# TEST_XKURT - Excess Kurtosis Test

Calculates the p-value of the statistical test for the population excess kurtosis (4th moment).

## Syntax

TEST_XKURT(X, Return_type, $\alpha$)

X
is the input data sample (one/two-dimensional array of cells (e.g., rows or columns)).
Return_type
is a switch to select the return output (1 = P-Value (default), 2 = Test Stats, 3 = Critical Value.
Method Description
1 P-Value.
2 Test Statistics (e.g., Z-score).
3 Critical Value.
$\alpha$
is the statistical significance of the test (i.e., alpha). If missing or omitted, an alpha value of 5% is assumed.

## Remarks

1. The data sample may include missing values (e.g., #N/A).
2. The test hypothesis for the population excess kurtosis: $$H_{o}: K=0$$ $$H_{1}: K\neq 0$$ Where:
• $H_{o}$ is the null hypothesis.
• $H_{1}$ is the alternate hypothesis.
3. For the case in which the underlying population distribution is normal, the sample excess kurtosis also has a normal sampling distribution: $$\hat K \sim N(0,\frac{24}{T})$$ Where:
• $\hat k$ is the sample excess kurtosis (i.e. 4th moment).
• $T$ is the number of non-missing values in the data sample.
• $N(.)$ is the normal (i.e., gaussian) probability distribution function.
4. Using a given data sample, the sample excess kurtosis is calculated as: $$\hat K (x)= \frac{\sum_{t=1}^T(x_t-\bar x)^4}{(T-1)\hat \sigma^4}-3$$ Where:
• $\hat K(x)$ is the sample excess kurtosis.
• $x_i$ is the i-th non-missing value in the data sample.
• $T$ is the number of non-missing values in the data sample.
• $\hat \sigma$ is the sample standard deviation.
5. The underlying population distribution is assumed normal (gaussian).
6. This is a two-sides (i.e., two-tails) test, so the computed p-value should be compared with half of the significance level ($\frac{\alpha}{2}$).