testing, Silver Spring

    MATH  Google Scholar 

  • Bersimis S, Koutras MV, Papadopoulos GK (2014) Waiting time for an almost perfect run and applications in statistical process control. Method Comput Appl 16:207–222

    MathSciNet  MATH  Google Scholar 

  • Bogush AJ (1973) Correlated clutter and resultant properties of binary signals. IEEE Trans Aerosp Electron 9:208–213

    Article  Google Scholar 

  • Boutsikas MV, Koutras MV (2000a) Generalized reliability bounds for coherent structures. J Appl Probab 37:778–794

    Article  MathSciNet  MATH  Google Scholar 

  • Boutsikas MV, Koutras MV (2000b) Reliability approximation for Markov chain imbeddable systems. Method Comput Appl 2:393–411

    MathSciNet  MATH  Google Scholar 

  • Boutsikas MV, Koutras MV (2006) On the asymptotic distribution of the discrete scan statistic. J Appl Probab 43:1137–1154

    Article  MathSciNet  MATH  Google Scholar 

  • Boutsikas MV, Koutras MV, Milienos FS (2009) Extreme value results for scan statistics. In: Glaz J, Pozdnyakov V, Wallenstein S (eds) Scan statistics. Statistics for industry and technology. Birkhäuser, Boston, pp 55–85

    Chapter  Google Scholar 

  • Buliev I, Pallikarakis N, Filos K (1994) Computer-based system for automated weaning of conscious patients from assisted mechanical ventilation: preliminary tests. Annuals of Technical University of Varna, Bulgaria

    Google Scholar 

  • Burghardt PD, Godbole AP, Prengaman AB (1994) A Poisson approximation for the number of k-matches. Stat Probab Lett 21:1–8

    Article  MathSciNet  MATH  Google Scholar 

  • Champ CW, Woodall WH (1987) Exact results for Shewhart control charts with supplementary runs rules. Technometrics 29:393–399

    Article  MATH  Google Scholar 

  • Chen J, Glaz J (1999) Approximations for the distribution and the moments of discrete scan statistics. In: Glaz J, Balakrishnan N (eds) Scan statistics and applications. Birkhäuser, Boston, pp 27–66

    Chapter  MATH  Google Scholar 

  • Dafnis SD, Philippou AN (2011) Distributions of patterns with applications in engineering. IAENG Int J Appl Math 41:68–75

    MathSciNet  MATH  Google Scholar 

  • Dafnis SD, Makri FS, Koutras MV (2021) Generalizations of runs and patterns distributions for sequences of binary trials. Method Comput Appl 23:165–185

    MathSciNet  Google Scholar 

  • Dermitzakis A, Buliev I, Filos K, Pallikarakis N (2008) Fuzzy logic controller for weaning of conscious patients from mechanical ventilation: a simulation study. In: 6th European symposium on biomedical engineering, Chania, Code: s3.08

    Google Scholar 

  • Dinneen GP, Reed IS (1956) An analysis of signal detection and location by digital methods. IRE Trans Inf Theory 2:29–38

    Article  Google Scholar 

  • Dojat M, Harf A, Touchard D, Lemaire F, Brochard L (2000) Clinical evaluation of a computer-controlled pressure support mode. Am J Respir Crit Care Med 161:1161–1166

    Article  Google Scholar 

  • El-Desouky BS, Hussen SA (1990) A generalization of a waiting time problem. Adv Appl Probab 22:758–760

    Article  MathSciNet  MATH  Google Scholar 

  • Esteban A, Alia I, Ibanez J, Benito S, Tobin MJ (1994) Modes of mechanical ventilation and weaning: a national survey of spanish hospitals. Chest 106:1188–1193

    Article  Google Scholar 

  • Feller W (1968) An introduction to probability theory and its applications. Wiley, New York

    MATH  Google Scholar 

  • Fu JC, Koutras MV (1994) Distribution theory of runs: a Markov chain approach. J Am Stat Assoc 89:1050–1058

    Article  MathSciNet  MATH  Google Scholar 

  • Fu JC, Koutras MV (1995) Reliability bounds for coherent structures with independent components. Stat Probab Lett 22:137–148

    Article  MathSciNet  MATH  Google Scholar 

  • Fu JC, Lou WW (2003) Distribution theory of runs and patterns and its applications: a finite Markov chain imbedding approach. World Scientific Publishing, River Edge

    Book  MATH  Google Scholar 

  • Furth R (1920) Schwankungerscheinungen in der Physik. Sammlung Vieweg, Braunschweig

    Book  MATH  Google Scholar 

  • Glaz J (1981) Clustering of events in a stochastic process. J Appl Probab 18:268–275

    Article  MathSciNet  MATH  Google Scholar 

  • Glaz J (1983) Moving window detection for discrete data. IEEE Trans Inf Theory 29:457–462

    Article  MATH  Google Scholar 

  • Glaz J, Balakrishnan N (1999) Scan statistics and applications. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  • Glaz J, Johnson BM (1984) Probability inequalities for multivariate distributions with dependence structures. J Am Stat Assoc 79:436–440

    Article  MathSciNet  MATH  Google Scholar 

  • Glaz J, Naus JI (1991) Tight bounds and approximations for scan statistic probabilities for discrete data. Ann Appl Probab 1:306–318

    Article  MathSciNet  MATH  Google Scholar 

  • Glaz J, Naus JI, Roos M, Wallenstein S (1994) Poisson approximations for the distribution and moments of ordered m-spacings. J Appl Probab 31:271–281

    Article  MathSciNet  MATH  Google Scholar 

  • Glaz J, Naus JI, Wallenstein S (2001) Scan statistics. Springer, New York

    Book  MATH  Google Scholar 

  • Goldstein L, Waterman MS (1992) Poisson, compound Poisson and process approximations for testing statistical significance in sequence comparisons. Bull Math Biol 54:785–812

    Article  MATH  Google Scholar 

  • Greenberg I (1970) The first occurrence of n successes in N trials. Technometrics 12:627–634

    Article  Google Scholar 

  • Hald A (1981) Statistical theory of sampling inspection by attributes. Academic Press, London

    MATH  Google Scholar 

  • Holst L, Janson S (1990) Poisson approximation using the Stein-Chen method and coupling – number of exceedances of gaussian random-variables. Ann Probab 18:713–723

    Article  MathSciNet  MATH  Google Scholar 

  • Hoover DR (1990) Subset complement addition upper bounds – an improved inclusion-exclusion method. J Stat Plan Inference 24:195–202

    Article  MATH  Google Scholar 

  • Janson S (1983) Limit theorems for some sequential occupancy problems. J Appl Probab 20:545–553

    Article  MathSciNet  MATH  Google Scholar 

  • Janson S (1984) Bounds on the distributions of extremal values of a scanning process. Stoch Process Appl 18:313–328

    Article  MathSciNet  MATH  Google Scholar 

  • Janson S (1987) Poisson convergence and Poisson processes with applications to random graphs. Stoch Process Appl 26:1–30

    Article  MathSciNet  MATH  Google Scholar 

  • Janson S (1994) Coupling and Poisson approximation. Acta Appl Math 34:7–15

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson NL (1968) Reliability determination of a r-successive-out-of-n: f system. Am Math Monthly 75:382–383

    Article  Google Scholar 

  • Johnson NL, Kemp AW, Kotz S (2005) Univariate discrete distributions. Wiley, New York

    Book  MATH  Google Scholar 

  • Karwe VV, Naus JI (1997) New recursive methods for scan statistic probabilities. Comput Stat Data Anal 23:389–402

    Article  MATH  Google Scholar 

  • Keilson J, Gerber H (1971) Some results for discrete unimodality. J Am Stat Assoc 66:386–389

    Article  MATH  Google Scholar 

  • Kemp AW (2004) Generalizations of classical discrete distributions. Encyclopedia of statistical sciences. Wiley, New York, pp 1–26

    Google Scholar 

  • Khoo MB, Quah S (2003) Incorporating runs rules into Hotelling’s χ2 control charts. Qual Eng 15:671–675

    Article  Google Scholar 

  • Khoo MB, Quah S, Low H (2005) Powerful rules for the Hotelling’s χ2 control chart. Qual Eng 17:139–149

    Article  Google Scholar 

  • Klein M (2000) Two alternatives to the Shewhart x control chart. J Qual Technol 32:427–431

    Article  Google Scholar 

  • Koutras MV (1996a) On a Markov chain approach for the study of reliability structures. J Appl Probab 33:357–367

    Article  MathSciNet  MATH  Google Scholar 

  • Koutras MV (1996b) On a waiting time distribution in a sequence of Bernoulli trials. Ann I Stat Math 48:789–806

    Article  MathSciNet  MATH  Google Scholar 

  • Koutras MV (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. Birkhäuser, Boston, pp 363–384

    Chapter  Google Scholar 

  • Koutras MV, Alexandrou VA (1995) Runs, scans and urn model distributions: a unified Markov chain approach. Ann I Stat Math 47:743–766

    Article  MathSciNet  MATH  Google Scholar 

  • Mrstik A (1978) Multistatic-radar binomial detection. IEEE Trans Aerosp Electron Syst:103–108

    Google Scholar 

  • Naus JI (1982) Approximations for distributions of scan statistics. J Am Stat Assoc 77:177–183

    Article  MathSciNet  MATH  Google Scholar 

  • Nelson JI (1978) Minimal-order models for false-alarm calculations on sliding windows. IEEE T Aero Elec Sys:351–363

    Google Scholar 

  • Papastavridis SG, Koutras MV (1993) Bounds for reliability of consecutive k-within-m-out-of-n: F systems. IEEE Trans Reliab 42:156–160

    Article  MATH  Google Scholar 

  • Philippou AN, Makri FS (1985) Longest success runs and Fibonacci-type polynomials’. Fib Q 23:338–346

    MathSciNet  MATH  Google Scholar 

  • Philippou AN, Makri FS (1986) Successes, runs and longest runs. Stat Probab Lett 4:101–105

    Article  MathSciNet  MATH  Google Scholar 

  • Philippou AN, Muwafi AA (1982) Waiting for the k-th consecutive success and the Fibonacci sequence of order k. Fib Q 20:28–32

    MathSciNet  MATH  Google Scholar 

  • Philippou AN, Georghiou C, Philippou GN (1983) A generalized geometric distribution and some of its properties. Stat Probab Lett 1:171–175

    Article  MathSciNet  MATH  Google Scholar 

  • Pozdnyakov V, Glaz J, Kulldorff M, Steele JM (2005) A martingale approach to scan statistics. Ann I Stat Math 57:21–37

    Article  MathSciNet  MATH  Google Scholar 

  • Rakitzis AC, Antzoulakos DL (2011) Chi-square control charts with runs rules. Methodol Comput Appl 13:657–669

    Article  MathSciNet  MATH  Google Scholar 

  • Roberts SW (1958) Properties of control chart zone tests. Bell Syst Tech J 37:83–114

    Article  Google Scholar 

  • Samuel-Cahn E (1983) Simple approximations to the expected waiting time for a cluster of any given size, for point processes. Adv Appl probab 15:21–38

    Article  MathSciNet  MATH  Google Scholar 

  • Saperstein B (1973) On the occurrence of n successes within N Bernoulli trials. Technometrics 15:809–818

    MathSciNet  MATH  Google Scholar 

  • Saperstein B (1976) The analysis of attribute moving averages: MIL-STD-105D reduced inspection plans. In: Sixth conference on stochastic processes and applications, Tel Aviv

    Google Scholar 

  • Shmueli G, Cohen A (2000) Run-related probability functions applied to sampling inspection. Technometrics 42:188–202

    Article  Google Scholar 

  • Shmueli G, Cohen A (2003) Run-length distribution for control charts with runs and scans rules. Commun Stat-Theor M 32:475–495

    Article  MathSciNet  MATH  Google Scholar 

  • Solov’ev A (1966) A combinatorial identity and its application to the problem concerning the first occurrence of a rare event. Theor Probab Appl 11:276–282

    Article  MATH  Google Scholar 

  • Starr SH, Freedman JE (1975) The effects of defruiting on the atcrbs. IEEE T Aero Elec Syst:474–484

    Google Scholar 

  • Wang YH (1986) Coupling methods in approximations. Can J Stat 14:69–74

    Article  MathSciNet  MATH  Google Scholar 

  • Wu TL, Glaz J, Fu JC (2013) Discrete, continuous and conditional multiple window scan statistics. J Appl Probab 50:1089–1101

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Work was funded by National Matching Funds 2016−2017 of the Greek government, and more specifically by the General Secretariat for Research and Technology (GSRT), related to EU project “ISMPH: Inference for a Semi-Markov Process” (GA No 329128).

Author information

Authors and Affiliations

  1. Department of Statistics and Insurance Science, School of Finance and Statistics, University of Piraeus, Piraeus, Greece

    Markos V. Koutras & Spiros D. Dafnis

Authors
  1. Markos V. Koutras
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Spiros D. Dafnis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Spiros D. Dafnis .

Editor information

Editors and Affiliations

  1. University of Connecticut Dept. Statistics, Storrs, CT, USA

    Joseph Glaz

  2. University of Pireaus Dept. Statistics & Insurance Sciences, Piraeus, Greece

    Markos V. Koutras

Section Editor information

  1. Department of Statistics, University of Connecticut, Storrs, CT, USA

    Joseph Glaz

  2. Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece

    M. V. Koutras

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Science+Business Media, LLC, part of Springer Nature

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Koutras, M.V., Dafnis, S.D. (2021). Waiting for Scans Containing Two Successes. In: Glaz, J., Koutras, M.V. (eds) Handbook of Scan Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8414-1_57-1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-8414-1_57-1

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-8414-1

  • Online ISBN: 978-1-4614-8414-1

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation