The Gaussian Data Assumption Leads to the Largest Cramér-Rao Bound [Lecture Notes]

The Gaussian data assumption is sometimes criticized as being unrealistic in certain applications. While this is a valid criticism, it is also true that the Gaussian assumption is a natural choice when nothing is known about the exact data distribution. The argument typically used to motivate this choice relies on the fact that the Gaussian distribution leads to the largest Cramer-Rao bound (CRB) in quite a general class of data distributions and for a significant set of parameter estimation problems. Consequently, the Gaussian CRB (i.e., the CRB that holds under the Gaussian assumption) is the worst-case one (over the distribution class), and therefore any optimal design based on attaining or minimizing it, including the parameter estimation operation itself, can be inter preted as being min-max optimal. In this lecture note, we provide a simple and yet quite general proof of the aforesaid fact that the Gaussian assump tion yields the largest CRB. This fact, which is sometimes considered to be a "folk theorem," is possibly known to many (see the cited works); however finding a proof of it in the literature, of comparable generality to that presented here, has eluded us.