Robust relative error estimation

TL;DR

This paper introduces a robust method for relative error estimation in regression that effectively handles outliers using a gamma-likelihood approach, with proven theoretical properties and demonstrated practical performance.

Contribution

It develops a gamma-likelihood based robust estimation method with a new MM algorithm and establishes its asymptotic properties, improving outlier resistance in relative error regression.

Findings

The proposed method reduces outlier sensitivity in relative error estimation.

The algorithm guarantees decrease of the objective function at each step.

Simulation and real data show improved robustness and accuracy.

Abstract

Relative error estimation has been recently used in regression analysis. A crucial issue of the existing relative error estimation procedures is that they are sensitive to outliers. To address this issue, we employ the $\gamma$-likelihood function, which is constructed through $\gamma$-cross entropy with keeping the original statistical model in use. The estimating equation has a redescending property, a desirable property in robust statistics, for a broad class of noise distributions. To find a minimizer of the negative $\gamma$-likelihood function, a majorize-minimization (MM) algorithm is constructed. The proposed algorithm is guaranteed to decrease the negative $\gamma$-likelihood function at each iteration. We also derive asymptotic normality of the corresponding estimator together with a simple consistent estimator of the asymptotic covariance matrix, so that we can readily construct approximate confidence sets. Monte Carlo simulation is conducted to investigate the effectiveness of the proposed procedure. Real data analysis illustrates the usefulness of our proposed procedure.