A Sufficiency Paradox: An Insufficient Statistic Preserving the Fisher Information

摘要

An example of a regular statistical experiment is constructed where an insufficient statistic preserves the Fisher information contained in the data. The data are a pair (Δ, X) where Δ is a binary random variable and, given Δ, X has a density f(x葇Δ) depending on a location parameter θ. The phenomenon is based on the fact that f(x|Δ) smoothly vanishes at one point; it can be eliminated by adding to the regularity of a statistical experiment positivity of the density function.