Lower Bounds on Estimator Error and the Threshold Effect

Rather than compute the exact error of a specific estimator, it is often more convenient to apply results which lower-bound the error of any estimator for a given problem. We present three such bounds and illustrate their applications to signal parameter estimation. The Cramèr-Rao bound is simple to compute and approximates the actual error under conditions of high signal-to-noise ratio. However, it fails to provide useful information for low signal-to-noise ratio conditions, and thus tighter bounds are required. The Barankin and Ziv-Zakai bounds approximate the Cramèr-Rao bound for high signal-to-noise ratio conditions but are significantly tighter than it for low signal-to-noise ratios. This situation demonstrates theoretically the well-known threshold effect for sonar time-delay estimation.