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Distribution theory of $$\delta $$ δ -record values: case $$\delta \ge 0$$ δ ≥ 0

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  • Fernando López-Blázquez
  • Begoña Salamanca-Miño

Abstract

A $$\delta $$ δ -record of a sequence of random variables is an observation that exceeds the current record by $$\delta $$ δ units. This concept is relatively recent, so there are still many unknown aspects that require further research. In this work, for $$\delta \ge 0$$ δ ≥ 0 , we investigate the distribution theory of $$\delta $$ δ -record values from an iid sample from an absolutely continuous parent. We obtain recurrent expressions for the density function of $$\delta $$ δ -records and the probability mass function of inter- $$\delta $$ δ -record times. In the particular case of the exponential distribution, we show that the sequence of $$\delta $$ δ -records are distributed as the points of a (delayed) renewal process. Finally, we point out some applications of $$\delta $$ δ -records to paralyzable counters, blocks in automobile traffic, and queuing theory. Copyright Sociedad de Estadística e Investigación Operativa 2015

Suggested Citation

  • Fernando López-Blázquez & Begoña Salamanca-Miño, 2015. "Distribution theory of $$\delta $$ δ -record values: case $$\delta \ge 0$$ δ ≥ 0," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 558-582, September.
    • Handle: RePEc:spr:testjl:v:24:y:2015:i:3:p:558-582
    DOI: 10.1007/s11749-014-0424-0
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    References listed on IDEAS

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    1. Li, Yun, 1999. "A note on the number of records near the maximum," Statistics & Probability Letters, Elsevier, vol. 43(2), pages 153-158, June.
    2. Gouet, Raúl & López, F. Javier & Maldonado, Lina P. & Sanz, Gerardo, 2014. "Statistical inference for the geometric distribution based on δ-records," Computational Statistics & Data Analysis, Elsevier, vol. 78(C), pages 21-32.
    3. Eliazar, Iddo, 2005. "On geometric record times," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 348(C), pages 181-198.
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