基于L (2) Wasserstein距离的正态性检验的Bootstrap方法.pdf

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应用概率统计第 38 卷 1. Introduction
In statistical theory and application, many statistical methods often require data to be
normally distributed, such as: Z test, t test, χ2 test, F test, variance analysis, correlation
analysis, parameter estimation and parameter hypothesis testing, etc. For instance, in
linear regression analysis, a basic assumption is that the residuals are normally distributed;
in t test, the first thing is to determine whether the sample population follows the normal
distribution.
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The project was supported by Beijing Union University Foundation (Grant No. 11202JA2007).
⋆Corresponding author, E-mail: ******@.
Received June 13, 2019. Revised December 12, Chinese Journal of Applied Probability and Statistics Vol. 38
The goodness-of-fit hypothesis includes two kinds: the simple null hypothesis and the
composite null hypothesis. The simple null hypothesis:
H0 : F (x) = F0(x),
where F0(x) is a completely known distribution function. The composite null hypothesis:
H0 : F (x) ∈ Ψ0,
where Ψ0 is a normal distribution family with unknown parameters.
Many tests have been developed to check the normality assumption. The goodness-
of-fit tests are very important normality testing methods. The goodness-of-fit tests are
defined in various ways, mainly in two types: one is empirical distribution function test,
the other is correlation and regression test.
The empirical distribution function test studies the differences between the edf Fn(x)
and the reference normal distribution F0(x), such as: Kolmogorov-Smirnov test (see [1,2]),
Cram´er-Von Mises test (see [3, 4]), Anderson-Darling test (see [5]). Kolmogorov-Smirnov
test:
√
Dn