When the Cramér-rao Inequality Provides No Information

We investigate a one-parameter family of probability densities (related to the Pareto distribution, which describes many natural phenomena) where the Cramér-Rao inequality provides no information. 1. Cramér-Rao Inequality One of the most important problems in statistics is estimating a population parameter from a finite sample. As there are often many different estimators, it is desirable to be able to compare them and say in what sense one estimator is better than another. One common approach is to take the unbiased estimator with smaller variance. For example, if X1, . . . , Xn are independent random variables uniformly distributed on [0, θ], Yn = maxi Xi and X = (X1 + · · ·+Xn)/n, then n+1 n Yn and 2X are both unbiased estimators of θ but the former has smaller variance than the latter and therefore provides a tighter estimate. Two natural questions are (1) which estimator has the minimum variance, and (2) what bounds are available on the variance of an unbiased estimator? The first question is very hard to solve in general. Progress towards its solution is given by the Cramér-Rao inequality, which provides a lower bound for the variance of an unbiased estimator (and thus if we find an estimator that achieves this, we can conclude that we have a minimum variance unbiased estimator). Date: February 5, 2008. 2000 Mathematics Subject Classification. 62B10 (primary), 62F12, 60E05 (secondary).