On delta-record observations: asymptotic rates for the counting process and elements of maximum likelihood estimation

Abstract

Given a sequence of random variables {X (n) ,na parts per thousand yen1} and delta aa"e, an observation X (n) is a delta-record if X (n) > max {X (1),aEuro broken vertical bar,X (n-1)}+delta. We obtain, for delta a parts per thousand currency sign0, weak and strong laws of large numbers for the counting process of delta-records among the first n observations from a sequence of independent identically distributed random variables, with common distribution F, possibly discontinuous. We provide examples of our results in the context of common probability distributions. Finally, we show how delta-records can be used for maximum likelihood estimation.