Estimation of the distribution of random shifts deformation

Consider discrete values of functions shifted by unobserved translation effects, which are independent realizations of a random variable with unknown distribution μ 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. Based on the results of Dalalyan et al . [7], 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.