Almost-exact parametric bootstrap calculation via the saddlepoint approximation

One impediment to wider use of bootstrap methods is the large amount of computer time often required to compute bootstrap estimates. This paper shows how the saddlepoint approximation method can be used in certain situations to give very accurate parametric bootstrap statistics with a much smaller amount of calculation. Essentially, the problem of computing an n-dimentional integral (where n is the sample size) by Monte Carlo is reduced to an integral whose dimension is that of the parameter space, which is usually much smaller. This allows accurate numerical integration techniques to be used in place of simulation. The method should be especially effective for maximum likelihood in linear exponential family problems; it is illustrated by application to a problem in reliability.