Kurtosis

The kurtosis is the fourth standardized moment, defined as

${\displaystyle \operatorname {Kurt} [X]=\operatorname {E} \left[\left({\frac {X-\mu }{\sigma }}\right)^{4}\right]={\frac {\operatorname {E} \left[(X-\mu )^{4}\right]}{\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{2}}}={\frac {\mu _{4}}{\sigma ^{4}}},}$

where μ₄ is the fourth central moment and σ is the standard deviation. Several letters are used in the literature to denote the kurtosis. A very common choice is κ, which is fine as long as it is clear that it does not refer to a cumulant. Other choices include γ₂, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis.

The kurtosis is bounded below by the squared skewness plus 1:[4]^:432

${\displaystyle {\frac {\mu _{4}}{\sigma ^{4}}}\geq \left({\frac {\mu _{3}}{\sigma ^{3}}}\right)^{2}+1,}$

where μ₃ is the third central moment. The lower bound is realized by the Bernoulli distribution. There is no upper limit to the kurtosis of a general probability distribution, and it may be infinite.

A reason why some authors favor the excess kurtosis is that cumulants are extensive. Formulea related to the extensive property are more naturally expressed in terms of the excess kurtosis. For example, let X₁, ..., X_n be independent random variables for which the fourth moment exists, and let Y be the random variable defined by the sum of the X_i. The excess kurtosis of Y is

${\displaystyle \operatorname {Kurt} [Y]-3={\frac {1}{\left(\sum _{j=1}^{n}\sigma _{j}^{\,2}\right)^{2}}}\sum _{i=1}^{n}\sigma _{i}^{\,4}\cdot \left(\operatorname {Kurt} \left[X_{i}\right]-3\right),}$

where ${\displaystyle \sigma _{i}}$ is the standard deviation of ${\displaystyle X_{i}}$. In particular if all of the X_i have the same variance, then this simplifies to

${\displaystyle \operatorname {Kurt} [Y]-3={1 \over n^{2}}\sum _{i=1}^{n}\left(\operatorname {Kurt} \left[X_{i}\right]-3\right).}$

The reason not to subtract off 3 is that the bare fourth moment better generalizes to multivariate distributions, especially when independence is not assumed. The cokurtosis between pairs of variables is an order four tensor. For a bivariate normal distribution, the cokurtosis tensor has off-diagonal terms that are neither 0 nor 3 in general, so attempting to "correct" for an excess becomes confusing. It is true, however, that the joint cumulants of degree greater than two for any multivariate normal distribution are zero.

For two random variables, X and Y, not necessarily independent, the kurtosis of the sum, X + Y, is

${\displaystyle {\begin{aligned}\operatorname {Kurt} [X+Y]={1 \over \sigma _{X+Y}^{4}}{\big (}&\sigma _{X}^{4}\operatorname {Kurt} [X]+4\sigma _{X}^{3}\sigma _{Y}\operatorname {Cokurt} [X,X,X,Y]\\&{}+6\sigma _{X}^{2}\sigma _{Y}^{2}\operatorname {Cokurt} [X,X,Y,Y]\\[6pt]&{}+4\sigma _{X}\sigma _{Y}^{3}\operatorname {Cokurt} [X,Y,Y,Y]+\sigma _{Y}^{4}\operatorname {Kurt} [Y]{\big )}.\end{aligned}}}$

Note that the binomial coefficients appear in the above equation.

The excess kurtosis is defined as kurtosis minus 3. There are 3 distinct regimes as described below.

Platykurtic

The coin toss is the most platykurtic distribution

A distribution with negative excess kurtosis is called platykurtic, or platykurtotic. "Platy-" means "broad".[10] In terms of shape, a platykurtic distribution has thinner tails. Examples of platykurtic distributions include the continuous and discrete uniform distributions, and the raised cosine distribution. The most platykurtic distribution of all is the Bernoulli distribution with p = 1/2 (for example the number of times one obtains "heads" when flipping a coin once, a coin toss), for which the excess kurtosis is −2. Such distributions are sometimes termed sub-Gaussian distribution, originally proposed by Jean-Pierre Kahane[11] and further described by Buldygin and Kozachenko.[12]

Given a sub-set of samples from a population, the sample excess kurtosis above is a biased estimator of the population excess kurtosis. An alternative estimator of the population excess kurtosis is defined as follows:

${\displaystyle {\begin{aligned}G_{2}&={\frac {k_{4}}{k_{2}^{2}}}\\[6pt]&={\frac {n^{2}\,[(n+1)\,m_{4}-3\,(n-1)\,m_{2}^{2}]}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {(n-1)^{2}}{n^{2}\,m_{2}^{2}}}\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,{\frac {m_{4}}{m_{2}^{2}}}-3\,(n-1)\right]\\[6pt]&={\frac {n-1}{(n-2)\,(n-3)}}\left[(n+1)\,g_{2}+6\right]\\[6pt]&={\frac {(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{4}}{\left[\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}\right]^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)\,(n-3)}}\\[6pt]&={\frac {(n+1)\,n}{(n-1)\,(n-2)\,(n-3)}}\;{\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{4}}{k_{2}^{2}}}-3\,{\frac {(n-1)^{2}}{(n-2)(n-3)}}\end{aligned}}}$

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, x_i is the i^th value, and ${\displaystyle {\bar {x}}}$ is the sample mean. Unfortunately, ${\displaystyle G_{2}}$ is itself generally biased. For the normal distribution it is unbiased.[3]

The sample kurtosis is a useful measure of whether there is a problem with outliers in a data set. Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods.

D'Agostino's K-squared test is a goodness-of-fit normality test based on a combination of the sample skewness and sample kurtosis, as is the Jarque–Bera test for normality.

For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see variance.

Pearson's definition of kurtosis is used as an indicator of intermittency in turbulence.[16]

A concrete example is the following lemma by , He, Zhang and Zhang[17]: Assume a random variable ${\displaystyle X}$ has expectation ${\displaystyle E[X]=\mu }$, variance ${\displaystyle E\left[(X-\mu )^{2}\right]=\sigma ^{2}}$ and kurtosis ${\displaystyle \kappa ={\tfrac {1}{\sigma ^{4}}}E\left[(X-\mu )^{4}\right]}$. Assume we sample ${\displaystyle n={\tfrac {2{\sqrt {3}}+3}{3}}\kappa \log {\tfrac {1}{\delta }}}$ many independent copies. Then

${\displaystyle \Pr \left[\max _{i=1}^{n}X_{i}\leq \mu \right]\leq \delta \quad {\text{and}}\quad \Pr \left[\min _{i=1}^{n}X_{i}\geq \mu \right]\leq \delta }$.

This shows that with ${\displaystyle \Theta (\kappa \log {\tfrac {1}{\delta }})}$ many samples, we will see one that is above the expectation with probability at least ${\displaystyle 1-\delta }$. In other words: If the kurtosis is large, we might see a lot values either all below or above the mean.

A different measure of "kurtosis" is provided by using L-moments instead of the ordinary moments.[19][20]

Wikimedia Commons has media related to Kurtosis.

• Kurtosis risk
• Maximum entropy probability distribution

• Kim, Tae-Hwan; White, Halbert (2003). "On More Robust Estimation of Skewness and Kurtosis: Simulation and Application to the S&P500 Index". Finance Research Letters. 1: 56–70. doi:10.1016/S1544-6123(03)00003-5. Alternative source (Comparison of kurtosis estimators)
• Seier, E.; Bonett, D.G. (2003). "Two families of kurtosis measures". Metrika. 58: 59–70. doi:10.1007/s001840200223.

Wikiversity has learning resources about Kurtosis

• Hazewinkel, Michiel, ed. (2001) [1994], "Excess coefficient", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Kurtosis calculator
• Free Online Software (Calculator) computes various types of skewness and kurtosis statistics for any dataset (includes small and large sample tests)..
• Kurtosis on the Earliest known uses of some of the words of mathematics
• Celebrating 100 years of Kurtosis a history of the topic, with different measures of kurtosis.