A Robbins-Monro procedure for the estimation of parametric deformations on random variables

TL;DR

This paper introduces a Robbins-Monro type recursive estimator for parametric deformations of i.i.d. random variables, utilizing Wasserstein distance, and also proposes a recursive density estimator that adapts based on parameter estimates, demonstrated through simulations.

Contribution

It presents a novel recursive estimation method for parametric deformations and density estimation that is computationally simple and effective, based on Wasserstein distance and kernel density ideas.

Findings

Effective recursive estimator for deformation parameters

Accurate density estimation incorporating parameter estimates

Successful simulation results for Box-Cox and arcsinh transformations

Abstract

The paper is devoted to the study of a parametric deformation model of independent and identically random variables. Firstly, we construct an efficient and very easy to compute recursive estimate of the parameter. Our stochastic estimator is similar to the Robbins-Monro procedure where the contrast function is the Wasserstein distance. Secondly, we propose a recursive estimator similar to that of Parzen-Rosenblatt kernel density estimator in order to estimate the density of the random variables. This estimate takes into account the previous estimation of the parameter of the model. Finally, we illustrate the performance of our estimation procedure on simulations for the Box-Cox transformation and the arcsinh transformation.