A sufficiency paradox: an insufficient statistic preserving the fisher information

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 (., X) where . is a binary random variable and, given ., X has a density f (x . .|.) depending on a location parameter .. The phenomenon is based on the fact that f (x|.) smoothly vanishes at one point; it can be eliminated by adding to the regularity of a statistical experiment positivity of the density function.