Multivariate tests-of-fit and uniform confidence bands using a weighted bootstrap

Many tests of fit procedures use the empirical distribution function (e.d.f.) of the data and have limiting distribution dependent on the data's underlying distribution or family of distribution functions. This paper uses a weighted bootstrap method based on independent random variables instead of sampling from the uniform. A proof of convergence of the weighted bootstrap is given using strong martingales. This approach is applied to the problem of obtaining uniform confidence bands for the distribution function of multivariate data. The multivariate two-sample problem, testing whether two independent random samples come from the same multivariate distribution function, is also discussed.