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L: logistic distribution, green curve, excess kurtosis = 1.2; N: normal distribution, black curve (inverted parabola in the log-scale plot), excess kurtosis = 0; C: raised cosine distribution, cyan curve, excess kurtosis = −0.593762... W: Wigner semicircle distribution, blue curve, excess kurtosis = −1; U: uniform distribution, magenta ...
The distribution of has no closed-form expression, but can be reasonably approximated by another log-normal distribution at the right tail. [36] Its probability density function at the neighborhood of 0 has been characterized [35] and it does not resemble any log-normal distribution.
If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all ...
The kurtosis of a frequency distribution is a measure of the proportion of extreme values (outliers), which appear at either end of the histogram. If the distribution is more outlier-prone than the normal distribution it is said to be leptokurtic; if less outlier-prone it is said to be platykurtic.
One disadvantage of L-moment ratios for estimation is their typically smaller sensitivity. 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.
Because the parameters of the Cauchy distribution do not correspond to a mean and variance, attempting to estimate the parameters of the Cauchy distribution by using a sample mean and a sample variance will not succeed. [19] For example, if an i.i.d. sample of size n is taken from a Cauchy distribution, one may calculate the sample mean as:
As the logistic distribution, which can be solved analytically, is similar to the normal distribution, it can be used instead. The blue picture illustrates an example of fitting the logistic distribution to ranked October rainfalls—that are almost normally distributed—and it shows the 90% confidence belt based on the binomial distribution.
The sample skewness g 1 and kurtosis g 2 are both asymptotically normal. However, the rate of their convergence to the distribution limit is frustratingly slow, especially for g 2 . For example even with n = 5000 observations the sample kurtosis g 2 has both the skewness and the kurtosis of approximately 0.3, which is not negligible.