Frequentist and Bayesian measures of confidence via multiscale bootstrap for testing three regions

TL;DR

This paper introduces a multiscale bootstrap method for computing both frequentist $p$-values and Bayesian posterior probabilities for complex hypotheses in multivariate normal models, extending applicability to two-sided tests and improving parameter estimation.

Contribution

It develops new parametric models for bootstrap probability scaling, enabling two-sided confidence measures and bridging frequentist and Bayesian approaches.

Findings

Extended bootstrap method to two-sided tests.

Improved parameter estimation with higher-order terms.

Interpreted Bayesian posterior as a frequentist $p$-value.

Abstract

A new computation method of frequentist $p$-values and Bayesian posterior probabilities based on the bootstrap probability is discussed for the multivariate normal model with unknown expectation parameter vector. The null hypothesis is represented as an arbitrary-shaped region. We introduce new parametric models for the scaling-law of bootstrap probability so that the multiscale bootstrap method, which was designed for one-sided test, can also computes confidence measures of two-sided test, extending applicability to a wider class of hypotheses. Parameter estimation is improved by the two-step multiscale bootstrap and also by including higher-order terms. Model selection is important not only as a motivating application of our method, but also as an essential ingredient in the method. A compromise between frequentist and Bayesian is attempted by showing that the Bayesian posterior probability with an noninformative prior is interpreted as a frequentist $p$-value of ``zero-sided'' test.