# Asymptotic normality of element-wise weighted total least squares estimator in a multivariate errors-in-variables model

## TL;DR

This paper investigates the asymptotic normality of a weighted total least squares estimator in a multivariate errors-in-variables model with complex error structures, providing conditions for its distribution as data size grows.

## Contribution

It establishes conditions under which the element-wise weighted total least squares estimator is asymptotically normal in a multivariable errors-in-variables model.

## Key findings

- Conditions for asymptotic normality are derived.
- The limiting Gaussian distribution has a nonsingular covariance.
- The model accounts for row-wise correlated errors and varying error covariances.

## Abstract

A multivariable measurement error model $AX \approx B$ is considered. Here $A$ and $B$ are input and output matrices of measurements and $X$ is a rectangular matrix of fixed size to be estimated. The errors in $[A,B]$ are row-wise independent, but within each row the errors may be correlated. Some of the columns are observed without errors and the error covariance matrices may differ from row to row. The total covariance structure of the errors is known up to a scalar factor. The fully weighted total least squares estimator of $X$ is studied. We give conditions for asymptotic normality of the estimator, as the number of rows in $A$ is increasing. We provide that the covariance structure of the limiting Gaussian random matrix is nonsingular.