# Excess coefficient

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coefficient of excess, excess

A scalar characteristic of the pointedness of the graph of the probability density of a unimodal distribution. It is used as a certain measure of the deviation of the distribution in question from the normal one. The excess $\gamma _ {2}$ is defined by the formula

$$\gamma _ {2} = \beta _ {2} - 3 ,$$

where $\beta _ {2} = \mu _ {4} / \mu _ {2} ^ {2}$ is the second Pearson coefficient (cf. Pearson distribution), and $\mu _ {2}$ and $\mu _ {4}$ are the second and fourth central moments of the probability distribution. In terms of the second- and fourth-order semi-invariants (cumulants) $\kappa _ {2}$ and $\kappa _ {4}$, the excess has the form

$$\gamma _ {2} = \frac{\kappa _ {4} }{\kappa _ {2} ^ {2} } .$$

If $\gamma _ {2} = 0$, then one says that the density of the probability distribution has normal excess, because for a normal distribution the excess is $\gamma _ {2} = 0$. When $\gamma _ {2} > 0$, one says that the probability distribution has positive excess, which corresponds, as a rule, to the fact that the graph of the density of the relevant distribution in a neighbourhood of the mode has a more pointed and higher vertex then a normal curve. When $\gamma _ {2} < 0$, one talks of a negative excess of the density, and then the probability density in a neighbourhood of the mode has a lower and flatter vertex than the density of a normal law.

If $X _ {1} \dots X _ {n}$ are independent random variables subject to one and same continuous probability law, then the statistic

$$\widehat \gamma _ {2} = \frac{1}{n ( s ^ {2} ) ^ {2} } \sum _ { i= } 1 ^ { n } ( X _ {i} - \overline{X}\; ) ^ {4} - 3$$

is called the sample excess, where

$$\overline{X}\; = \frac{1}{n} \sum _ { i= } 1 ^ { n } X _ {i} ,\ \ s ^ {2} = \frac{1}{n} \sum _ { i= } 1 ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2} .$$

The sample excess $\widehat \gamma _ {2}$ is used as a statistical point estimator of $\gamma _ {2}$ when the distribution law of the $X _ {i}$ is not known. In the case of a normal distribution of the random variables $X _ {1} \dots X _ {n}$, the sample excess $\widehat \gamma _ {2}$ is asymptotically normally distributed, as $n \rightarrow \infty$, with parameters

$${\mathsf E} \widehat \gamma _ {2} = - \frac{6}{n+} 1$$

and

$$\mathop{\rm Var} \widehat \gamma _ {2} = \frac{2 4 n ( n - 2 ) ( n - 3 ) }{( n + 1 ) ^ {2} ( n + 3 ) ( n + 5 ) } =$$

$$= \ \frac{24}{n} \left [ 1 - \frac{225}{15n+} 24 + O \left ( \frac{1}{n ^ {3} } \right ) \right ] .$$

This is the reason why, when the observed value of $\widehat \gamma _ {2}$ differs substantially from $0$, one must assume that the distribution of the $X _ {i}$ is not normal. This is used in practice to verify the hypothesis $H _ {0}$: $\gamma _ {2} \neq 0$, which is equivalent to the fact that the distribution of the $X _ {i}$ deviates from the normal distribution.

#### References

 [1] M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , 3. Design and analysis , Griffin (1969) [2] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) [3] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)

#### Comments

The (coefficient of) excess is usually called the coefficient of kurtosis, or simply the kurtosis.

A density of normal, positive or negative excess is usually called a density of zero, positive or negative kurtosis, while a density of positive (negative) kurtosis is also said to be leptokurtic (respectively, platykurtic).

How to Cite This Entry:
Excess coefficient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Excess_coefficient&oldid=46870
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article