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.