Maximum—Entropy Moments—Based Approximations of the Cramer–Rao Bound
Nicolás von Ellenrieder, Carlos Horacio Muravchik, Marcelo de Souza Lauretto, Carlos Alberto de Bragança Pereira, Julio Michael Stern · AIP conference proceedings · 2008
In parametric estimation problems, under the appropriate hypotheses the Cramér–Rao bound (CRB) gives the greatest lower bound of the variance of any unbiased estimator. To compute the CRB it is necessary to know the distribution of the concerned random variables, but in some practical situations this distribution may not be known. In this work we present an approximation to the CRB when only some low order moments of the random variables are known. This approximation is obtained by finding the maximum entropy solution to the moment problem. An alternative approximation is introduced, which makes use of the expression derived for the maximum entropy distribution, but it does not require solving the moment problem, which is the most computationally demanding task in the first approximation.We present some examples to compare the true CRB and the proposed approximations. As expected, the approximations seem to converge to the real bound when an increasing number of moments is used. For the studied cases a good approximation of the CRB is obtained with a relatively small number of moments.