def shapiro(x):
  """
  Perform the Shapiro-Wilk test for normality.

  The Shapiro-Wilk test tests the null hypothesis that the
  data was drawn from a normal distribution.

  Parameters
  ----------
  x : array_like
    Array of sample data.

  Returns
  -------
  W : float
    The test statistic.
  p-value : float
    The p-value for the hypothesis test.

def kstest(rvs, cdf, args=(), N=20, alternative='two-sided', mode='approx'):
  """
  Perform the Kolmogorov-Smirnov test for goodness of fit.

  This performs a test of the distribution G(x) of an observed
  random variable against a given distribution F(x). Under the null
  hypothesis the two distributions are identical, G(x)=F(x). The
  alternative hypothesis can be either 'two-sided' (default), 'less'
  or 'greater'. The KS test is only valid for continuous distributions.

  Parameters
  ----------
  rvs : str, array or callable
    If a string, it should be the name of a distribution in `scipy.stats`.
    If an array, it should be a 1-D array of observations of random
    variables.
    If a callable, it should be a function to generate random variables;
    it is required to have a keyword argument `size`.
  cdf : str or callable
    If a string, it should be the name of a distribution in `scipy.stats`.
    If `rvs` is a string then `cdf` can be False or the same as `rvs`.
    If a callable, that callable is used to calculate the cdf.
  args : tuple, sequence, optional
    Distribution parameters, used if `rvs` or `cdf` are strings.
  N : int, optional
    Sample size if `rvs` is string or callable. Default is 20.
  alternative : {'two-sided', 'less','greater'}, optional
    Defines the alternative hypothesis (see explanation above).
    Default is 'two-sided'.
  mode : 'approx' (default) or 'asymp', optional
    Defines the distribution used for calculating the p-value.

     - 'approx' : use approximation to exact distribution of test statistic
     - 'asymp' : use asymptotic distribution of test statistic

  Returns
  -------
  statistic : float
    KS test statistic, either D, D+ or D-.
  pvalue : float
    One-tailed or two-tailed p-value.

def normaltest(a, axis=0, nan_policy='propagate'):
  """
  Test whether a sample differs from a normal distribution.

  This function tests the null hypothesis that a sample comes
  from a normal distribution. It is based on D'Agostino and
  Pearson's [1]_, [2]_ test that combines skew and kurtosis to
  produce an omnibus test of normality.


  Parameters
  ----------
  a : array_like
    The array containing the sample to be tested.
  axis : int or None, optional
    Axis along which to compute test. Default is 0. If None,
    compute over the whole array `a`.
  nan_policy : {'propagate', 'raise', 'omit'}, optional
    Defines how to handle when input contains nan. 'propagate' returns nan,
    'raise' throws an error, 'omit' performs the calculations ignoring nan
    values. Default is 'propagate'.

  Returns
  -------
  statistic : float or array
    ``s^2 + k^2``, where ``s`` is the z-score returned by `skewtest` and
    ``k`` is the z-score returned by `kurtosistest`.
  pvalue : float or array
    A 2-sided chi squared probability for the hypothesis test.

def anderson(x, dist='norm'):
  """
  Anderson-Darling test for data coming from a particular distribution

  The Anderson-Darling tests the null hypothesis that a sample is
  drawn from a population that follows a particular distribution.
  For the Anderson-Darling test, the critical values depend on
  which distribution is being tested against. This function works
  for normal, exponential, logistic, or Gumbel (Extreme Value
  Type I) distributions.

  Parameters
  ----------
  x : array_like
    array of sample data
  dist : {'norm','expon','logistic','gumbel','gumbel_l', gumbel_r',
    'extreme1'}, optional
    the type of distribution to test against. The default is 'norm'
    and 'extreme1', 'gumbel_l' and 'gumbel' are synonyms.

  Returns
  -------
  statistic : float
    The Anderson-Darling test statistic
  critical_values : list
    The critical values for this distribution
  significance_level : list
    The significance levels for the corresponding critical values
    in percents. The function returns critical values for a
    differing set of significance levels depending on the
    distribution that is being tested against.

Critical values provided are for the following significance levels:

  normal/exponenential
    15%, 10%, 5%, 2.5%, 1%
  logistic
    25%, 10%, 5%, 2.5%, 1%, 0.5%
  Gumbel
    25%, 10%, 5%, 2.5%, 1%

import numpy as np
from scipy import stats

a = np.random.normal(0,2,50)
b = np.linspace(0, 10, 100)

# Shapiro-Wilk test
S,p = stats.shapiro(a)
print('the shapiro test result is:',S,',',p)

# kstest（K-S檢驗）
K,p = stats.kstest(a, 'norm')
print(K,p)

# normaltest
N,p = stats.normaltest(b)
print(N,p)

# Anderson-Darling test
A,C,p = stats.anderson(b,dist='norm')
print(A,C,p)