Kurtosis Measure

A Kurtosis Measure is a measure of "tailedness" or degree of "peakness" of a probability distribution.

• AKA: Kurtosis Excess.
• Context:
  • It can be defined as [math]\displaystyle{ \kappa(X)= \mu_4(X)/\mu_2(X)^2 }[/math] and the excess as [math]\displaystyle{ \gamma_2(X)=\kappa(X) -3 }[/math] where [math]\displaystyle{ \mu_i }[/math] is the ith central moment and [math]\displaystyle{ X }[/math] is a random variable. Note that [math]\displaystyle{ \mu_2=\sigma }[/math] where [math]\displaystyle{ \sigma }[/math] is the variance.
  • If [math]\displaystyle{ Z = (X - \mu)/ \sigma, }[/math] then [math]\displaystyle{ \kappa = E( Z^4 ) = \operatorname{var}( Z^2 ) + [ E( Z^2 ) ]^2 = \operatorname{var}( Z^2 ) + 1, }[/math] where E is the expectation. This is the Moors (1986) interpertation, kurtosis is measure of Z around its expectation.
• Example(s):
  • A Laplace Distribution has an excess kurtosis = 3
  • A Normal Distribution has an excess kurtosis = 0
  • An Uniform Distribution has an excess kurtosis = −1.2.
  • …
• Counter-Example(s):
  • L-kurtosis.
  • Skewness Measure.
• See: Normal Distribution, Central Moment, Leptokurtic Distribution, Platykurtic Distribution, Mesokurtic Distribution, Probability Distribution, Statistical Deviation, Statistical Dispersion.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Kurtosis Retrieved 2016-08-21
  • In probability theory and statistics, kurtosis (from κυρτός, kyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. In a similar way to the concept of skewness, kurtosis is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Depending on the particular measure of kurtosis that is used, there are various interpretations of kurtosis, and of how particular measures should be interpreted.

    The standard measure of kurtosis, originating with Karl Pearson, is based on a scaled version of the fourth moment of the data or population. This number measures heavy tails, and not peakedness; hence, the historical "peakedness" definition is wrong. For this measure, higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations.

    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(...)

The kurtosis is the fourth standardized moment, defined as

[math]\displaystyle{ \operatorname{Kurt}[X] = \frac{\mu_4}{\sigma^4} = \frac{\operatorname{E}[(X-\mu)^4]}{(\operatorname{E}[(X-\mu)^2])^2}, }[/math]

where μ₄ is the fourth moment about the mean and σ is the standard deviation.

Several letters are used in the literature to denote the kurtosis. A very common choice is κ, which is fine as long as it is clear that it does not refer to a cumulant. Other choices include γ₂, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis.

• (Weisstein, 2016) ⇒ Weisstein, Eric W., "Kurtosis." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/Kurtosis.html Retrieved 2016-08-21
  • QUOTE: Kurtosis is the degree of peakedness of a distribution, defined as a normalized form of the fourth central moment [math]\displaystyle{ \mu_4 }[/math] of a distribution. There are several flavors of kurtosis commonly encountered, including the kurtosis proper, denoted beta_2 (Abramowitz and Stegun 1972, p. 928) or alpha_4 (Kenney and Keeping 1951, p. 27; Kenney and Keeping 1961, pp. 99-102) and defined by

[math]\displaystyle{ \beta_2=(\mu_4)/(\mu_2^2)\;, \quad (1) }[/math]

where [math]\displaystyle{ \mu_i }[/math] denotes the ith central moment (and in particular, [math]\displaystyle{ \mu_2 }[/math] is the variance). This form is implemented in the Wolfram Language as Kurtosis[dist].

The kurtosis "excess" (Kenney and Keeping 1951, p. 27) is denoted [math]\displaystyle{ \gamma_2 }[/math] (Abramowitz and Stegun 1972, p. 928) or [math]\displaystyle{ b_2 }[/math], is defined by

[math]\displaystyle{ \gamma_2=(\mu_4)/(\mu_2^2)-3 \;,\quad (2) }[/math]

and is implemented in the Wolfram Language as KurtosisExcess[dist]. Kurtosis excess is commonly used because [math]\displaystyle{ \gamma_2 }[/math] of a normal distribution is equal to 0, while the kurtosis proper is equal to 3.

Unfortunately, Abramowitz and Stegun (1972) confusingly refer to [math]\displaystyle{ \beta_2 }[/math] as the "excess or kurtosis."

A distribution with a high peak ([math]\displaystyle{ \gamma_2\gt 0 }[/math]) is called leptokurtic, a flat-topped curve ([math]\displaystyle{ \gamma_2\lt 0 }[/math]) is called platykurtic, and the normal distribution ([math]\displaystyle{ \gamma_2=0 }[/math]) is called mesokurtic.

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