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Technical Studies Reference


Kurtosis

This study calculates and displays the Kurtosis of the data specified by the Input Data Input. Kurtosis is a statistical measure of the peakedness or flatness of a distribution relative to the Normal Distribution, which has a Kurtosis of \(3\). More precisely, this study calculates what is known as Excess Kurtosis, which is the Kurtosis in excess of \(3\).

Let \(X\) be a random variable denoting the Input Data, and let \(X_t\) be its value at Index \(t\). Let the \(n\) be the Length Input. We denote the Kurtosis at Index \(t\) as K_t(X,n), and we compute it over a moving window of Length \(n\) for \(t \geq n - 1\). The method of computation depends on the setting of the Kurtosis Type Input. There are four such settings, and we explain each one below. In each case, the function \(MA_t(X,n)\) refers to the Simple Moving Average.

Case 1: Population Excess Kurtosis

\(\displaystyle{K_t(X,n) = \frac{E_t\left[(X - MA(X,n))^4\right]}{\left(E_t\left[(X - MA(x,n))^2\right]\right)^2} - 3}\)

In the above formula, the function \(E_t[]\) refers to the Expected Value at Index \(t\). The numerator and denominator are the Fourth and Second Central Moments of \(X\), respectively, and they are given explicitly by the following formulas.

\(\displaystyle{E_t\left[(X - MA(X,n))^4\right] = \frac{1}{n}\sum_{i = t - n + 1}^t(X_i - MA_t(X,n))^4}\)
\(\displaystyle{E_t\left[(X - MA(X,n))^2\right] = \frac{1}{n}\sum_{i = t - n + 1}^t(X_i - MA_t(X,n))^2}\)

Case 2: Sample Excess Kurtosis

\(\displaystyle{K_t(X,n) = \frac{(n - 1)(n + 1)}{(n - 2)(n - 3)}\cdot\frac{E_t\left[(X - MA(X,n))^4\right]}{\left(E_t\left[(X - MA(x,n))^2\right]\right)^2} - \frac{3(n - 1)^2}{(n - 2)(n - 3)}}\)

The Sample Excess Kurtosis is an unbiased estimator of the Population Kurtosis.

Regardless of whether one is calculating Excess Kurtosis for populations or samples, the value is interpreted as follows.

  • If \(K_t(X,n) > 0\), then the distribution is more peaked than a Normal Distribution with the same mean and variance.
  • If \(K_t(X,n) = 0\), then the distribution is a Normal Distribution.
  • If \(K_t(X,n) < 0\), then the distribution is less peaked than a Normal Distribution with the same mean and variance.

Inputs

  • Input Data
  • Length
  • Kurtosis Type: This is a custom Input that determines the method of calculation of the Kurtosis.

*Last modified Friday, 25th January, 2019.