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Considering the problem of risk-sensitive parameter estimation, we propose afairly wide family of lower bounds on the exponential moments of the quadraticerror, both in the Bayesian and the non--Bayesian regime. This family ofbounds, which is based on a change of measures, offers considerable freedom inthe choice of the reference measure, and our efforts are devoted to explorethis freedom to a certain extent. Our focus is mostly on signal models that arerelevant to communication problems, namely, models of a parameter-dependentsignal (modulated signal) corrupted by additive white Gaussian noise, but themethodology proposed is also applicable to other types of parametric families,such as models of linear systems driven by random input signals (white noise,in most cases), and others. In addition to the well known motivations of therisk-sensitive cost function (i.e., the exponential quadratic cost function),which is most notably, the robustness to model uncertainty, we also view thiscost function as a tool for studying fundamental limits concerning the tailbehavior of the estimation error. Another interesting aspect, that wedemonstrate in a certain parametric model, is that the risk-sensitive costfunction may be subjected to phase transitions, owing to some analogies withstatistical mechanics.

Lower Bounds on Exponential Moments of the Quadratic Error in Parameter Estimation