Sign-perturbed sums (SPS): A method for constructing exact finite-sample confidence regions for general linear systems

In this paper we propose an algorithm for constructing non-asymptotic confidence regions for parameters of general linear systems under mild statistical assumptions. The constructed regions are centered around the prediction error estimate and are guaranteed to contain the "true" parameter with a user-chosen exact probability. Our main assumption is that the noise terms are independent and symmetrically distributed about zero, but they do not have to be stationary, nor do their variances and distributions have to be known. The construction of the region is based on the uniform ordering property of some carefully selected sign-perturbed sums (SPS) which, as we prove, rigorously guarantees the confidence probability for every finite dataset. The paper also investigates weighted estimates and presents a simulation example on an ARMA process that compares our exact confidence regions with the approximate ones based on the asymptotic theory.