【Stats】Jarque Bera test正态性检验

Jarque Bera test

JB检验主要检验样本数据的skewness和kurtosis是否与正态分布相匹配，统计检验的结果是非负的，如果检验值比0大很多，那么表示样本数据不符合normal distribution.
JB检验的统计量定义为
$$JB=\frac{n}{6}(S^2+\frac{1}{4}(K?3{)}^2)$$
其中，$n$表示样本数量，或者自由度(d.o.f)，$S$表示样本skewness，$K$表示样本kurtosis:
$$?\begin{array}{l}S=\frac{{\hat{\mu }}_3}{{\hat{\sigma }}^3}=\frac{\frac{1}{n}{\sum }_{i=1}^n(x_i?\bar{x}{)}^3}{(\frac{1}{n}{\sum }_{i=1}^n(x_i?\bar{x}{)}^2{)}^{\frac{3}{2}}}\\ K=\frac{{\hat{\mu }}_4}{{\hat{\sigma }}_4}=\frac{\frac{1}{n}{\sum }_{i=1}^n(x_i?\bar{x}{)}^4}{(\frac{1}{n}{\sum }_{i=1}^n(x_i?\bar{x}){)}^2}\end{array}$$
如果数据采样自一个正态分布的总体，那么JB检验渐近于$\chi (2)$分布，假设检验的零假设为一个联合假设(joint hypothesis)：偏度为0且超额峰度为0.

If the data comes from a normal distribution, the JB statistic asymptotically has a chi-squared distribution with two degrees of freedom，so the statistic can be used to test the hypothesis that the data are from a normal distribution. The null hypothesis is a joint hypothesis of the skewness being zero and the excess kurtosis being zero. Samples from a normal distribution have an expected skewness of 0 and an expected excess kurtosis of 0. As the definition of JB shows, any deviation from this increases the JB statistic.

Scipy.stats.jarque_bera

scipy.stats.jarque_bera(x)

Parameter: x: array_like  // obsverations of a random variable
Returns:jb_value: float, the test statisticp: the p-value for the hypothesis test

statsmodels.stats.stattools.jarque_bera

``statsmodels.stats.stattools.jarque_bera(resids, axis=0)

Parameters:resids: array_like // data to test for normalityaxis: int
Returns:JB: float, the Jarque-Bera test statisticJBpv: float, the pvalue of test statisticskew: estimated skewness of the datakurtosis: estimated kurtosis of the data

industry 12 portfolio JB test

对美国12个行业指数returns进行JB检验

def JB_test(df, ind_names):if df is None or ind_names is None: raise Exception('Data Frame is None')for ind_name in ind_names:ind_returns = df[ind_name]jb_ans = JB(ind_returns)# print(jb_ans)_jbstats, _pvalue = jb_ans.statistic, jb_ans.pvalueprint(f'{ind_name}\t{_jbstats}\t{_pvalue}')

参考资料

python数据正态性检验
Jarque Bera test. wiki
JB正态性检验
scipy jarque_bera