Asymptotic normality of total least squares estimator in a multivariate errors-in-variables model $AX=B$

TL;DR

This paper investigates the asymptotic normality of the total least squares estimator in a multivariate errors-in-variables model, providing conditions under which the estimator converges to a Gaussian distribution as data size grows.

Contribution

It establishes the asymptotic normality of the total least squares estimator in a multivariate errors-in-variables model with uncorrelated, row-wise independent errors, extending understanding of its statistical properties.

Findings

Conditions for asymptotic normality are derived.

The covariance of the limit Gaussian is nonsingular under mild assumptions.

Results enable construction of confidence intervals for linear functionals of X.

Abstract

We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of $X$, which, in the case of normal errors, coincides with the maximum likelihood one. We give conditions for asymptotic normality of the estimator when the number of rows in $A$ is increasing. Under mild assumptions, the covariance structure of the limit Gaussian random matrix is nonsingular. For normal errors, the results can be used to construct an asymptotic confidence interval for a linear functional of $X$.