Excess coefficient

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).