Equivariant adjusted least squares estimator in two-line fitting model

TL;DR

This paper develops and compares several equivariant estimators for the two-line fitting problem with Gaussian noise, providing conditions for their consistency and asymptotic normality.

Contribution

It introduces new equivariant estimators for the two-line fitting model and analyzes their theoretical properties and numerical performance.

Findings

Projections of the adjusted least squares estimator are consistent and asymptotically normal.

Numerical comparisons show the effectiveness of the proposed estimators.

Conditions for estimator consistency and asymptotic normality are established.

Abstract

We consider the two-line fitting problem. True points lie on two straight lines and are observed with Gaussian perturbations. For each observed point, it is not known on which line the corresponding true point lies. The parameters of the lines are estimated. This model is a restriction of the conic section fitting model because a couple of two lines is a degenerate conic section. The following estimators are constructed: two projections of the adjusted least squares estimator in the conic section fitting model, orthogonal regression estimator, parametric maximum likelihood estimator in the Gaussian model, and regular best asymptotically normal moment estimator. The conditions for the consistency and asymptotic normality of the projections of the adjusted least squares estimator are provided. All the estimators constructed in the paper are equivariant. The estimators are compared numerically.