Confidence Analysis of Linear Unbiased Estimates under Uncertain Unimodal Noise Distributions

Abstract

The estimation problem for a linear parametric function in a linear regression model with uncertain symmetric unimodal noise distributions and given noise covariances is solved. The quality of estimation is determined by the probability that the error exceeds specified limits. For taking the uncertainty into account, a minimax optimization problem is formulated. The main result of this paper is that the Gauss inequality defines a tight upper bound of the error probability for any linear unbiased estimate. The worst-case distribution of the noise vector is constructed by using a linear transformation of a Gaussian random vector and a uniform random variable. The estimate that is minimax with respect to the probability criterion coincides with the least-squares estimate. The obtained solution is illustrated by the example of estimating motion parameters for several hypotheses about noise distributions.