Equal Fisher Information in Order Statistics

摘要

Any collection of order statistics from two different probability distributions may contain equal Fisher information about a scalar parameter. We derive a necessary and sufficient condition under which two distributions have equal Fisher information in any order statistics. Hence this condition can be used to define an equivalence relation on parametric distributions. Within the location (scale) family of distributions, we show that this equivalence relation uniquely determines the parametric family by the values of the Fisher information about the location (scale) parameter in any order statistics. The results are used to derive some location-scale distribution and obtain a simple characterization in terms of the Fisher information in the sequence of the minimum order statistics.