\[ \begin{cases}\text{H}_0: \gamma_1 = 0 \\\text{H}_A: \gamma_1 \neq 0\end{cases} \]
The Skewness parameter \(\gamma_1\) (Section 59.5) is tested with the normal variant of the D’Agostino Skewness Test (D’Agostino 1970) as described in Section 59.6.
\[ \begin{cases}\text{H}_0: \beta_2 = 3 \\\text{H}_A: \beta_2 \neq 3\end{cases} \]
or, formulated in terms of “excess kurtosis”
\[ \begin{cases}\text{H}_0: \gamma_2 = 0 \\\text{H}_A: \gamma_2 \neq 0\end{cases} \]
The Kurtosis parameter \(\gamma_2\) (Section 59.9) is tested with the normal variant of the D’Agostino Kurtosis Test (which was adapted by Anscombe-Glynn (Anscombe and Glynn 1983)) as described in Section 59.10.
This test is usually performed with the Null value which corresponds to the Kurtosis of the Normal Distribution as described in Section 20.17.
\[ \begin{cases}\text{H}_0: \gamma_1 = 0 \text{ and } \gamma_2 = 0 \\\text{H}_A: \gamma_1 \neq 0 \text{ or } \gamma_2 \neq 0\end{cases} \]
The Jarque-Bera test (Jarque and Bera 1980) is based on the sum of squared Skewness and squared Kurtosis tests as described in Section 59.12.
These tests are often used as diagnostics (for example, to screen for departures from Normality or to flag tail/shape asymmetry before choosing a downstream method).
For the role-based threshold framework, see Chapter 104.
The software can be found on the public website (https://compute.wessa.net/rwasp_skewness_kurtosis.wasp) and on RFC (“Hypotheses / Empirical Tests”).
This R module contains the following fields:
The D’Agostino Skewness Test shows that the Skewness parameter \(\gamma_1 = 0.57707\) is significantly different from \(0\) (given a type I error of 1%) because the p-value \(= 0.005445 < 0.01\). The p-value was computed based on the Standard Normal Table for the z-value \(= 2.77940\) (cfr. Appendix E): p-value \(= 1 - 2 \times 0.49728\).
The result from the Anscombe-Glynn Test shows a Kurtosis value \(\beta_2 = 2.60620\) which has a z-score, i.e. \(z = -0.96443\), which can be found in the Standard Normal Table (cfr. Appendix E): p-value \(= 1 - 2 \times 0.33147\).
The Jarque-Bera Test shows a test statistic JB = 8.9225 which has a \(\chi^2\)-distribution with 2 degrees of freedom. A rough approximation of the corresponding p-value can be found in Appendix G.
To compute the Skewness & Kurtosis Tests on your local machine, the following script can be used in the R console:
D'Agostino skewness test
data: x
skew = 0.15243, z = 0.79241, p-value = 0.4281
alternative hypothesis: data have a skewness
Anscombe-Glynn kurtosis test
data: x
kurt = 2.69255, z = -0.67057, p-value = 0.5025
alternative hypothesis: kurtosis is not equal to 3
[1] 0.8021199
Jarque-Bera Normality Test
data: x
JB = 1.1716, p-value = 0.5567
alternative hypothesis: greatergeary(x) reports Geary’s normality/kurtosis-related coefficient test (Geary 1935) (based on the ratio of mean absolute deviation to standard deviation). For a normal distribution this coefficient is approximately \(\\sqrt{2/\\pi} \\approx 0.798\); strong deviations suggest non-normal tail behavior.
Generally speaking, the Skewness and Kurtosis Tests assume that the sample has a relatively large size. Especially the Jarque-Bera Test is known to be problematic when the sample size is small.
There are several alternatives for the Skewness and Kurtosis Tests shown above: