Kurtosis

Comparing Distributions in Terms of Kurtosis

A distribution such as the Laplace is said to have higher kurtosis than the normal distribution because it has more mass toward the center and heavier tails [see Figure 1(a)]. To visually compare the density functions of two symmetric distributions in terms of kurtosis, these should have the same center and variance. Figure 1(b) displays the corresponding cumulative version or distribution functions (CDFs). Willem R. van Zwet defined in 1964 a criterion to compare and order symmetric distributions based on their CDFs. According to this criterion, the normal has indeed no larger kurtosis than the Laplace distribution. However, not all the symmetric distributions are ordered.

Measuring Kurtosis in Distributions

In Greek, kurtos means convex; the mathematician Heron in the first century used the word kurtosis to mean curvature. Kurtosis was defined, as a statistical term, by Karl Pearson around 1905 as the measure

to compare other distributions with the normal distribution in terms of the frequency toward the mean μ (σ is the standard deviation and β2 = 3 for the normal distribution). It was later that ordering criteria based on the distribution functions were defined; in addition, more flexible definitions acknowledging that kurtosis is related [Page 677][Page 678]to both peakedness and tail weight were proposed, and it was accepted that kurtosis could be measured in several ways. New measures of kurtosis, to be considered valid, have to agree with the orderings defined over distributions by the criteria based on distribution functions. It is said that some kurtosis measures, such as α2, naturally have an averaging effect that prevents them from being as informative as the CDFs. Two distributions can have the same value of α2 and still look different. Because kurtosis is related to the peak and tails of a distribution, in the case of nonsymmetric distributions, kurtosis and skewness tend to be associated, particularly if they are represented by measures that are highly sensitive to the tails.

Figure 1 Density Functions and CDFs for Normal and Laplace Distributions

Notes: (a) Density functions. (b) Cumulative distribution functions.

Figure 2 Histogram and Normal Probability Plot for a Sample From a Laplace Distribution

Notes: (a) Histogram of data. (b) Probability plot of data. Normal − −95% CI.

Two of the several kurtosis measures that have been defined as alternatives to α2 are as

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