67
Views
9
CrossRef citations to date
0
Altmetric
Original Articles

Waiting Times and Number of Appearances of Runs: A Unified Approach

Pages 1289-1315 | Published online: 02 Sep 2006

References

  • Aki , S. and Hirano , K. 1988 . “ Some characteristics of the binomial distribution of order k and related distributions ” . In Statistical Theory and Data Analysis II Edited by: Matusita , K. 211 – 222 . Amsterdam : Elsevier Science Publishers .
  • Aki , S. and Hirano , K. 1993 . “ Discrete distributions related to succession events in a two-state markov chain ” . In Statistical Science and Data Analysis Edited by: Matusita , K. , Puri , M. L. and Hayakawa , T. 467 – 474 . Zeist : VSP International Science Publishers .
  • Aki , S. and Hirano , K. 1994 . Distributions of numbers of failures and successes until the first consecutive k successes . Ann. Inst. Statist. Math. , 46 : 193 – 202 .
  • Aki , S. and Hirano , K. 2000 . Number of success-runs of specified length until certain stopping time rules and generalized binomial distributions of order k . Ann. Inst. Statist. Math. , 52 : 767 – 777 .
  • Antzoulakos , D. L. 1999 . On waiting time problems associated with runs in markov dependent trials . Ann. Inst. Statist. Math. , 51 : 323 – 330 .
  • Antzoulakos , D. L. and Chadjiconstantinidis , S. 2001 . Distributions of numbers of success runs of fixed length in markov dependent trials . Ann. Inst. Statist. Math. , 53 : 599 – 619 .
  • Antzoulakos , D. L. and Philippou , A. N. 1996 . Distributions of the numbers of successes and failures until the occurrence of the r-th overlapping and non-overlapping success run of length k . J. Indian Soc. Stat. Opers. Res. , Vol. XVII : 19 – 30 .
  • Antzoulakos , D. L. and Philippou , A. N. 1997 . Probability distribution functions of succession quotas in the case of markov dependent trials . Ann. Inst. Statist. Math. , 49 : 531 – 539 .
  • Balakrishnan , N. and Koutras , M. V. 2002 . “ Runs and Scans with Applications ” . New York : John Wiley .
  • Balasubramanian , K. , Viveros , R. and Balakrishnan , N. 1993 . Sooner and later waiting time problems for markovian bernoulli trials . Statist. Probab. Lett. , 18 : 153 – 161 .
  • Chadjiconstantinidis , S. , Antzoulakos , D. L. and Koutras , M. V. 2000 . Joint distributions of successes, failures and patterns in enumeration problems . Adv. Appl. Prob. , 32 : 866 – 884 .
  • Ebneshahrashoob , M. and Sobel , M. 1990 . Sooner and later waiting time problems for bernoulli trials: frequency and run quotas . Statist. Probab. Lett. , 9 : 5 – 11 .
  • Feller , W. 1968 . An introduction to probability theory and its applications, , 3rd ed. Vol. I. , New York : John Wiley .
  • Fu , J. C. and Koutras , M. V. 1994 . Distribution theory of runs: a markov chain approach . J. Amer. Statist. Assoc. , 89 : 1050 – 1058 .
  • Gibbons , J. D. 1971 . Nonparametric statistical inference New York : McGraw-Hill .
  • Han , Q. and Aki , S. 1999 . Joint distributions of runs in a sequence of multi-state trials . Ann. Inst. Statist. Math. , 51 : 419 – 447 .
  • Han , Q. and Aki , S. 2000a . Waiting time problems in a two-state markov chain . Ann. Inst. Statist. Math. , 52 : 778 – 789 .
  • Han , S. and Aki , S. 2000b . A unified approach to binomial-type distributions of order k . Commun. Statist.—Theory Meth. , 29 ( 8 ) : 929 – 1943 .
  • 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 .
  • Inoue , K. and Aki , S. 2002 . Generalized binomial and negative binomial distributions of order k by the ℓ-overlapping enumeration scheme . Ann. Inst. Statist. Math. , to appear
  • Johnson , N. L. , Kotz , S. and Kemp , A. W. 1992 . Univariate discrete distributions New York : John Wiley .
  • Kolev , N. and Minkova , L. 1999 . Quotas on runs of successes and failures in a multi-state markov chain . Commun. Statist.—Theory Meth. , 28 ( 9 ) : 2235 – 2248 .
  • Koutras , M. V. 1997a . Waiting time distributions associated with runs of fixed length in two-state markov chains . Ann. Inst. Statist. Math. , 49 : 123 – 139 .
  • Koutras , M. V. 1997b . “ Waiting times and number of appearances of events in a sequence of discrete random variables ” . In Advances in Combinatorial Methods and Applications to Probability and Statistics Edited by: Balakrishnan , N. 363 – 384 . Boston : Birkhauser .
  • 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 .
  • Ling , K. D. 1988 . On binomial distributions of order k . Statist. Probab. Lett. , 6 : 247 – 250 .
  • Mohanty , S. G. 1994 . Success runs of length k in markov dependent trials . Ann. Inst. Statist. Math. , 46 : 777 – 796 .
  • Panjer , H. H. and Willmot , G. E. 1992 . Insurance Risk Models, American Society of Actuaries Illinois : Schaumburg .
  • 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 .
  • Uchida , M. and Aki , S. 1995 . Sooner and later waiting time problems in a two-state markov chain . Ann. Inst. Statist. Math. , 47 : 415 – 433 .

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.