RESAMPLING - CONSISTENCY OF SUBSTITUTION ESTIMATORS

On the basis of N i.i.d. random variables with a common unknown distri bution P we wish to estimate a functional tau(N)(P) An obvious and ver y general approach to this problem is to find an estimator (P) over ca p(N) of P first, and then construct a so-called substitution estimator tau(N)((P) over cap(N)) of tau(N)(P). In this paper we investigate ho w to choose the estimator (P) over cap(N) so that the substitution est imator tau(N)((P) over cap(N)) Will be consistent. Although our setup covers a broad class of estimation problems, the main substitution est imator we have in mind is a general version of the bootstrap where res ampling is done from an estimated distribution (P) over cap(N). We do not focus in advance on a particular estimator (P) over cap(N), such a s, 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 pape r is that the bootstrap is an exceptionally flexible method which come s 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. Unfor tunately, this may not always be available.