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An Asymptotic Conditional Test of Independence in Bernoulli Sequences Using the Number of Runs Given the Number of Successes

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  • Sungsu Kim

    (University of Louisiana at Lafayette)

  • Chong Jin Park

    (San Diego State University)

Abstract

In this paper, we prove the asymptotic normality of the conditional distribution of the number of runs given the number of successes for a sequence of independent Bernoulli random variables. In our proof, the Frobenius-Harper technique is used to represent the number of runs as the sum of independent and not necessarily identically distributed Bernoulli random variables. Then, an asymptotic conditional test for independence is provided. Our simulation results exhibit that the test based on conditional distribution performs better that one based on unconditional distribution, over the entire range of success probability and first order correlation. In addition, the UMVUEs of the factorial moments and the probabilities of the number of runs are presented in this paper.

Suggested Citation

  • Sungsu Kim & Chong Jin Park, 2021. "An Asymptotic Conditional Test of Independence in Bernoulli Sequences Using the Number of Runs Given the Number of Successes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 143-154, February.
    • Handle: RePEc:spr:sankha:v:83:y:2021:i:1:d:10.1007_s13171-019-00176-1
    DOI: 10.1007/s13171-019-00176-1
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    References listed on IDEAS

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    1. D. Antzoulakos & S. Bersimis & M. Koutras, 2003. "On the distribution of the total number of run lengths," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(4), pages 865-884, December.
    2. Philippou, Andreas N. & Makri, Frosso S., 1986. "Successes, runs and longest runs," Statistics & Probability Letters, Elsevier, vol. 4(2), pages 101-105, March.
    3. Philippou, Andreas N. & Makri, Frosso S., 1986. "Successes, runs and longest runs," Statistics & Probability Letters, Elsevier, vol. 4(4), pages 211-215, June.
    4. Kim, Sungsu & Park, Chongjin & Oh, Jungtaek, 2013. "On waiting time distribution of runs of ones or zeros in a Bernoulli sequence," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 339-344.
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