The weighted likelihood bootstrap and an algorithm for prepivoting

Abstract

The method of bootstrapping, which has transformed the theory and practice of frequentist statistical inference, is applicable within the Bayesian paradigm. Rather than simulating data that might have been observed, this Bayesian extension, called the weighted likelihood bootstrap, involves simulating parameters corresponding to distributions that might have generated the observed data. The weighted likelihood bootstrap is an extension of earlier work by D. Rubin (Annals of Statistics, 1981) from purely nonparametric models into semi and fully parametric models for data. The resulting simulation, which is viewed as simply a Monte Carlo approximation to a posterior distribution of interest, has desirable asymptotic properties. This simulation method produces easily generated samples from a posterior under an effective prior which can be identified either exactly or approximately in certain models. The simulation is straightforward, requiring only an algorithm for maximum likelihood estimation. It is also closely related to frequentist bootstrapping procedures. The weighted likelihood bootstrap is applied to a wide variety of statistical models.The prepivoting procedure is studied in a general modeling framework and an efficient Monte Carlo algorithm, called bootstrap recycling, is introduced. This algorithm is shown to be simulation consistent; that is, it produces a closer approximation to the right answer as the amount of computing resources gets large. This new algorithm, which is an alternative to the iterated bootstrap, is applied to the likelihood ratio test of a sparse contingency table, and to the construction of likelihood based confidence sets in a complex stochastic model.