Shapiro-Wilk Test for Normality

总结自：

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

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

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: $W=\frac{{\left(\sum _{i=1}^n{a}_i{x}_{(i)}\right)}^2}{\sum _{i=1}^n(x_i-\bar{x}{)}^2}$, where
  • $x_{(i)}$ is the $i^{th}$ order statistic, i.e., the $i^{th}$-smallest number in the sample;
  • $\bar{x}=(x_1+\cdots +x_n)/n$ is the sample mean;
  • the constants $a_i$ are given by $(a_1,\dots ,a_n)=\frac{m^{\mathsf{T}}{V}^{-1}}{(m^{\mathsf{T}}{V}^{-1}{V}^{-1}m{)}^{1/2}}$ where
    • $m=(m_1,\dots ,m_n{)}^{\mathsf{T}}$,
    • and $m_1,\dots ,m_n$ 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.