Unitarily invariant errors-in-variables estimation

• Published in: Statistical papers (Berlin, Germany) Vol. 57; no. 4; pp. 1041 - 1057
• Main Author: Pesta, Michal
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2016; Springer Nature B.V
• Subjects: Approximation; Disturbances; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Error analysis; Errors; Estimators; Euclidean space; Finance; Insurance; Invariants; Least squares method; Management; Mathematical models; Mathematics; Mathematics and Statistics; Norms; Operations Research/Decision Theory; Optimization; Probability Theory and Stochastic Processes; Random variables; Regular Article; Statistics; Statistics for Business; Studies; Errors-in-variables model; 15B51; Total least squares; 62F10; Equivariance; Invariance; Unitarily invariant matrix norm; 62H12; 62J99
• ISSN: 0932-5026; 1613-9798
• DOI: 10.1007/s00362-016-0800-9

Linear relations, containing measurement errors in the input and output data, are considered. Parameters of these errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input–output disturbances, i.e., penalizing the orthogonal squared misfit. This approach corresponds to minimizing the Frobenius norm of the error matrix. An extension of the traditional TLS estimator in the EIV model—the EIV estimator—is proposed in the way that a general unitarily invariant norm of the error matrix is minimized. Such an estimator is highly non-linear. Regardless of the chosen unitarily invariant matrix norm, the corresponding EIV estimator is shown to coincide with the TLS estimator. Its existence and uniqueness is discussed. Moreover, the EIV estimator is proved to be scale invariant, interchange, direction, and rotation equivariant.