Multistep-Multiscale Bootstrap Resampling
Approximately unbiased tests based on the bootstrap probabilities are considered for the exponential family of distributions with unknown expectation parameter vector, where the null hypothesis is represented as an arbitrary-shaped region with smooth boundaries. The newly developed multistep-multiscale bootstrap resampling calculates an approximately unbiased p-value with asymptotic third-order accuracy.
This p-value is equivalent to those obtained by the double bootstrap (Hall 1992) and the p*-formula (Barndorff-Nielsen 1986), yet the implementation of the new algorithm leads to simpler and faster computation. In fact, the simplest form (one-step multiscale bootstrap) has been already used in Bioinformatics applications (Shimodaira 2002), and the paper (Shimodaira and Hasegawa 2001) for the software implementing the algorithm is highly cited.
The multistep-multiscale bootstrap resampling is described in Shimodaira (2004a) [PDF]. The technical details are given in Shimodaira (2004b) [PDF]. The main result is proved in Shimodaira (2003) [HTML], which is a computer program for Mathematica with MathTensor. The multiscale bootstrap of Shimodaira (2002) is a special case of the multistep-multiscale bootstrap, i.e., the one-step multiscale bootstrap.
The abstract is available in Japanese language Shimodaira (2004c) [PDF]. See Japanese page.
- MAIN TEXT: Shimodaira (2004a) [PDF] (published in Annals of Statistics).
- SUPPLEMENTARY MATERIALS: Shimodaira (2004b) [PDF] and Shimodaira (2003) [HTML].
- PDFs (Shimodaira 2003, 2004a, 2004b) 1.3MB
- HTML (Shimodaira 2003) 2.4MB
- Notebook (Shimodaira 2003) 0.2MB
- H. Shimodaira and M. Hasegawa (2001). CONSEL: for assessing the confidence of phylogenetic tree selection, Bioinformatics, 17, 1246-1247. [PDF] [SOFTWARE]
- H. Shimodaira (2002). An approximately unbiased test of phylogenetic tree selection, Systematic Biology, 51, 492-508. [PDF]
- H. Shimodaira (2003). Asymptotic analysis of the bootstrap methods. Mathematica notebook document. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan. (Available in [HTML], [PDF], and [Notebook.])
- H. Shimodaira (2004a). Approximately unbiased tests of regions using multistep-multiscale bootstrap resampling, Annals of Statistics, 32, 2616-2641. [PDF]. (Earlier version of this paper is Research Reports B-402. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan. [Older PDF])
- H. Shimodaira (2004b). Technical details of the multistep-multiscale bootstrap resampling. Research Reports B-403. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan. [PDF]
- H. Shimodaira (2004c). Geometry of bootstrap method and scale transformation (in Japanese). Special lecture at Mathematical Society of Japan. Hokkaido, Japan. [PDF]