# 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 ($\frac{\alpha}{2}$).

## Examples

Example 1:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A B
Date Data
1/1/2008 #N/A
1/2/2008 -2.83
1/3/2008 -0.95
1/4/2008 -0.88
1/5/2008 1.21
1/6/2008 -1.67
1/7/2008 0.83
1/8/2008 -0.27
1/9/2008 1.36
1/10/2008 -0.34
1/11/2008 0.48
1/12/2008 -2.83
1/13/2008 -0.95
1/14/2008 -0.88
1/15/2008 1.21
1/16/2008 -1.67
1/17/2008 -2.99
1/18/2008 1.24
1/19/2008 0.64

Formula Description (Result)
=KURT($B$2:$B$20) Sample excess kurtosis (-1.0517)
=TEST_XKURT($B$2:$B$20) p-value of the test when excess kurtosis = 0 (0.171)