Semiparametric regression estimation using noisy nonlinear non invertible functions of the observations

TL;DR

This paper develops a semiparametric regression framework for noisy, nonlinear, non-invertible functions of observations, with applications to bearings-only tracking, establishing estimator properties and efficiency bounds.

Contribution

It introduces a novel semiparametric estimation approach for complex nonlinear models, proving consistency, asymptotic normality, and efficiency of estimators under mild conditions.

Findings

Least squares estimator is consistent and asymptotically normal.

The model exhibits local asymptotic normality, enabling efficiency bounds.

Parametric likelihood estimator is shown to be regular and efficient.

Abstract

We investigate a semiparametric regression model where one gets noisy non linear non invertible functions of the observations. We focus on the application to bearings-only tracking. We first investigate the least squares estimator and prove its consistency and asymptotic normality under mild assumptions. We study the semiparametric likelihood process and prove local asymptotic normality of the model. This allows to define the efficient Fisher information as a lower bound for the asymptotic variance of regular estimators, and to prove that the parametric likelihood estimator is regular and asymptotically efficient. Simulations are presented to illustrate our results.