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TEST XKURT

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

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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.

MethodDescription
1P-Value
2Test Statistics (e.g. Z-score)
3Critical 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 ($\frac{\alpha}{2}$).

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