Jarque-Bera Test

Jarque-Bera Test

The Jarque-Bera (JB) test checks whether a sample comes from a normal distribution by examining skewness and excess kurtosis. A normal distribution has skewness = 0 and excess kurtosis = 0.

The JB statistic is: $JB = \frac{n}{6} \left( S^2 + \frac{K^2}{4} \right)$

where:

• $n$ is the sample size
• $S$ is the sample skewness: $\frac{\frac{1}{n}\sum(x_i - \bar{x})^3}{\hat{\sigma}^3}$
• $K$ is the excess kurtosis: $\frac{\frac{1}{n}\sum(x_i - \bar{x})^4}{\hat{\sigma}^4} - 3$
• $\hat{\sigma}$ is the population standard deviation (divide by $n$)

Under the null hypothesis of normality, $JB$ follows a chi-squared distribution with 2 degrees of freedom. Large $JB$ values reject normality.

Symmetric data ($S = 0$) and mesokurtic data ($K = 0$) yield small JB. Skewed or heavy-tailed data yield large JB.

Your Task

Implement jarque_bera(xs) returning the JB statistic.