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The name of a family of continuous probability distributions (Pearson distributions) whose densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071920/p0719201.png" /> satisfy the differential equation
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<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/p/p071/p071920/p0719202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071920/p0719203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071920/p0719204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071920/p0719205.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071920/p0719206.png" /> are real numbers. More precisely, Pearson curves are graphs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071920/p0719207.png" /> as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071920/p0719208.png" />. The distributions that are solutions to (*) coincide with limiting forms of the [[Hypergeometric distribution|hypergeometric distribution]]. Pearson curves are classified in accordance with the character of the roots of the equation
+
The name of a family of continuous probability distributions (Pearson distributions) whose densities  $  p( x) $
 +
satisfy the ''Pearson differential equation''
  
<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/p/p071/p071920/p0719209.png" /></td> </tr></table>
+
$$ \tag{* }
 +
\frac{dp(x)}{dx} = \frac{x-a}{b _ {0} + b _ {1} x + b _ {2} x  ^ {2} } \, p( x),
 +
$$
 +
 
 +
where the parameters  $  a $,
 +
$  b _ {0} $,
 +
$  b _ {1} $,
 +
and  $  b _ {2} $
 +
are real numbers. More precisely, Pearson curves are graphs of  $  p( x) $
 +
as a function of  $  x $.  
 +
The distributions that are solutions to (*) coincide with limiting forms of the [[Hypergeometric distribution|hypergeometric distribution]]. Pearson curves are classified in accordance with the character of the roots of the equation
 +
 
 +
$$
 +
b _ {0} + b _ {1} x + b _ {2} x  ^ {2}  =  0.
 +
$$
  
 
The family of Pearson curves consists of 12 types plus the [[Normal distribution|normal distribution]]. Many very important distributions in mathematical statistics may be obtained by a transformation from (*).
 
The family of Pearson curves consists of 12 types plus the [[Normal distribution|normal distribution]]. Many very important distributions in mathematical statistics may be obtained by a transformation from (*).
Line 13: Line 36:
 
Type I:
 
Type I:
  
<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/p/p071/p071920/p07192010.png" /></td> </tr></table>
+
$$
 +
p( x)  = k \left ( 1 +
 +
\frac{x}{a _ {1} }
 +
\right ) ^ {m _ {1} } \left ( 1 -  
 +
\frac{x}{a _ {2} }
 +
\right ) ^ {m _ {2} } ,
 +
$$
  
<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/p/p071/p071920/p07192011.png" /></td> </tr></table>
+
$$
 +
- a _ {1}  <   x  < a _ {2} ,\  m _ {1} , m _ {2}  > - 1,
 +
$$
  
 
with as a particular case the [[Beta-distribution|beta-distribution]] of the first kind.
 
with as a particular case the [[Beta-distribution|beta-distribution]] of the first kind.
Line 21: Line 52:
 
Type II:
 
Type II:
  
<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/p/p071/p071920/p07192012.png" /></td> </tr></table>
+
$$
 +
p( x)  = k \left ( 1 -
 +
\frac{x  ^ {2} }{a  ^ {2} }
 +
\right )  ^ {m} ,\ \
 +
- < < a,\ \
 +
> - 1
 +
$$
  
 
(a version of a type-I Pearson curve); a particular case is the [[Uniform distribution|uniform distribution]].
 
(a version of a type-I Pearson curve); a particular case is the [[Uniform distribution|uniform distribution]].
Line 27: Line 64:
 
Type III:
 
Type III:
  
<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/p/p071/p071920/p07192013.png" /></td> </tr></table>
+
$$
 +
p( x)  = k \left ( 1 +
 +
\frac{x}{a}
 +
\right ) ^ {\mu a } e ^ {- \mu x } ,\ \
 +
- a  < < \infty ,\ \
 +
\mu , a  > 0;
 +
$$
  
 
particular cases are the [[Gamma-distribution|gamma-distribution]] and the [[Chi-squared distribution| "chi-squared"  distribution]].
 
particular cases are the [[Gamma-distribution|gamma-distribution]] and the [[Chi-squared distribution| "chi-squared"  distribution]].
Line 33: Line 76:
 
Type IV:
 
Type IV:
  
<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/p/p071/p071920/p07192014.png" /></td> </tr></table>
+
$$
 +
p( x)  = k \left ( 1 + \frac{x  ^ {2} }{a  ^ {2} } \right )  ^ {-m} e ^ {- \mu \  \mathop{\rm arctg} ( x/a) } ,
 +
$$
  
<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/p/p071/p071920/p07192015.png" /></td> </tr></table>
+
$$
 +
- \infty < < \infty ,\  a , \mu  > 0.
 +
$$
  
 
Type V:
 
Type V:
  
<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/p/p071/p071920/p07192016.png" /></td> </tr></table>
+
$$
 +
p( x)  = kx  ^ {-q} e ^ {- ( \alpha /x) } ,\ \
 +
0 < < \infty ,\ \
 +
\alpha  > 0 ,\ \
 +
> 1
 +
$$
  
 
(which can be reduced by transformation to type III).
 
(which can be reduced by transformation to type III).
Line 45: Line 97:
 
Type VI:
 
Type VI:
  
<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/p/p071/p071920/p07192017.png" /></td> </tr></table>
+
$$
 +
p( x)  = kx ^ {- q _ {1} } ( x- a) ^ {q _ {2} } ,
 +
$$
  
<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/p/p071/p071920/p07192018.png" /></td> </tr></table>
+
$$
 +
< x  <  \infty ,\  q _ {1}  <  1,\  q _ {2}  > - 1 ,\  q _ {1}  > q _ {2} - 1;
 +
$$
  
particular cases are the beta-distribution of the second kind and the [[Fisher-F-distribution|Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071920/p07192019.png" />-distribution]].
+
particular cases are the beta-distribution of the second kind and the [[Fisher-F-distribution|Fisher $  F $-
 +
distribution]].
  
 
Type VII:
 
Type VII:
  
<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/p/p071/p071920/p07192020.png" /></td> </tr></table>
+
$$
 +
p( x)  = k \left ( 1 +
 +
\frac{x  ^ {2} }{a  ^ {2} }
 +
\right )  ^ {-m} ,\ \
 +
- \infty  < < \infty ,\ \
 +
>
 +
\frac{1}{2}
 +
;
 +
$$
  
 
a particular case is the [[Student distribution|Student distribution]].
 
a particular case is the [[Student distribution|Student distribution]].
Line 59: Line 124:
 
Type VIII:
 
Type VIII:
  
<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/p/p071/p071920/p07192021.png" /></td> </tr></table>
+
$$
 +
p( x)  = k \left ( 1+
 +
\frac{x}{a}
 +
\right )  ^ {-m} ,\ \
 +
- < x  \leq  0 \ \
 +
m  >  1 .
 +
$$
  
 
Type IX:
 
Type IX:
  
<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/p/p071/p071920/p07192022.png" /></td> </tr></table>
+
$$
 +
p( x)  = k \left ( 1 +
 +
\frac{x}{a}
 +
\right )  ^ {m} ,\ \
 +
- a  < x  \leq  0,\ \
 +
> - 1.
 +
$$
  
 
Type X:
 
Type X:
  
<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/p/p071/p071920/p07192023.png" /></td> </tr></table>
+
$$
 +
p( x)  = ke ^ {- ( x- m)/ \sigma } ,\ \
 +
m  \leq  x  < \infty ,\ \
 +
\sigma  > 0,
 +
$$
  
 
i.e. an [[Exponential distribution|exponential distribution]].
 
i.e. an [[Exponential distribution|exponential distribution]].
Line 73: Line 154:
 
Type XI:
 
Type XI:
  
<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/p/p071/p071920/p07192024.png" /></td> </tr></table>
+
$$
 +
p( x)  = kx  ^ {-m} ,\ \
 +
b  \leq  x  < \infty ,\ \
 +
> 1;
 +
$$
  
 
a particular case is the [[Pareto distribution|Pareto distribution]].
 
a particular case is the [[Pareto distribution|Pareto distribution]].
Line 79: Line 164:
 
Type XII:
 
Type XII:
  
<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/p/p071/p071920/p07192025.png" /></td> </tr></table>
+
$$
 +
p( x)  = \left (
 +
\frac{1 +
 +
\frac{x}{a _ {1} }
 +
}{1-
 +
\frac{x}{a _ {2} }
 +
}
 +
\right )  ^ {m} ,\ \
 +
- a _ {1}  < < a _ {2} ,\ \
 +
| m |  < 1
 +
$$
  
 
(a version of Type I).
 
(a version of Type I).
Line 87: Line 182:
 
Any Pearson curve is uniquely determined by the first four of its moments:
 
Any Pearson curve is uniquely determined by the first four of its moments:
  
<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/p/p071/p071920/p07192026.png" /></td> </tr></table>
+
$$
 +
\alpha _ {k}  = \int\limits _ {- \infty } ^ { {+ }  \infty } x  ^ {k} p( x)  dx,
 +
$$
  
 
if these are finite. This property of the family of Pearson curves is used for the approximate description of empirical distributions (cf. [[Empirical distribution|Empirical distribution]]).
 
if these are finite. This property of the family of Pearson curves is used for the approximate description of empirical distributions (cf. [[Empirical distribution|Empirical distribution]]).
Line 93: Line 190:
 
The method of fitting Pearson curves to some empirical distribution is as follows. Independent observations are used to calculate the first four sample moments, and then the suitable type of Pearson curve is determined, using the method of moments to find the unknown parameters. In the general case, the method of moments is not an efficient method for obtaining estimators of Pearson curves. L.N. Bol'shev (1963) provided a new solution to the problem of a more accurate approximation to distributions by means of Pearson curves using asymptotic transformations.
 
The method of fitting Pearson curves to some empirical distribution is as follows. Independent observations are used to calculate the first four sample moments, and then the suitable type of Pearson curve is determined, using the method of moments to find the unknown parameters. In the general case, the method of moments is not an efficient method for obtaining estimators of Pearson curves. L.N. Bol'shev (1963) provided a new solution to the problem of a more accurate approximation to distributions by means of Pearson curves using asymptotic transformations.
  
The curves were introduced by K. Pearson in 1894.
+
The curves were introduced by [[Pearson, Karl|K. Pearson]] in 1894.
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.P. Elderton,  "Frequency curves and correlation" , Harren  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Stuart,   J.K. Ord,  "Kendall's advanced theory of statistics" , Griffin  (1987)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.N. Bol'shev,  "Asymptotically Pearson transformations"  ''Theory Probab. Appl.'' , '''8''' :  2  (1963)  pp. 121–146  ''Teor. Veroyatnost. Primenen.'' , '''8''' :  2  (1963)  pp. 129–155</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
 
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,  S. Kotz,  "Distributions in statistics" , '''1. Continuous univariate distributions''' , Wiley  (1970)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  W.P. Elderton,  "Frequency curves and correlation" , Harren  (1953)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  A. Stuart,  J.K. Ord,  "Kendall's advanced theory of statistics" , Griffin  (1987)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  L.N. Bol'shev,  "Asymptotically Pearson transformations"  ''Theory Probab. Appl.'' , '''8''' :  2  (1963)  pp. 121–146  ''Teor. Veroyatnost. Primenen.'' , '''8''' :  2  (1963)  pp. 129–155</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,  S. Kotz,  "Distributions in statistics" , '''1. Continuous univariate distributions''' , Wiley  (1970)</TD></TR>
 +
</table>

Latest revision as of 07:20, 24 March 2023


The name of a family of continuous probability distributions (Pearson distributions) whose densities $ p( x) $ satisfy the Pearson differential equation

$$ \tag{* } \frac{dp(x)}{dx} = \frac{x-a}{b _ {0} + b _ {1} x + b _ {2} x ^ {2} } \, p( x), $$

where the parameters $ a $, $ b _ {0} $, $ b _ {1} $, and $ b _ {2} $ are real numbers. More precisely, Pearson curves are graphs of $ p( x) $ as a function of $ x $. The distributions that are solutions to (*) coincide with limiting forms of the hypergeometric distribution. Pearson curves are classified in accordance with the character of the roots of the equation

$$ b _ {0} + b _ {1} x + b _ {2} x ^ {2} = 0. $$

The family of Pearson curves consists of 12 types plus the normal distribution. Many very important distributions in mathematical statistics may be obtained by a transformation from (*).

W. Elderton (1938) gave a systematic description of the types of Pearson curves. In simplified form, the classification by types is as follows:

Type I:

$$ p( x) = k \left ( 1 + \frac{x}{a _ {1} } \right ) ^ {m _ {1} } \left ( 1 - \frac{x}{a _ {2} } \right ) ^ {m _ {2} } , $$

$$ - a _ {1} < x < a _ {2} ,\ m _ {1} , m _ {2} > - 1, $$

with as a particular case the beta-distribution of the first kind.

Type II:

$$ p( x) = k \left ( 1 - \frac{x ^ {2} }{a ^ {2} } \right ) ^ {m} ,\ \ - a < x < a,\ \ m > - 1 $$

(a version of a type-I Pearson curve); a particular case is the uniform distribution.

Type III:

$$ p( x) = k \left ( 1 + \frac{x}{a} \right ) ^ {\mu a } e ^ {- \mu x } ,\ \ - a < x < \infty ,\ \ \mu , a > 0; $$

particular cases are the gamma-distribution and the "chi-squared" distribution.

Type IV:

$$ p( x) = k \left ( 1 + \frac{x ^ {2} }{a ^ {2} } \right ) ^ {-m} e ^ {- \mu \ \mathop{\rm arctg} ( x/a) } , $$

$$ - \infty < x < \infty ,\ a , \mu > 0. $$

Type V:

$$ p( x) = kx ^ {-q} e ^ {- ( \alpha /x) } ,\ \ 0 < x < \infty ,\ \ \alpha > 0 ,\ \ q > 1 $$

(which can be reduced by transformation to type III).

Type VI:

$$ p( x) = kx ^ {- q _ {1} } ( x- a) ^ {q _ {2} } , $$

$$ a < x < \infty ,\ q _ {1} < 1,\ q _ {2} > - 1 ,\ q _ {1} > q _ {2} - 1; $$

particular cases are the beta-distribution of the second kind and the Fisher $ F $- distribution.

Type VII:

$$ p( x) = k \left ( 1 + \frac{x ^ {2} }{a ^ {2} } \right ) ^ {-m} ,\ \ - \infty < x < \infty ,\ \ m > \frac{1}{2} ; $$

a particular case is the Student distribution.

Type VIII:

$$ p( x) = k \left ( 1+ \frac{x}{a} \right ) ^ {-m} ,\ \ - a < x \leq 0 \ \ m > 1 . $$

Type IX:

$$ p( x) = k \left ( 1 + \frac{x}{a} \right ) ^ {m} ,\ \ - a < x \leq 0,\ \ m > - 1. $$

Type X:

$$ p( x) = ke ^ {- ( x- m)/ \sigma } ,\ \ m \leq x < \infty ,\ \ \sigma > 0, $$

i.e. an exponential distribution.

Type XI:

$$ p( x) = kx ^ {-m} ,\ \ b \leq x < \infty ,\ \ m > 1; $$

a particular case is the Pareto distribution.

Type XII:

$$ p( x) = \left ( \frac{1 + \frac{x}{a _ {1} } }{1- \frac{x}{a _ {2} } } \right ) ^ {m} ,\ \ - a _ {1} < x < a _ {2} ,\ \ | m | < 1 $$

(a version of Type I).

The most important distributions for applications are the Types I, III, VI, and VII.

Any Pearson curve is uniquely determined by the first four of its moments:

$$ \alpha _ {k} = \int\limits _ {- \infty } ^ { {+ } \infty } x ^ {k} p( x) dx, $$

if these are finite. This property of the family of Pearson curves is used for the approximate description of empirical distributions (cf. Empirical distribution).

The method of fitting Pearson curves to some empirical distribution is as follows. Independent observations are used to calculate the first four sample moments, and then the suitable type of Pearson curve is determined, using the method of moments to find the unknown parameters. In the general case, the method of moments is not an efficient method for obtaining estimators of Pearson curves. L.N. Bol'shev (1963) provided a new solution to the problem of a more accurate approximation to distributions by means of Pearson curves using asymptotic transformations.

The curves were introduced by K. Pearson in 1894.

References

[1] W.P. Elderton, "Frequency curves and correlation" , Harren (1953)
[2] A. Stuart, J.K. Ord, "Kendall's advanced theory of statistics" , Griffin (1987)
[3] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[4] L.N. Bol'shev, "Asymptotically Pearson transformations" Theory Probab. Appl. , 8 : 2 (1963) pp. 121–146 Teor. Veroyatnost. Primenen. , 8 : 2 (1963) pp. 129–155
[a1] N.L. Johnson, S. Kotz, "Distributions in statistics" , 1. Continuous univariate distributions , Wiley (1970)
How to Cite This Entry:
Pearson curves. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pearson_curves&oldid=28553
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article