Asymptotic Concentration

Suppose the problem is to estimate based on observations from a model governed by the parameter (). What is the best asymptotic performance of an estimator sequence Tn for

To simplify the situation, we shall in most of this chapter assume that the sequence converges in distribution under every possible value of. Next we rephrase the question as: What are the best possible limit distributions? In analogy with the CramerRao theorem a “best” limit distribution is referred to as an asymptotic lower bound. Under certain restrictions the normal distribution with mean zero and covariance the inverse Fisher information is an asymptotic lower bound for estimating in a smooth parametric model. This is the main result of this chapter, but it needs to be qualified.

The notion of a “best” limit distribution is understood in terms of concentration. If the limit distribution is a priori assumed to be normal, then this is usually translated into asymptotic unbiasedness and minimum variance. The statement that converges in distribution to a distribution can be roughly understood in the sense that eventually Tn is approximately normally distributed with mean and variance given by

Because Tn is meant to estimate, optimal choices for the asymptotic mean and variance are and variance as small as possible. These choices ensure not only that the asymptotic mean square error is small but also that the limit distribution is maximally concentrated near zero. For instance, the probability of the interval is maximized by choosingminimal.

We do not wish to assume a priori that the estimators are asymptotically normal. That normal limits are best will actually be an interesting conclusion.