Kurtosis

excess kurtosisleptokurticplatykurticleptokurtoticmesokurticPlatykurtic distributionchance of crisesfat tailsheavy tailskurtosis excess
In probability theory and statistics, kurtosis (from κυρτός, kyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable.wikipedia
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Moment (mathematics)

momentsmomentraw moment
The standard measure of kurtosis, originating with Karl Pearson, is based on a scaled version of the fourth moment of the data or population.
If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
In probability theory and statistics, kurtosis (from κυρτός, kyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable.
Kurtosis: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.

L-moment

probability weighted moments
Alternative measures of kurtosis are: the L-kurtosis, which is a scaled version of the fourth L-moment; measures based on four population or sample quantiles.
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).

Skewness

skewedskewskewed distribution
In a similar way to the concept of skewness, kurtosis is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population.
The last equality expresses skewness in terms of the ratio of the third cumulant κ 3 to the 1.5th power of the second cumulant κ 2 . This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant.

Fat-tailed distribution

fat tailfat tailsfatter tails
In terms of shape, a leptokurtic distribution has fatter tails.
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution.

Standard deviation

standard deviationssample standard deviationsigma
where μ 4 is the fourth central moment and σ is the standard deviation.
where γ 2 denotes the population excess kurtosis.

Bernoulli distribution

BernoulliBernoulli random variableBernoulli random variables
The lower bound is realized by the Bernoulli distribution.
The kurtosis goes to infinity for high and low values of p, but for p=1/2 the two-point distributions including the Bernoulli distribution have a lower excess kurtosis than any other probability distribution, namely −2.

Logistic distribution

logisticbell-shaped curvelogistical
Examples of leptokurtic distributions include the Student's t-distribution, Rayleigh distribution, Laplace distribution, exponential distribution, Poisson distribution and the logistic distribution.
It resembles the normal distribution in shape but has heavier tails (higher kurtosis).

Cokurtosis

The cokurtosis between pairs of variables is an order four tensor.
* Kurtosis is a special case of the cokurtosis when the two random variables are identical:

Pearson distribution

Pearson Type III distributionPearson's system of continuous curvesPearson
Consider the Pearson type VII family, which is a special case of the Pearson type IV family restricted to symmetric densities.
However, it was not known how to construct probability distributions in which the skewness (standardized third cumulant) and kurtosis (standardized fourth cumulant) could be adjusted equally freely.

Standardized moment

standardized central moments
The kurtosis is the fourth standardized moment, defined as

Rayleigh distribution

RayleighRayleigh distributeddistribution
Examples of leptokurtic distributions include the Student's t-distribution, Rayleigh distribution, Laplace distribution, exponential distribution, Poisson distribution and the logistic distribution.
The excess kurtosis is given by:

Student's t-distribution

Student's ''t''-distributiont''-distributiont-distribution
Examples of leptokurtic distributions include the Student's t-distribution, Rayleigh distribution, Laplace distribution, exponential distribution, Poisson distribution and the logistic distribution.
The skewness is 0 if \nu > 3 and the excess kurtosis is if \nu > 4.

Shape parameter

shape
where a is a scale parameter and m is a shape parameter.
Most simply, they can be estimated in terms of the higher moments, using the method of moments, as in the skewness (3rd moment) or kurtosis (4th moment), if the higher moments are defined and finite.

Multivariate normal distribution

multivariate normalbivariate normal distributionjointly normally distributed
It is true, however, that the joint cumulants of degree greater than two for any multivariate normal distribution are zero.
Mardia's test is based on multivariate extensions of skewness and kurtosis measures.

Generalized normal distribution

exponential power distributionGEDgeneralized error distribution
*e.g., exponential power distributions with sufficiently large shape parameter b
It is a useful way to parametrize a continuum of symmetric, platykurtic densities spanning from the normal to the uniform density, and a continuum of symmetric, leptokurtic densities spanning from the Laplace to the normal density.

Heavy-tailed distribution

heavy tailsheavy-tailedheavy tail
One is that kurtosis measures both the "peakedness" of the distribution and the heaviness of its tail.
Leptokurtic distribution

Jarque–Bera test

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.
In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution.

Probability density function

probability densitydensity functiondensity
The probability density function is given by
For instance, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution of the probability.

D'Agostino's K-squared test

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.
The test is based on transformations of the sample kurtosis and skewness, and has power only against the alternatives that the distribution is skewed and/or kurtic.

Kurtosis risk

kurtosis
Kurtosis risk
Kurtosis risk applies to any kurtosis-related quantitative model that assumes the normal distribution for certain of its independent variables when the latter may in fact have kurtosis much greater than does the normal distribution.

Central moment

central momentsmoment about the meanmoments about the mean
where μ 4 is the fourth central moment and σ is the standard deviation. where m 4 is the fourth sample moment about the mean, m 2 is the second sample moment about the mean (that is, the sample variance), x i is the i th value, and is the sample mean.
The third and fourth central moments are used to define the standardized moments which are used to define skewness and kurtosis, respectively.

Normality test

Non-normality of errorsnormality testingstatistical test for the normality
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.
Historically, the third and fourth standardized moments (skewness and kurtosis) were some of the earliest tests for normality.

Variance

sample variancepopulation variancevariability
where m 4 is the fourth sample moment about the mean, m 2 is the second sample moment about the mean (that is, the sample variance), x i is the i th value, and is the sample mean.
Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance), and introduces bias.

Hyperbolic secant distribution

GHS distributionhyperbolic secantsech distribution
S: hyperbolic secant distribution, orange curve, excess kurtosis = 2
However, the hyperbolic secant distribution is leptokurtic; that is, it has a more acute peak near its mean, and heavier tails, compared with the standard normal distribution.