Bootstrap goodness-of-fit tests with estimated parameters based on empirical transforms

Several test statistics have been proposed recently which employ a weighted distance that depends on an empirical transform, as well as on estimated parameters. The empirical characteristic function is a typical example, but alternative empirical transforms have also been employed, such as the empirical Laplace transform when dealing with non-negative random variables or the empirical probability generating function corresponding to discrete observations. We propose a general formulation that covers most of the transform-based test statistics which have appeared in the literature. Under this formulation, the asymptotic properties of the test statistics, such as the limiting null distribution and the consistency under general alternatives, are derived. Since large-sample critical values are extremely complicated (if not impossible) to compute, two effective bootstrap versions of the test procedures are derived, which can be used to approximate the critical values, for any given sample size, and to calculate the power under contiguous alternatives. The validity of these bootstrap procedures is shown analytically.