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Excess kurtosis, typically compared to a value of 0, characterizes the “tailedness” of a distribution. A univariate normal distribution has an excess kurtosis of 0. Negative excess kurtosis indicates a platykurtic distribution, which doesn’t necessarily have a flat top but produces fewer or less extreme outliers than the normal distribution.
The normal distribution has a skewness of zero. But in reality, data points may not be perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be positive or negative. D'Agostino's K-squared test is a goodness-of-fit normality test based on sample skewness and sample kurtosis.
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. Kurtosis risk is commonly referred to as "fat tail" risk. The "fat tail" metaphor explicitly describes the ...
The constant 3 ensures that Gaussian signals have zero kurtosis, Super-Gaussian signals have positive kurtosis, and Sub-Gaussian signals have negative kurtosis. The denominator is the variance of , and ensures that the measured kurtosis takes account of signal variance. The goal of projection pursuit is to maximize the kurtosis, and make the ...
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. [when defined as?] In common usage, the terms fat-tailed and heavy-tailed are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed ...
The kurtosis is here defined to be the standardised fourth moment around the mean. The value of b lies between 0 and 1. [26] The logic behind this coefficient is that a bimodal distribution with light tails will have very low kurtosis, an asymmetric character, or both – all of which increase this coefficient. The formula for a finite sample ...
The kurtosis goes to infinity for high and low values of , but for = / the two-point distributions including the Bernoulli distribution have a lower excess kurtosis, namely −2, than any other probability distribution.
The second is the traditional kurtosis, or fourth standardized moment: β 2 = γ 2 + 3. (Modern treatments define kurtosis γ 2 in terms of cumulants instead of moments, so that for a normal distribution we have γ 2 = 0 and β 2 = 3. Here we follow the historical precedent and use β 2.)