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Statistical inference for discrete patterns in dependent sequences

Research Project

Project/Area Number 14540114
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKansai University (2004-2005)
Osaka University (2002-2003)

Principal Investigator

AKI Sigeo  Kansai University, Faculty of Engineering, Professor, 工学部, 教授 (90132696)

Co-Investigator(Kenkyū-buntansha) 谷口 正信  大阪大学, 大学院・基礎工学研究科, 助教授 (00116625)
稲垣 宣生  大阪大学, 大学院・基礎工学研究科, 教授 (10000184)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsdiscrete patterns / discrete distribution theory / probability generating function / Markov chain / urn models / system reliability / waiting time problem / conditional expectation / 条件付き確率 / 離散確率分布 / 起動試験 / 工学的システムの信頼性 / start-up demonstration test / 幾何分布 / 条件付確率 / 壷のモデル
Research Abstract

We have studied the distributions of numbers of occurrences of discrete patterns and waiting time distributions for them in various dependent sequences. The following results have been derived.
1. By using a combinatorial method it is shown that for every finite pattern, the distribution of the waiting time for the reversed pattern coincides with that of the waiting time for the original pattern in a multi-state dependent sequence with a certain type of exchangeability. The corresponding results for the waiting time for the r-th occurrence of the pattern, and for the number of occurrences of a specified pattern in n trials are also studied. Illustrative examples based on urn models are also given.
2. Several waiting time random variables for a duplication within a memory window of size k in a sequence of {1,2,...,m}-valued random variables are investigated. The exact distributions of the waiting time random variables are derived by the method of conditional probability generating functio … More ns. In particular, the exact distribution of the waiting time for the first k-match is obtained when the underlying sequence is generated by higher order Markov dependent trials. Examples for numerical calculations are also given.
3. We consider waiting time problems for two-dimensional pattern in a sequence of i.i.d. random vectors each of whose entries is 0 or 1. A general method for obtaining the exact distribution of the waiting time for the first occurrence of the pattern in the sequence is presented. The method is an extension of the method of conditional probability generating functions and it is very suitable for computations with computer algebra systems as well as usual numerical computations. Computational results applied to computation of exact system reliability are also given.
4. Let k and m be positive integers with k>m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k-1 is obtained.
5. Various new distributions related to runs and patterns in dependent sequences are obtained by using the stepwise smoothing formula for conditional expectations. Less

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (23 results)

All 2005 2004 2003 2002 Other

All Journal Article (18 results) Publications (5 results)

  • [Journal Article] A generalized Polya urn model and related multivariate distributions2005

    • Author(s)
      Inoue, K, Aki, S
    • Journal Title

      Annals of the Institute of statistical Mathematics 57・1

      Pages: 49-59

    • NAID

      40006712676

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Joint distributions of numbers of success runs of specified lengths in linear and circular sequences2005

    • Author(s)
      Inoue, K, Aki, S
    • Journal Title

      Annals of the Institute of statistical Mathematics 57・2

      Pages: 353-368

    • NAID

      40006751309

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A generalized Polya urn model and related multivariate distributions2005

    • Author(s)
      Inoue, K., Aki, S.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 57-1

      Pages: 49-59

    • NAID

      40006712676

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Joint distributions of numbers of success runs of specified lengths in linear and circular sequences2005

    • Author(s)
      Inoue, K., Aki, S.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 57-2

      Pages: 353-368

    • NAID

      40006751309

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A generalized Polya urn model and related multivariate distributions2005

    • Author(s)
      Inoue, K., Aki, S.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 57・1

      Pages: 49-59

    • NAID

      40006712676

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Joint distributions of numbers of success runs of specified lengths in linear and circular sequences2005

    • Author(s)
      inoue, K., Aki, S.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 57・2

      Pages: 353-368

    • NAID

      40006751309

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Waiting time problems for a two-dimensional pattern2004

    • Author(s)
      Aki.S., Hirano, K.
    • Journal Title

      Annals of the Insti tute of statistical Mathematics 56・1

      Pages: 169-182

    • NAID

      40006149248

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Waiting time problems for a two-dimensional pattern2004

    • Author(s)
      Aki, S., Hirano, K.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 56-1

      Pages: 169-182

    • NAID

      40006149248

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Waiting time problems for a two-dimensional pattern2004

    • Author(s)
      Aki, S., Hirano, K.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 56・1

      Pages: 169-182

    • NAID

      40006149248

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On k-match problems2003

    • Author(s)
      Hirano, K., Aki, S.
    • Journal Title

      Journal of Statistical Planning and Inference 109

      Pages: 67-79

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generalized binomial and negative binoiial distributions of order k by the l-overlapping enumeration scheme2003

    • Author(s)
      Inoue, K., Aki, S.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 55・1

      Pages: 153-167

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Nuiber of occurrences of subpattern until the first appearance of a pattern and geometric distribution2003

    • Author(s)
      Hirano, K., Aki, S.
    • Journal Title

      Statistics & Probability Letters 65

      Pages: 259-262

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generalized binomial and negative binomial distributions of order k by the l-overlapping enumeration scheme2003

    • Author(s)
      Inoue, K., Aki, S.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 55-1

      Pages: 153-167

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Number of occurrences of subpattern until the first appearance of a pattern and geometric distribution2003

    • Author(s)
      Hirano, K., Aki, S.
    • Journal Title

      Statistics & Probability Letters 65

      Pages: 259-262

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Generalized waiting time problems associated with pattern in Polya's urn scheme2002

    • Author(s)
      Inoue, K, Aki, S
    • Journal Title

      Annals of the Institute of Statistical Mathematics 54・3

      Pages: 681-688

    • NAID

      40005584327

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On waiting time for reversed patterns in random sequences2002

    • Author(s)
      Aki.S., Hirano, K.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 54・4

      Pages: 713-718

    • NAID

      40005626539

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generalized waiting time problems associated with pattern in Polya's urn scheme2002

    • Author(s)
      Inoue, K., Aki, S.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 54-3

      Pages: 681-688

    • NAID

      40005584327

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On waiting time for reversed patterns in random sequences2002

    • Author(s)
      Aki, S., Hirano, K.
    • Journal Title

      Annals of the Institute of Statistical Mathematics 54-4

      Pages: 713-718

    • NAID

      40005626539

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] Aki, S.: "Generalized binomial and negative binomial distributions of order k by the l-overlapping enumeration scheme"Annals of the Institute of Statistical Mathematics. 55・1. 153-167 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Aki, S.: "Waiting time problems for a two-dimensional pattern"Annals of the Institute of Statistical Mathematics. (発表予定). (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Inoue, K., Aki, S.: "Generalized waiting time problems associated with pattern in Polya's urn scheme"Annals of the Institute of Statistical Mathematics. 54・3. 681-688 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Aki, S., Hirano, K.: "On waiting time for reversed patterns in random sequences"Annals of the Institute of Statistical Mathematics. 54・4. 713-718 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hirano, K., Aki, S.: "On k-match problems"Journal of Statistical Planning and Inference. 109. 67-79 (2003)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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