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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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.G. Kendall,   A. Stuart,   "The advanced theory of statistics. Distribution theory" , '''3. Design and analysis''' , Griffin  (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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)</TD></TR></table>

====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).