Kurtosis - Estimators of Population Kurtosis

Estimators of Population Kurtosis

Given a sub-set of samples from a population, the sample excess kurtosis above is a biased estimator of the population excess kurtosis. The usual estimator of the population excess kurtosis (used in DAP/SAS, Minitab, PSPP/SPSS, and Excel but not by BMDP) is G2, defined as follows:

\begin{align} G_2 & = \frac{k_4}{k_{2}^2} \\ & = \frac{n^2\,((n+1)\,m_4 - 3\,(n-1)\,m_{2}^2)}{(n-1)\,(n-2)\,(n-3)} \; \frac{(n-1)^2}{n^2\,m_{2}^2} \\ & = \frac{n-1}{(n-2)\,(n-3)} \left( (n+1)\,\frac{m_4}{m_{2}^2} - 3\,(n-1) \right) \\ & = \frac{n-1}{(n-2) (n-3)} \left( (n+1)\,g_2 + 6 \right) \\ & = \frac{(n+1)\,n\,(n-1)}{(n-2)\,(n-3)} \; \frac{\sum_{i=1}^n (x_i - \bar{x})^4}{\left(\sum_{i=1}^n (x_i - \bar{x})^2\right)^2} - 3\,\frac{(n-1)^2}{(n-2)\,(n-3)} \\ & = \frac{(n+1)\,n}{(n-1)\,(n-2)\,(n-3)} \; \frac{\sum_{i=1}^n (x_i - \bar{x})^4}{k_{2}^2} - 3\,\frac{(n-1)^2}{(n-2) (n-3)} \end{align}

where k4 is the unique symmetric unbiased estimator of the fourth cumulant, k2 is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance), m4 is the fourth sample moment about the mean, m2 is the second sample moment about the mean, xi is the ith value, and is the sample mean. Unfortunately, is itself generally biased. For the normal distribution it is unbiased.

For computationally efficient ways of calculating the sample kurtosis see Algorithms for calculating higher-order statistics.

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