Shapiro-Wilk Test for Normality

总结自：

• Wikipedia: Shapiro–Wilk test
• Perform a Shapiro-Wilk Normality Test

• Shapiro: [ʃəˈpirəu]
• normality: [nɔ:ˈmæləti]

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Given a sample $ x_1, \cdots, x_n $,

• $ H_0 $: the sample come from a normally distributed population
• $ H_a $: the sample does not come from a normally distributed population
• The test statistic is: $ W = \frac{\left(\sum_{i=1}^n a_i x_{(i)}\right)^2}{\sum_{i=1}^n (x_i-\overline{x})^2} $, where
  • $ x_{(i)} $ is the $ i^{th} $ order statistic, i.e., the $ i^{th} $-smallest number in the sample;
  • $ \overline{x} = \left( x_1 + \cdots + x_n \right) / n $ is the sample mean;
  • the constants $ a_i $ are given by $ (a_1,\dots,a_n) = {m^{\mathsf{T}} V^{-1} \over (m^{\mathsf{T}} V^{-1}V^{-1}m)^{1/2}} $ where
    • $ m = (m_1,\dots,m_n)^{\mathsf{T}} $,
    • and $ m_1,\ldots,m_n $ are the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution,
    • and $ V $ is the covariance matrix of those order statistics.