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On the joint distribution of success runs of several lengths in the sequence of MBT and its applications

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  • Kirtee Kamalja

Abstract

Let $$X_1 ,X_2 ,\ldots ,X_n $$ X 1 , X 2 , … , X n be a sequence of Markov Bernoulli trials (MBT) and $$\underline{X}_n =( {X_{n,k_1 } ,X_{n,k_2 } ,\ldots ,X_{n,k_r } })$$ X ̲ n = ( X n , k 1 , X n , k 2 , … , X n , k r ) be a random vector where $$X_{n,k_i } $$ X n , k i represents the number of occurrences of success runs of length $$k_i \,( {i=1,2,\ldots ,r})$$ k i ( i = 1 , 2 , … , r ) . In this paper the joint distribution of $$\underline{X}_n $$ X ̲ n in the sequence of $$n$$ n MBT is studied using method of conditional probability generating functions. Five different counting schemes of runs namely non-overlapping runs, runs of length at least $$k$$ k , overlapping runs, runs of exact length $$k$$ k and $$\ell $$ ℓ -overlapping runs (i.e. $$\ell $$ ℓ -overlapping counting scheme), $$0\le \ell >k$$ 0 ≤ ℓ > k are considered. The pgf of joint distribution of $$\underline{X}_n $$ X ̲ n is obtained in terms of matrix polynomial and an algorithm is developed to get exact probability distribution. Numerical results are included to demonstrate the computational flexibility of the developed results. Various applications of the joint distribution of $$\underline{X}_n $$ X ̲ n such as in evaluation of the reliability of $$( {n,f,k})\!\!:\!\!G$$ ( n , f , k ) : G and $$>n,f,k>\!:\!\!G$$ > n , f , k > : G system, in evaluation of quantities related to start-up demonstration tests, acceptance sampling plans are also discussed. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Kirtee Kamalja, 2014. "On the joint distribution of success runs of several lengths in the sequence of MBT and its applications," Statistical Papers, Springer, vol. 55(4), pages 1179-1206, November.
  • Handle: RePEc:spr:stpapr:v:55:y:2014:i:4:p:1179-1206
    DOI: 10.1007/s00362-013-0560-8
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    References listed on IDEAS

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    1. Qing Han & Sigeo Aki, 2000. "Waiting Time Problems in a Two-State Markov Chain," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 778-789, December.
    2. Anant Godbole & Stavros Papastavridis & Robert Weishaar, 1997. "Formulae and Recursions for the Joint Distribution of Success Runs of Several Lengths," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(1), pages 141-153, March.
    3. Kiyoshi Inoue & Sigeo Aki, 2003. "Generalized binomial and negative binomial distributions of orderk by thel-overlapping enumeration scheme," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(1), pages 153-167, March.
    4. Sigeo Aki & Katuomi Hirano, 2000. "Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 767-777, December.
    5. Han, Qing & Aki, Sigeo, 1998. "Formulae and recursions for the joint distributions of success runs of several lengths in a two-state Markov chain," Statistics & Probability Letters, Elsevier, vol. 40(3), pages 203-214, October.
    6. Ling, K. D., 1988. "On binomial distributions of order k," Statistics & Probability Letters, Elsevier, vol. 6(4), pages 247-250, March.
    7. K. Kotwal & R. Shinde, 2006. "Joint distributions of runs in a sequence of higher-order two-state Markov trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(3), pages 537-554, September.
    8. N. Balakrishnan & P. Chan, 2000. "Start-Up Demonstration Tests with Rejection of Units upon Observing d Failures," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(1), pages 184-196, March.
    9. Qing Han & Sigeo Aki, 1999. "Joint Distributions of Runs in a Sequence of Multi-State Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(3), pages 419-447, September.
    10. Kiyoshi Inoue & Sigeo Aki, 2005. "Joint distributions of numbers of success runs of specified lengths in linear and circular sequences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(2), pages 353-368, June.
    11. Chang, Gerard J. & Cui, Lirong & Hwang, Frank K., 1999. "Reliabilities for (n,f,k) systems," Statistics & Probability Letters, Elsevier, vol. 43(3), pages 237-242, July.
    12. Balasubramanian, K. & Viveros, R. & Balakrishnan, N., 1993. "Sooner and later waiting time problems for Markovian Bernoulli trials," Statistics & Probability Letters, Elsevier, vol. 18(2), pages 153-161, September.
    13. Kiyoshi Inoue & Sigeo Aki, 2007. "Joint Distributions of Numbers of Runs of Specified Lengths in a Sequence of Markov Dependent Multistate Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(3), pages 577-595, September.
    14. Shinde, R.L. & Kotwal, K.S., 2006. "On the joint distribution of runs in the sequence of Markov-dependent multi-state trials," Statistics & Probability Letters, Elsevier, vol. 76(10), pages 1065-1074, May.
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