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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 plot of excess kurtosis as a function of the variance and the mean shows that the minimum value of the excess kurtosis (−2, which is the minimum possible value for excess kurtosis for any distribution) is intimately coupled with the maximum value of variance (1/4) and the symmetry condition: the mean occurring at the midpoint (μ = 1/2).
The skewness value can be positive, zero, negative, or undefined. For a unimodal distribution (a distribution with a single peak), negative skew commonly indicates that the tail is on the left side of the distribution, and positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness ...
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 ...
Skewness risk can arise in any quantitative model that assumes a symmetric distribution (such as the normal distribution) but is applied to skewed data. Ignoring skewness risk, by assuming that variables are symmetrically distributed when they are not, will cause any model to understate the risk of variables with high skewness.
The values γ 1 and γ 2 are the random variable's skewness and (excess) kurtosis respectively. The value(s) in each set of brackets are the terms for that level of polynomial estimation, and all must be calculated and combined for the Cornish–Fisher expansion at that level to be valid.
In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis).
where is the beta function, is the location parameter, > is the scale parameter, < < is the skewness parameter, and > and > are the parameters that control the kurtosis. and are not parameters, but functions of the other parameters that are used here to scale or shift the distribution appropriately to match the various parameterizations of this distribution.