  The raking-ratio method is a statistical and computational method which
adjusts the empirical measure to match the true probability of sets of a finite
partition. We study the asymptotic behavior of the raking-ratio empirical
process indexed by a class of functions when the auxiliary information is given
by estimates. We suppose that these estimates result from the learning of the
probability of sets of partitions from another sample larger than the sample of
the statistician, as in the case of two-stage sampling surveys. Under some
metric entropy hypothesis and conditions on the size of the information source
sample, we establish the strong approximation of this process and show in this
case that the weak convergence is the same as the classical raking-ratio
empirical process. We also give possible statistical applications of these
results like the strengthening of the $Z$-test and the chi-square goodness of
fit test.
