On Consistency and Asymptotic Normality of Least Absolute Deviation Estimators for 2-dimensional Sinusoidal Model

TL;DR

This paper introduces robust least absolute deviation estimators for 2D sinusoidal models, demonstrating their strong consistency, asymptotic normality, and practical advantages over least squares methods, especially with outliers or heavy-tailed noise.

Contribution

It develops and analyzes LAD estimators for 2D sinusoidal models, proving their asymptotic properties and validating their effectiveness through simulations and real data analysis.

Findings

LAD estimators are strongly consistent and asymptotically normal.

LAD outperforms least squares in the presence of outliers.

Simulation and real data confirm practical advantages of LAD.

Abstract

Estimation of the parameters of a 2-dimensional sinusoidal model is a fundamental problem in digital signal processing and time series analysis. In this paper, we propose a robust least absolute deviation (LAD) estimators for parameter estimation. The proposed methodology provides a robust alternative to non-robust estimation techniques like the least squares estimators, in situations where outliers are present in the data or in the presence of heavy tailed noise. We study important asymptotic properties of the LAD estimators and establish the strong consistency and asymptotic normality of the LAD estimators of the signal parameters of a 2-dimensional sinusoidal model. We further illustrate the advantage of using LAD estimators over least squares estimators through extensive simulation studies. Data analysis of a 2-dimensional texture data indicates practical applicability of the proposed LAD approach.