Search results
Results from the WOW.Com Content Network
For a sample of n values, a method of moments estimator of the population excess kurtosis can be defined as = = = (¯) [= (¯)] 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.
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 ...
Let X and Y each be normally distributed with correlation coefficient ρ. The cokurtosis terms are (,,,) = +(,,,) = (,,,) =Since the cokurtosis depends only on ρ, which is already completely determined by the lower-degree covariance matrix, the cokurtosis of the bivariate normal distribution contains no new information about the distribution.
For instance, the Laplace distribution has a kurtosis of 6 and weak exponential tails, but a larger 4th L-moment ratio than e.g. the student-t distribution with d.f.=3, which has an infinite kurtosis and much heavier tails.
In the following, { x i } denotes a sample of n observations, g 1 and g 2 are the sample skewness and kurtosis, m j ’s are the j-th sample central moments, and ¯ is the sample mean. Frequently in the literature related to normality testing, the skewness and kurtosis are denoted as √ β 1 and β 2 respectively.
The null hypothesis is a joint hypothesis of the skewness being zero and the excess kurtosis being zero. Samples from a normal distribution have an expected skewness of 0 and an expected excess kurtosis of 0 (which is the same as a kurtosis of 3). As the definition of JB shows, any deviation from this increases the JB statistic.
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.
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.