Examples of "kurtosis"
Another approach is using negentropy instead of kurtosis. Negentropy is a robust method for kurtosis, as kurtosis is very sensitive to outliers.
The "excess kurtosis" is defined as kurtosis minus 3. There are 3 distinct regimes as described below.
A necessary but not sufficient condition for a symmetrical distribution to be bimodal is that the kurtosis be less than three. Here the kurtosis is defined to be the standardised fourth moment around the mean. The reference given prefers to use the "excess kurtosis" – the kurtosis less 3.
Given a sub-set of samples from a population, the sample excess kurtosis above is a biased estimator of the population excess kurtosis. An alternative estimator of the population excess kurtosis is defined as follows:
where formula_64 is the sample mean of formula_65, the extracted signals. 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 formula_65, and ensures that the measured kurtosis takes account of signal variance. The goal of projection pursuit is to maximize the kurtosis, and make the extracted signal as non-normal as possible.
If a Doppler spectrum would be exactly normal distributed its kurtosis would be 3. If in general the kurtosis is positive the spectrum is called leptokurtic, or leptokurtotic.
where "γ" denotes the population excess kurtosis. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data.
The exact interpretation of the Pearson measure of kurtosis (or excess kurtosis) used to be disputed, but is now settled. As Westfall (2014) notes, "...its only unambiguous interpretation is in terms of tail extremity; i.e., either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution)." The logic is simple: Kurtosis is the average (or expected value) of the standardized data raised to the fourth power. Any standardized values that are less than 1 (i.e., data within one standard deviation of the mean, where the "peak" would be), contribute virtually nothing to kurtosis, since raising a number that is less than 1 to the fourth power makes it close to zero. The only data values (observed or observable) that contribute to kurtosis in any meaningful way are those outside the region of the peak; i.e., the outliers. Therefore kurtosis measures outliers only; it measures nothing about the "peak."
where "γ" denotes the population excess kurtosis. The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data.
The excess kurtosis is just a correction to make the kurtosis of the normal distribution equal to zero, and it is the following,
skewness = formula_387, and excess kurtosis = formula_388
Ignoring kurtosis risk will cause any model to understate the risk of variables with high kurtosis. For instance, Long-Term Capital Management, a hedge fund cofounded by Myron Scholes, ignored kurtosis risk to its detriment. After four successful years, this hedge fund had to be bailed out by major investment banks in the late 1990s because it understated the kurtosis of many financial securities underlying the fund's own trading positions.
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 situation of having more observations at either extreme than the tails of the normal distribution would suggest; therefore, the tails are "fatter".
An example of a beta distribution near the upper boundary (excess kurtosis − (3/2) skewness = 0) is given by α = 0.1, β = 1000, for which the ratio (excess kurtosis)/(skewness) = 1.49835 approaches the upper limit of 1.5 from below. An example of a beta distribution near the lower boundary (excess kurtosis + 2 − skewness = 0) is given by α= 0.0001, β = 0.1, for which values the expression (excess kurtosis + 2)/(skewness) = 1.01621 approaches the lower limit of 1 from above. In the infinitesimal limit for both α and β approaching zero symmetrically, the excess kurtosis reaches its minimum value at −2. This minimum value occurs at the point at which the lower boundary line intersects the vertical axis (ordinate). (However, in Pearson's original chart, the ordinate is kurtosis, instead of excess kurtosis, and it increases downwards rather than upwards).
In 1986 Moors gave an interpretation of kurtosis. Let
where "γ" is the skewness and "κ" is the kurtosis. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of "b" lies between 0 and 1. The logic behind this coefficient is that a bimodal distribution will have very low kurtosis, an asymmetric
where formula_33. The kurtosis excess may also be written as:
The kurtosis is the fourth standardized moment, defined as
The average of these values is 18.05 and the excess kurtosis is thus 18.05 − 3 = 15.05. This example makes it clear that data near the "middle" or "peak" of the distribution do not contribute to the kurtosis statistic, hence kurtosis does not measure "peakedness." It is simply a measure of the outlier, 999 in this example.
The kurtosis of any univariate normal distribution is 3. It is common to compare the kurtosis of a distribution to this value. Distributions with kurtosis less than 3 are said to be "platykurtic", although this does not imply the distribution is "flat-topped" as sometimes reported. Rather, it means the distribution produces fewer and less extreme outliers than does the normal distribution. An example of a platykurtic distribution is the uniform distribution, which does not produce outliers. Distributions with kurtosis greater than 3 are said to be "leptokurtic". An example of a leptokurtic distribution is the Laplace distribution, which has tails that asymptotically approach zero more slowly than a Gaussian, and therefore produces more outliers than the normal distribution. It is also common practice to use an adjusted version of Pearson's kurtosis, the excess kurtosis, which is the kurtosis minus 3, to provide the comparison to the normal distribution. Some authors use "kurtosis" by itself to refer to the excess kurtosis. For the reason of clarity and generality, however, this article follows the non-excess convention and explicitly indicates where excess kurtosis is meant.