Shannon Information Properties of the Endpoints of Record Coverage

Abstract

This paper addresses the largest and the smallest observations, at the times when a new record of either kind (upper or lower) occurs, which are it called the current upper and lower record, respectively. We examine the entropy properties of these statistics, especially the difference between entropy of upper and lower bounds of record coverage. The results are presented for some common parametric families of distributions. Several upper and lower bounds, in terms of the entropy of parent distribution, for the entropy of current records are obtained. It is shown that mutual information, as well as Kullback–Leibler distance between the endpoints of record coverage, Kullback–Leibler distance between data distribution, and current records, are all distribution-free.

Keywords:

• Kullback–Leibler distance
• Mutual information
• Order statistics
• Record range
• Record values

Mathematics Subject Classification:

• Primary 62G30
• Secondary 62B10, 94A17

Acknowledgments

The research was done while the first author was on sabbatical leave at The Australian National University, Canberra. Much appreciation goes to Professor Peter Hall for his hospitality. The authors would like to thank the referee for his/her careful reading and useful comments that improved presentation of the manuscript. This paper was partially supported by the Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad.

Notes

The first author is a member of Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad.