Kurtosis

In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations. A high kurtosis distribution has a sharper "peak" and fatter "tails", while a low kurtosis distribution has a more rounded peak with wider "shoulders". The fourth standardized moment is defined as μ4 / σ4, where μ4 is the fourth moment about the mean and σ is the standard deviation. This is sometimes used as the definition of kurtosis in older works, but is not the definition used here. Kurtosis is more commonly defined as μ4 / σ4 − 3, which is also known as kurtosis excess. The minus 3 at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero. Another reason can be seen by looking at the formula for the kurtosis of the sum of random variables. If Y is the sum of n independent random variables, all with the same distribution as X, then KurtY = KurtX / n, while the formula would be more complicated if kurtosis were defined as μ4 / σ4. A normal distribution has a kurtosis of zero (distributions with zero kurtosis are called mesokurtic). A distribution with positive kurtosis is called leptokurtic, and one with negative kurtosis platykurtic. For a sample of n values the sample kurtosis is
g_2 = \frac{m_4}{m_{2}^2} -3 = \frac{n\,\sum_{i=1}^n (x_i - \overline{x})^4}{\left(\sum_{i=1}^n (x_i - \overline{x})^2\right)^2} - 3
where m4 is the fourth sample moment about the mean, m2 is the second sample moment about the mean (that is, the sample variance), xi is the ith value, and \overline{x} is the sample mean. Given a sub-set of samples from a population, the sample kurtosis above is a biased estimator of the population kurtosis. An unbiased estimator of the population kurtosis is G2, defined as follows: G_2 \!\!\!\!>
math>= \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)}
where k4 is the unique symmetric unbiased estimator of the fourth cumulant, k2 is the unbiased estimator of the population variance, m4 is the fourth sample moment about the mean, m2 is the sample variance, xi is the ith value, and \bar{x} is the sample mean.

See also

References

  • Joanes, D. N. & Gill, C. A. (1998) Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society (Series D): The Statistician 47 (1), 183–189. doi:10.1111/1467-9884.00122

External links

 

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