asymptotic analysis of the bootstrap methods 20030721

Hidetoshi Shimodaira

Department of Mathematical and Computing Sciences

Tokyo Institute of Technology

Ookayama, Meguro, Tokyo 152-8552, JAPAN.

shimo (a) is.titech.ac.jp

http://www.is.titech.ac.jp/~shimo/

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Asymptotic Analysis of the Bootstrap Methods

This documentation consists of three parts, namely "Exponential Family of Distributions", "Tube-Coordinates and -formula", and "Bootstrap Methods". Each part is an independent *Mathematica* session, and should be run separately.

The calculation is third-order accurate, correct up to terms ignoring the error of . We use "o" to indicate quantities. So, indicates and indicates , etc. We repeatedly use a simplification function to keep only up to terms.

In the first two parts, the tensor notation is heavily used. The add-on package "MathTensor" is required to run the *Mathematica* session by yourself.

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Exponential Family of Distributions
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Tube-Coordinates and -formula
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Bootstrap Methods
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Converted by *Mathematica*
July 21, 2003