[1807.08216] Sign-Perturbed Sums: A New System Identification Approach for Constructing Exact Non-Asymptotic Confidence Regions in Linear Regression Models
Simulation experiments demonstrated that the SPS method works well, and that the confidence regions have similar size and shape as the heuristic ellipsoids of the asymptotic theory or the exact ellipsoids based on the F-distribution when the noise is i.i.d

Abstract: We propose a new system identification method, called Sign-Perturbed Sums
(SPS), for constructing non-asymptotic confidence regions under mild
statistical assumptions. SPS is introduced for linear regression models,
including but not limited to FIR systems, and we show that the SPS confidence
regions have exact confidence probabilities, i.e., they contain the true
parameter with a user-chosen exact probability for any finite data set.
Moreover, we also prove that the SPS regions are star convex with the
Least-Squares (LS) estimate as a star center. The main assumptions of SPS are
that the noise terms are independent and symmetrically distributed about zero,
but they can be nonstationary, and their distributions need not be known. The
paper also proposes a computationally efficient ellipsoidal outer approximation
algorithm for SPS. Finally, SPS is demonstrated through a number of simulation
experiments.

Figure 1. 95% confidence regions, n = 25, m = 100. Figure 2. 95% confidence regions, n = 400, m = 100. Figure 3. 95% confidence regions using various norms, n = 400, m = 100. (Second Order FIR System)Figure 4. 95% confidence regions, n = 25, m = 100. The solid line gives the SPS region. The dashed line gives the confidence ellipsoid based on the F-distribution. (Comparing with Exact Confidence Ellipsoids Based on the F-distribution)Figure 5. 95% confidence regions, n = 200, m = 100. The solid line gives the SPS region. The dashed line gives the confidence ellipsoid based on the F-distribution. (Comparing with Exact Confidence Ellipsoids Based on the F-distribution)Figure 6. 95% confidence regions, n = 25. The true system is a third order system, while the model is second order. (Comparing with Exact Confidence Ellipsoids Based on the F-distribution)Figure 7. 95% confidence regions, n = 200, m = 100. The solid line gives the standard SPS region and the dashed line gives the block SPS region. The dash dotted line gives the confidence ellipsoid based on asymptotic system identification theory. (Comparing with Exact Confidence Ellipsoids Based on the F-distribution)›