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 $X$.
Highlights
We deal with overdetermined system of linear equations AX ≈ B, which is common in linear parameter estimation problem [9]
In [5] a more general, element-wise weighted total least squares (TLS) estimator was studied, where the errors in [A, B] were row-wise independent, but within each row, the entries could be observed without errors, and, the error covariance matrix could differ from row to row
The TLS estimator Xtls is finite iff there exists an unconstrained minimum of the function (2.7), and Xtls is a minimum point of that function
Summary
We deal with overdetermined system of linear equations AX ≈ B, which is common in linear parameter estimation problem [9]. We show that under mild conditions, the normalized estimator converges in distribution to a Gaussian random matrix with nonsingular covariance structure. For normal errors, the latter structure can be estimated consistently based on the observed matrix [A, B]. The latter structure depends continuously on some nuisance parameters of the model, and we derive consistent estimators for those parameters. By Ip we denote the unit matrix of size p
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