  Consider discrete values of functions shifted by unobserved translation
effects, which are independent realizations of a random variable with unknown
distribution $\mu$, modeling the variability in the response of each
individual. Our aim is to construct a nonparametric estimator of the density of
these random translation deformations using semiparametric preliminary
estimates of the shifts. Building on results of Dalalyan et al. (2006),
semiparametric estimators are obtained in our discrete framework and their
performance studied. From these estimates we construct a nonparametric
estimator of the target density. Both rates of convergence and an algorithm to
construct the estimator are provided.
