  Sign-Perturbed Sums (SPS) is a system identification method that constructs
confidence regions for the unknown system parameters. In this paper, we study
SPS for ARX systems, and establish that the confidence regions are guaranteed
to include the true model parameter with exact, user-chosen, probability under
mild statistical assumptions, a property that holds true for any finite number
of observed input-output data. Furthermore, we prove the strong consistency of
the method, that is, as the number of data points increases, the confidence
region gets smaller and smaller and will asymptotically almost surely exclude
any parameter value different from the true one. In addition, we also show
that, asymptotically, the SPS region is included in an ellipsoid which is
marginally larger than the confidence ellipsoid obtained from the asymptotic
theory of system identification. The results are theoretically proven and
illustrated in a simulation example.
