On statistics which are almost sufficient from the viewpoint of the Fisher metrics

A statistic on a statistical model is sufficient if it has no information loss, namely, the Fisher metric of the induced model coincides with that of the original model due to Kullback and Ay-Jost-L\^e-Schwachh\"ofer. We introduce a quantitatively weak version of sufficient statistics such that the Fisher metric of the induced model is bi-Lipschitz equivalent to that of the original model. We characterize such statistics in terms of the conditional probability or by the existence of a certain decomposition of the density function in a way similar to characterizations of sufficient statistics due to Fisher-Neyman and Ay-Jost-L\^e-Schwachh\"ofer.