Bias and covariance of the least squares estimate in a structured errors-in-variables problem

Abstract

A structured errors-in-variables (EIV) problem arising in metrology is studied. The observations of a sensor response are subject to perturbation. The input estimation from the transient response leads to a structured EIV problem. Total least squares (TLS) is a typical estimation method to solve EIV problems. The TLS estimator of an EIV problem is consistent, and can be computed efficiently when the perturbations have zero mean, and are independently and identically distributed (i.i.d). If the perturbation is additionally Gaussian, the TLS solution coincides with maximum-likelihood (ML). However, the computational complexity of structured TLS and total ML prevents their real-time implementation. The least-squares (LS) estimator offers a suboptimal but simple recursive solution to structured EIV problems with correlation, but the statistical properties of the LS estimator are unknown. To know the LS estimate uncertainty in EIV problems, either structured or not, to provide confidence bounds for the estimation uncertainty, and to find the difference from the optimal solutions, the bias and variance of the LS estimates should be quantified. Expressions to predict the bias and variance of LS estimators applied to unstructured and structured EIV problems are derived. The predicted bias and variance quantify the statistical properties of the LS estimate and give an approximation of the uncertainty and the mean squared error for comparison to the Cramér–Rao lower bound of the structured EIV problem.