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The use of the bootstrap test lends both elegance and simplicity to the analysis of complicated statistical problems. Such a numerical approach yields an estimated P-value as a binomial proportion. The purpose of this article is to recommend that confidence intervals for the bootstrap P-value be calculated routinely. In constructing these intervals, due consideration needs to be given to the P-values that may be close to or equal to zero. We consider three instructive examples and related methods for designing the bootstrap sample size. [ABSTRACT FROM AUTHOR]

AbstractWe propose a new distributional regression model for a multivariate response vector based on a copula process over the covariate space. It uses the implicit copula of a Gaussian multivariate regression, which we call a “regression copula process”. To allow for large covariate vectors the regression coefficients are regularized using a novel multivariate extension of the horseshoe prior. Bayesian inference and distributional predictions are evaluated using efficient variational inference methods, allowing application to large datasets. An advantage of the approach is that the marginal distributions of the response vector can be estimated separately and accurately, resulting in predictive distributions that are marginally calibrated. Two applications of the methodology illustrate its efficacy. The first is the econometric modeling and prediction of half-hourly regional Australian electricity prices. The second is the evaluation of multivariate posteriors in likelihood-free inference (LFI) for a model of tree species abundance. This extends a previous univariate regression copula LFI method for evaluating univariate posteriors. In both applications, we demonstrate that our new approach exhibits a desirable marginal calibration property. [ABSTRACT FROM AUTHOR]