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''coefficient of excess, excess''
 
''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e0368001.png" /> is defined by the formula
+
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
 +
 
 +
$$
 +
\gamma _ {2}  =  \beta _ {2} - 3 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e0368002.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e0368003.png" /> is the second Pearson coefficient (cf. [[Pearson distribution|Pearson distribution]]), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e0368004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e0368005.png" /> are the second and fourth central moments of the probability distribution. In terms of the second- and fourth-order semi-invariants (cumulants) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e0368006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e0368007.png" />, the excess has the form
+
$$
 +
\gamma _ {2}  =
 +
\frac{\kappa _ {4} }{\kappa _ {2}  ^ {2} }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e0368008.png" /></td> </tr></table>
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e0368009.png" />, then one says that the density of the probability distribution has normal excess, because for a normal distribution the excess is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680010.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680011.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680012.png" />, 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
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680013.png" /> are independent random variables subject to one and same continuous probability law, then the statistic
+
$$
 +
\widehat \gamma  _ {2}  =
 +
\frac{1}{n ( s  ^ {2} )  ^ {2} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680014.png" /></td> </tr></table>
+
\sum _ { i= } 1 ^ { n }  ( X _ {i} - \overline{X}\; )  ^ {4} - 3
 +
$$
  
 
is called the sample excess, where
 
is called the sample excess, where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680015.png" /></td> </tr></table>
+
$$
 +
\overline{X}\;  =
 +
\frac{1}{n}
 +
\sum _ { i= } 1 ^ { n }  X _ {i} ,\ \
 +
s  ^ {2}  =
 +
\frac{1}{n}
 +
\sum _ { i= } 1 ^ { n }
 +
( X _ {i} - \overline{X}\; )  ^ {2} .
 +
$$
  
The sample excess <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680016.png" /> is used as a statistical point estimator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680017.png" /> when the distribution law of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680018.png" /> is not known. In the case of a normal distribution of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680019.png" />, the sample excess <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680020.png" /> is asymptotically normally distributed, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680021.png" />, with parameters
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680022.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} \widehat \gamma  _ {2}  = -  
 +
\frac{6}{n+}
 +
1
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680023.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Var}  \widehat \gamma  _ {2}  =
 +
\frac{2 4 n ( n - 2 ) ( n - 3 ) }{( n + 1 )  ^ {2} ( n + 3 ) ( n + 5 ) }
 +
=
 +
$$
 +
 
 +
$$
 +
= \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680024.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680025.png" /> differs substantially from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680026.png" />, one must assume that the distribution of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680027.png" /> is not normal. This is used in practice to verify the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680028.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680029.png" />, which is equivalent to the fact that the distribution of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036800/e03680030.png" /> deviates from the normal distribution.
+
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====
 
====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>
 
<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====
 
====Comments====

Revision as of 19:38, 5 June 2020


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