Camber (statistics)

The curvature , kyrtosis , kurtosis or kurtose ( Greek κύρτωσις kýrtōsis "curvature", "arching") is a measure of the steepness or "pointedness" of a (unimodal) probability function , statistical density function or frequency distribution . The curvature is the standardized (central) 4th order moment . Distributions with little curvature scatter relatively evenly; In the case of distributions with high curvature, the scatter results more from extreme, but rare, events.

The excess indicates the difference between the curvature of the function under consideration and the curvature of the density function of a normally distributed random variable .

Bulge

Empirical curvature

The following formula is used to calculate the curvature of an empirical frequency distribution: ${\displaystyle x_{1},\ldots ,x_{n}}$

${\displaystyle w={\frac {1}{n}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\bar {x}}}{s}}\right)^{4}}$

So that the curvature is independent of the unit of measurement of the variable, the observation values ​​are calculated using the arithmetic mean and the standard deviation ${\displaystyle x_{i}}$ ${\displaystyle {\bar {x}}}$ ${\displaystyle s}$

${\displaystyle z_{i}={\frac {x_{i}-{\bar {x}}}{s}}}$

standardized . The standardization applies

${\displaystyle {\bar {z}}={\frac {1}{n}}\sum _{i=1}^{n}z_{i}=0,\quad s_{z}^{2}={\frac {1}{n}}\sum _{i=1}^{n}z_{i}^{2}=1\quad {\text{und}}\quad w={\frac {1}{n}}\sum _{i=1}^{n}z_{i}^{4}.}$

The curvature can only assume non-negative values. A value indicates that the standardized observations are concentrated near the mean; H. the distribution is flat-peaked (see picture), for the distribution is pointed-peak compared to a normal distribution. ${\displaystyle w<3}$ ${\displaystyle z_{i}}$ ${\displaystyle w>3}$

Curvature of a random variable

Analogously to the empirical curvature of a frequency distribution, the curvature or kurtosis of the density function or probability function of a random variable is defined as its fourth central moment normalized to the fourth power of the standard deviation . ${\displaystyle X}$ ${\displaystyle \sigma }$ ${\displaystyle \mu _{4}(X)}$

${\displaystyle \beta _{2}={\frac {\mu _{4}(X)}{\mu _{2}(X)^{2}}}={\frac {\mu _{4}(X)}{\sigma ^{4}(X)}}={\frac {\operatorname {E} [(X-\mu )^{4}]}{(\operatorname {E} [(X-\mu )^{2}])^{2}}}}$

with the expected value . ${\displaystyle \mu =\operatorname {E} [X]}$

As a representation by means of the cumulative results ${\displaystyle \kappa _{i}}$

${\displaystyle \beta _{2}={\frac {\kappa _{4}}{\kappa _{2}^{2}}}+3={\frac {\kappa _{4}}{\operatorname {Var} (X)^{2}}}+3}$

Estimating the curvature of a population

To estimate the unknown curvature of a population using sample data ( the sample size), the expected value and the variance must be estimated from the sample; H. the theoretical moments are replaced by the empirical ones: ${\displaystyle \omega }$ ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle n}$

${\displaystyle {\hat {\omega }}={\frac {1}{n}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\bar {x}}}{s}}\right)^{4}}$

with the sample mean and the sample standard deviation . ${\displaystyle {\bar {x}}}$ ${\displaystyle s}$

excess

In order to be able to better assess the extent of the bulge, it is compared with the bulge of a normal distribution for which applies. The excess (also: overkurtosis) is therefore defined as ${\displaystyle \beta _{2}=3}$

${\displaystyle \gamma =\operatorname {Exzess} =\beta _{2}-3}$

Using the cumulative results

${\displaystyle \gamma ={\frac {\kappa _{4}}{\operatorname {Var} (X)^{2}}}}$

It is not uncommon for the bulge to be incorrectly referred to as excess.

Types of excess

According to their excess, distributions are divided into:

• ${\displaystyle \mathrm {Exzess} =0}$: normal peak or mesokurtic . The normal distribution has the kurtosis and, accordingly, the excess . ${\displaystyle \beta _{2}=3}$ ${\displaystyle 0}$
• ${\displaystyle \mathrm {Exzess} >0}$: steep-peaked , super-Gaussian or leptokurtic . In comparison to the normal distribution, these are more acute distributions, i.e. H. Distributions with strong peaks.
• ${\displaystyle \mathrm {Exzess} <0}$: flat- peaked , subgaussian or platykurtic . One speaks of a distribution that is flattened compared to the normal distribution.

See also

• Skewness (statistics)
• Radius of curvature

Individual evidence

1. ↑ ^{a b} Bernd Rönz, Hans G. Strohe (1994), Lexicon Statistics , Gabler Verlag, p. 115