On estimating population characteristics from record‐breaking observations. i. parametric results

Abstract

Abstract Consider an experiment in which only record‐breaking values (e.g., values smaller than all previous ones) are observed. The data available may be represented as X 1 , K 1 , X 2 , K 2 , …, where X 1 , X 2 , … are successive minima and K 1 , K 2 , … are the numbers of trials needed to obtain new records. We treat the problem of estimating the mean of an underlying exponential distribution, and we consider both fixed sample size problems and inverse sampling schemes. Under inverse sampling, we demonstrate certain global optimality properties of an estimator based on the “total time on test” statistic. Under random sampling, it is shown than an analogous estimator is consistent, but can be improved for any fixed sample size.