Jarque–Bera test

The Jarque - Bera test is a statistical test, which checks based on the skewness and kurtosis in the data, whether a normal distribution. Therefore, there is a special fit test. The test was proposed by Carlos M. Jarque and Anil K. Bera.

Definition

The test statistic JB the Jarque-Bera test is defined as

This number is the observations; with the skewness and kurtosis denoted by.

The skewness in the data is defined as follows:

For symmetric distributions such as the normal distribution is the theoretical value of the skewness is zero.

Kurtosis is a measure of the curvature of a distribution, in the normal distribution has a value of three. Values ​​are above, indicate that the distribution of fat has ends, i.e. that the density of distribution at the edges, for example, outside of normal ± 2σ barrier, and greater for in the central regions is less than that the normal distribution. This applies, for example the t-distribution. Kurtosis is defined as follows:

And said third and fourth central moment represent the average of the sample and the second torque, so the variance of symbol.

It is, i.e., the test statistic is asymptotic chi-squared distribution with two degrees of freedom.

The pair of hypotheses is:

At a significance level applies: For values ​​of the test statistic about 4.6 of the normal distribution, the hypothesis will be discarded; for the significance levels and results in the barriers 6 and 7.8.