Estimation of the distribution of random shifts deformation

TL;DR

This paper develops a nonparametric method to estimate the distribution of random translation deformations in shifted functions, using semiparametric preliminary estimates and providing convergence rates and an algorithm.

Contribution

It introduces a novel nonparametric density estimator for random shifts based on semiparametric preliminary estimates in a discrete setting.

Findings

Derived convergence rates for the estimator

Provided an explicit algorithm for implementation

Extended previous results to a discrete framework

Abstract

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.