Abstract: We develop analytical approximations to bootstrap distribution functions of statistics that are smooth functions of vector means. In particular, our technique is applicable in the case of bootstrap inference for a population mean, for a Studentized mean, and for other more complex situations, such as inference for population variances or correlation coefficients. The approximations are based on the application of a tail probability approximation of DiCiccio and Martin (1991) to a saddlepoint approximation for the joint density of several means. Our method extends the work of Davison and Hinkley (1988), who proposed the use of saddlepoint methods to replace bootstrap resampling primarily in the case of linear statistics, and the work of Daniels and Young (1991), who considered the problem of inference for a Studentized mean. Our technique produces accurate approximations over the entire range of the distribution function and is easy to implement. It has two critical advantages over standard resampling techniques: it can yield significant computational savings; and it is more accurate than standard resampling approaches based on 5,000 or 10,000 resamples. We illustrate these points by applying our technique to estimate the bootstrap distribution of a bivariate correlation coefficient for both real and simulated data; and the method performs well in each case. Finally, we illustrate the power and flexibility of our technique in the very complex problem of estimating the bootstrap distribution of a Studentized, transformed correlation coefficient.
Key words and phrases: Asymptotic approximations, correlation coefficient, cumulant generating function, Fisher's z transformation, iterated bootstrap, moment generating function, normal approximation, pivotal statistic, resampling, simulation, Studentized statistic, tail probability approximations, vector means.