Abstract
A \(\delta \)-record of a sequence of random variables is an observation that exceeds the current record by \(\delta \) units. This concept is relatively recent, so there are still many unknown aspects that require further research. In this work, for \(\delta \ge 0\), we investigate the distribution theory of \(\delta \)-record values from an iid sample from an absolutely continuous parent. We obtain recurrent expressions for the density function of \(\delta \)-records and the probability mass function of inter-\(\delta \)-record times. In the particular case of the exponential distribution, we show that the sequence of \(\delta \)-records are distributed as the points of a (delayed) renewal process. Finally, we point out some applications of \(\delta \)-records to paralyzable counters, blocks in automobile traffic, and queuing theory.
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Acknowledgments
The authors are grateful to the Editors and the referees for their helpful comments and remarks. This research has been supported by Grant MTM2010-16949 of the Spanish Ministry of Education and Science and Grants FQM331, FQM5849 of Junta de Andalucía, Spain.
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López-Blázquez, F., Salamanca-Miño, B. Distribution theory of \(\delta \)-record values: case \(\delta \ge 0\) . TEST 24, 558–582 (2015). https://doi.org/10.1007/s11749-014-0424-0
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DOI: https://doi.org/10.1007/s11749-014-0424-0