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A probability distribution with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060780/l0607801.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060780/l0607802.png" /> is scale parameter, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060780/l0607803.png" /> is a shift and
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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/l/l060/l060780/l0607804.png" /></td> </tr></table>
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The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060780/l0607805.png" /> satisfies the differential equation
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A probability distribution with distribution function $  \psi ( a x + b ) $,
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where  $  a $
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is scale parameter,  $  b $
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is a shift 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/l/l060/l060780/l0607806.png" /></td> </tr></table>
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$$
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\psi ( x)  =
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\frac{1}{1 + e  ^ {-x} }
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.
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$$
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The function  $  \psi ( x) $
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satisfies the differential equation
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$$
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\frac{d \psi }{d x }
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  = \
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\psi ( 1 - \psi ) .
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$$
  
 
The logistic distribution is close to the [[Normal distribution|normal distribution]]:
 
The logistic distribution is close to the [[Normal distribution|normal distribution]]:
  
<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/l/l060/l060780/l0607807.png" /></td> </tr></table>
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$$
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\sup _ { x } \
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| \psi ( 1 . 7  x ) - \Phi ( x) |  < 0 . 0 1 ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060780/l0607808.png" /> is the normal distribution function with mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060780/l0607809.png" /> and variance 1. To test the hypothesis of coincidence of the distribution functions of two samples of a logistic distribution with possibly different shifts the [[Wilcoxon test|Wilcoxon test]] (the [[Mann–Whitney test|Mann–Whitney test]]) is asymptotically optimal. The logistic distribution is sometimes more convenient than the normal distribution in data processing and the interpretation of inferences. In applications the multi-dimensional logistic distribution is also used.
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where $  \Phi ( x) $
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is the normal distribution function with mean 0 $
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and variance 1. To test the hypothesis of coincidence of the distribution functions of two samples of a logistic distribution with possibly different shifts the [[Wilcoxon test|Wilcoxon test]] (the [[Mann–Whitney test|Mann–Whitney test]]) is asymptotically optimal. The logistic distribution is sometimes more convenient than the normal distribution in data processing and the interpretation of inferences. In applications the multi-dimensional logistic distribution is also used.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1979)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.R. Cox,  D.V. Hinkley,  "Theoretical statistics" , Chapman &amp; Hall  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1979)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.R. Cox,  D.V. Hinkley,  "Theoretical statistics" , Chapman &amp; Hall  (1974)</TD></TR></table>
 
 
  
 
====Comments====
 
====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">[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 17:36, 12 January 2021


A probability distribution with distribution function $ \psi ( a x + b ) $, where $ a $ is scale parameter, $ b $ is a shift and

$$ \psi ( x) = \frac{1}{1 + e ^ {-x} } . $$

The function $ \psi ( x) $ satisfies the differential equation

$$ \frac{d \psi }{d x } = \ \psi ( 1 - \psi ) . $$

The logistic distribution is close to the normal distribution:

$$ \sup _ { x } \ | \psi ( 1 . 7 x ) - \Phi ( x) | < 0 . 0 1 , $$

where $ \Phi ( x) $ is the normal distribution function with mean $ 0 $ and variance 1. To test the hypothesis of coincidence of the distribution functions of two samples of a logistic distribution with possibly different shifts the Wilcoxon test (the Mann–Whitney test) is asymptotically optimal. The logistic distribution is sometimes more convenient than the normal distribution in data processing and the interpretation of inferences. In applications the multi-dimensional logistic distribution is also used.

References

[1] M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979)
[2] D.R. Cox, D.V. Hinkley, "Theoretical statistics" , Chapman & Hall (1974)

Comments

References

[a1] N.L. Johnson, S. Kotz, "Distributions in statistics" , 1. Continuous univariate distributions , Wiley (1970)
How to Cite This Entry:
Logistic distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logistic_distribution&oldid=16368
This article was adapted from an original article by A.I. Orlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article