Enhancements

As can be predicted, the bigger B, the number of bootstrap resamples, the better the approximations are, thus much recent work has been devoted to `making the most' of the computations :

• By choosing better quantities to bootstrap than the original estimate, through various transforms some of which require a double level bootstrap. (such as those in sections 2.1, 2.2 and 2.5 )
• By using techniques inspired from Monte Carlo methods : importance sampling, antithetic sampling. balanced sampling.
• By transforming the results obtained after an ordinary bootstrap process : bias correction EFRON 1982 [], accelerated bias correction EFRON 1987 [] for instance.

Preliminaries : Pivotal quantities
Construction of confidence intervals is based (ideally) on a pivotal quantity , that is a function of the sample and the parameter whose distribution is independant of the parameter , the sample, or any other unknown parameter. Thus for instance no knowledge is needed about the parameters and of a normal variable to construct confidence intervals for both of these parameters, because one has two pivotal quantities available :

Let denote the largest quantile of the distribution of the pivot , and the parameter space, set of all possible values for then

is a confidence set whose level is . Thus if , and denote the relevant quantiles for the distributions cited above the corresponding confidence intervals will be

and

Such an ideal situation is of course rare and one has to do with approximate pivots, (sometimes called roots). Therefore errors will be made on the level ( ) of the confidence sets. Level error is sensitive to how far is from being a pivot.

We will present several approaches for finding roots One approach is to studentize the root that is divide it by an estimate of its standard deviation. Another is to use a cdf transform constructed from a first set of bootstrap simulations.