Approximately Unbiased Tests for Singular Surfaces via Multiscale Bootstrap Resampling

A class of approximately unbiased tests based on bootstrap probabilities is considered for the normal model with unknown expectation parameter vector, where the null hypothesis is represented as an arbitrary-shaped region with possibly singular boundary surfaces. We alter the sample size n of replicated datasets from the sample size n of the observed dataset, and calculate bootstrap probabilities at several n values. As shown in a previous paper, this multiscale bootstrap gives a bias-corrected p-value with third-order accuracy by differentiating bootstrap z-value with respect to √ n. However, the asymptotic theory is justified only for smooth boundary surfaces, and a breakdown of the theory is observed for cone shaped regions derived from inequality constraints. In this paper, we develop a linear theory, where the Fourier transformation of the boundary surface, instead of the Taylor series, plays an important role. A low-pass filter and its inverse filter represent calculation of bootstrap probability and an unbiased p-value, respectively. It turns out that the unbiased p-value is expressed as the bootstrap probability with n = −n. The obtained class of p-values includes, as special cases, the bootstrap probability and the third-order accurate p-value, and is equivalent in a certain sense to the bootstrap iteration. A new geometrical insight into the controversy over unbiasedness and monotonicity is given by showing that monotone tests, not the approximately unbiased tests, can be counterin-tuitive. The proposed procedure is illustrated in examples of phylogenetic inference and multiple comparisons.