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] - 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]