  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.
