Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials

ABSTRACT

In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1)* i.e*. we have a sequence {(X_{n}/Y_{n}), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (*X*^{0}_{n,k10},*X*^{1}_{n,k11};*Y*^{0}_{n,k20},*Y*^{1}_{n,k21}) where for *i*=0,1,*X*^{i}_{n,k1i} denotes the number of occurrences of *i*-runs of length *k*^{1}_{i} in the first component and *Y*^{i}_{n,k2i} denotes the number of occurrences of *i*-runs of length *k*^{2}_{i} in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ_{1} and Λ_{2} of lengths *k*_{1} and *k*_{2} respectively and obtain the pgf of joint distribution of (*X*_{n,Λ 1},*Y*_{n,Λ2} ) using method of conditional probability generating functions where *X*_{n,Λ1}(*Y*_{n,Λ2}) denotes the number of occurrences of pattern Λ_{1}(Λ_{2} ) of length *k*_{1} (*k*_{2}) in the first (second) *n* components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.

In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1)

KEYWORDS

Markov Dependent Bivariate trials, Conditional Probability Generating Function, Joint Distribution

Markov Dependent Bivariate trials, Conditional Probability Generating Function, Joint Distribution

Cite this paper

nullK. Kamalja and R. Shinde, "Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials,"*Open Journal of Statistics*, Vol. 1 No. 2, 2011, pp. 115-127. doi: 10.4236/ojs.2011.12014.

nullK. Kamalja and R. Shinde, "Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials,"

References

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[2] N. Balakrishnan and P. S. Chan, “Start-up Demonstration Tests with Rejection of Units upon Observing d Failures,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 1, 2000, pp. 184-196. doi:10.1023/A:1004101402897

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[5] S. Aki and K. Hirano, “Number of Success-Runs of Specified Length until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 4, 2000, pp. 767-777. doi:10.1023/A:1017585512412

[6] A. M. Mood, “The Distribution Theory of Runs,” Annals of Mathematical Statistics, Vol. 11, No. 4, 1940, pp. 367-392. doi:10.1214/aoms/1177731825

[7] S. Aki and K. Hirano, “Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials,” Annals of Institute of Statistical Mathematics, Vol. 51, No. 1, 1999, pp. 17-29. doi:10.1023/A:1003874900507

[8] M. Uchida, “On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables,” Annals of Institute of Statistical Mathematics, Vol. 50, No. 4, 1998, pp. 655-671. doi:10.1023/A:1003756712643

[9] D. L. Antzoulakos, “Waiting Times for Patterns in a Sequence of Multistate Trials,” Journal of Applied Probability, Vol. 38, No. 2, 2001, pp. 508-518. doi:10.1239/jap/996986759

[10] K. Inoue and S. Aki, “Generalized Waiting Time Problems Associated with Pattern in Polya’s Urn Scheme,” Annals of Institute of Statistical Mathematics, Vol. 54, No. 3, 2002, pp. 681-688. doi:10.1023/A:1022431631697

[11] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Patterns in the Sequence of Markov Dependent Multi- State Trials,” 2010 Submitted.

[12] S. Aki and K. Hirano, “Waiting Time Problems for a Two-Dimensional Pattern,” Annals of Institute of Statistical Mathematics, Vol. 56, No. 1, 2004, 169-182. doi:10.1007/BF02530530

[13] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Runs in a Sequence of Higher-Order Two-State Markov trials,” Annals of Institute of Statistical Mathematics, Vol. 58, No. 1, 2006, pp. 537-554. doi:10.1007/s10463-005-0024-6

[14] A. A. Salvia and W. C. Lasher, “2-Dimensional Consecutive k-out-of-n: F Models,” IEEE Transactions on Reliability, Vol. R-39, No. 3, 1990, pp. 382-385. doi:10.1109/24.103023

[15] R. L. Shinde and K. S. Kotwal, “On the Joint Distribution of Runs in the Sequence of Markov Dependent Multi- State Trials,” Statistics & Probability Letters, Vol. 76, No. 10, 2006, pp. 1065-1074. doi:10.1016/j.spl.2005.12.005

[1] S. J. Schwager, “Run Probabilities in Sequences of Markov-Dependent Trials,” Journal of American Statistical Association, Vol. 78, No. 381, 1983, pp. 168-175. doi:10.2307/2287125

[2] N. Balakrishnan and P. S. Chan, “Start-up Demonstration Tests with Rejection of Units upon Observing d Failures,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 1, 2000, pp. 184-196. doi:10.1023/A:1004101402897

[3] W. Feller, “An Introduction to Probability Theory and Its Applications,” 3rd Edition, John Wiley & Sons, Hoboken, 1968.

[4] K. D. Ling, “On Binomial Distributions of Order k,” Statistics & Probability Letters, Vol. 6, No. 4, 1988, pp. 247-250. doi:10.1016/0167-7152(88)90069-7

[5] S. Aki and K. Hirano, “Number of Success-Runs of Specified Length until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 4, 2000, pp. 767-777. doi:10.1023/A:1017585512412

[6] A. M. Mood, “The Distribution Theory of Runs,” Annals of Mathematical Statistics, Vol. 11, No. 4, 1940, pp. 367-392. doi:10.1214/aoms/1177731825

[7] S. Aki and K. Hirano, “Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials,” Annals of Institute of Statistical Mathematics, Vol. 51, No. 1, 1999, pp. 17-29. doi:10.1023/A:1003874900507

[8] M. Uchida, “On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables,” Annals of Institute of Statistical Mathematics, Vol. 50, No. 4, 1998, pp. 655-671. doi:10.1023/A:1003756712643

[9] D. L. Antzoulakos, “Waiting Times for Patterns in a Sequence of Multistate Trials,” Journal of Applied Probability, Vol. 38, No. 2, 2001, pp. 508-518. doi:10.1239/jap/996986759

[10] K. Inoue and S. Aki, “Generalized Waiting Time Problems Associated with Pattern in Polya’s Urn Scheme,” Annals of Institute of Statistical Mathematics, Vol. 54, No. 3, 2002, pp. 681-688. doi:10.1023/A:1022431631697

[11] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Patterns in the Sequence of Markov Dependent Multi- State Trials,” 2010 Submitted.

[12] S. Aki and K. Hirano, “Waiting Time Problems for a Two-Dimensional Pattern,” Annals of Institute of Statistical Mathematics, Vol. 56, No. 1, 2004, 169-182. doi:10.1007/BF02530530

[13] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Runs in a Sequence of Higher-Order Two-State Markov trials,” Annals of Institute of Statistical Mathematics, Vol. 58, No. 1, 2006, pp. 537-554. doi:10.1007/s10463-005-0024-6

[14] A. A. Salvia and W. C. Lasher, “2-Dimensional Consecutive k-out-of-n: F Models,” IEEE Transactions on Reliability, Vol. R-39, No. 3, 1990, pp. 382-385. doi:10.1109/24.103023

[15] R. L. Shinde and K. S. Kotwal, “On the Joint Distribution of Runs in the Sequence of Markov Dependent Multi- State Trials,” Statistics & Probability Letters, Vol. 76, No. 10, 2006, pp. 1065-1074. doi:10.1016/j.spl.2005.12.005