Resampling: consistency of substitution estimators

On the basis of N i.i.d. random variables with a common unknown distribution P we wish to estimate a functional $\tau_N(P)$. An obvious and very general approach to this problem is to find an estimator $\hat{P}_N$ of P first, and then construct a so-called substitution estimator $\tau_N (\hat{P}_N)$ of $\tau_N(P)$. In this paper we investigate how to choose the estimator $\hat{P}_N$ so that the substitution estimator $\tau_N (\hat{P}_N)$ will be consistent. Although our setup covers a broad class of estimation problems, the main substitution estimator we have in mind is a general version of the bootstrap where resampling is done from an estimated distribution $\hat{P}_N$. We do not focus in advance on a particular estimator $\hat{P}_N$, such as, for example, the empirical distribution, but try to indicate which resampling distribution should be used in a particular situation. The conclusion that we draw from the results and the examples in this paper is that the bootstrap is an exceptionally flexible method which comes into its own when full use is made of its flexibility. However, the choice of a good bootstrap method in a particular case requires rather precise information about the structure of the problem at hand. Unfortunately, this may not always be available.