Jarque–Bera test

In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. The test is named after Carlos Jarque and Anil K. Bera. The test statistic is always nonnegative. If it is far from zero, it signals the data does not have a normal distribution. The test statistic JB is defined as J B = n 6 ( S 2 + 1 4 ( K − 3 ) 2 ) {\displaystyle {\mathit {JB}}={\frac {n}{6}}\left(S^{2}+{\frac {1}{4}}(K-3)^{2}\right)} where n is the number of observations (or degrees of freedom in general); S is the sample skewness, K is the sample kurtosis : S = μ ^ 3 σ ^ 3 = 1 n ∑ i = 1 n ( x i − x ¯ ) 3 ( 1 n ∑ i = 1 n ( x i − x ¯ ) 2 ) 3 / 2 , {\displaystyle S={\frac {{\hat {\mu }}_{3}}{{\hat {\sigma }}^{3}}}={\frac {{\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{3}}{\left({\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}\right)^{3/2}}},} K = μ ^ 4 σ ^ 4 = 1 n ∑ i = 1 n ( x i − x ¯ ) 4 ( 1 n ∑ i = 1 n ( x i − x ¯ ) 2 ) 2 , {\displaystyle K={\frac {{\hat {\mu }}_{4}}{{\hat {\sigma }}^{4}}}={\frac {{\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{4}}{\left({\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}\right)^{2}}},} where μ ^ 3 {\displaystyle {\hat {\mu }}_{3}} and μ ^ 4 {\displaystyle {\hat {\mu }}_{4}} are the estimates of third and fourth central moments, respectively, x ¯ {\displaystyle {\bar {x}}} is the sample mean, and σ ^ 2 {\displaystyle {\hat {\sigma }}^{2}} is the estimate of the second central moment, the variance. If the data comes from a normal distribution, the JB statistic asymptotically has a chi-squared distribution with two degrees of freedom, so...