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A particular [[Probability distribution|probability distribution]] of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w0973701.png" />, characterized by the distribution function
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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/w/w097/w097370/w0973702.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w0973703.png" /> is a parameter of the shape of the distribution curve, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w0973704.png" /> is a scale parameter and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w0973705.png" /> is a shift parameter. The family of distributions (*) was named after W. Weibull [[#References|[1]]], who was the first to use it in the approximation of extremal data on the tensile strength of steel during fatigue testing and to propose methods for estimating the parameters of the distribution (*). Weibull's distribution belongs to the limit distributions of the third kind for the extremal terms of a series of order statistics. It is extensively used to describe the laws governing the breakdown of such things as ball bearings, vacuum instruments and electronic components. The [[Exponential distribution|exponential distribution]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w0973706.png" />) and the [[Rayleigh distribution|Rayleigh distribution]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w0973707.png" />) are special cases of the Weibull distribution. The distribution functions (*) do not belong to the Pearson family. There are auxiliary tables, from which the Weibull distribution functions may be calculated ([[#References|[2]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w0973708.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w0973709.png" />-level quantile is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737010.png" />,
+
A particular [[Probability distribution|probability distribution]] of random variables  $  X _ {w} $,  
 +
characterized by the distribution function
  
<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/w/w097/w097370/w09737011.png" /></td> </tr></table>
+
$$ \tag{* }
 +
F _ {w} ( t, p, \sigma , \mu )  = \
 +
\left \{
  
<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/w/w097/w097370/w09737012.png" /></td> </tr></table>
+
\begin{array}{ll}
 +
1- \mathop{\rm exp} \left \{ - \left (
 +
\frac{t- \mu } \sigma
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737013.png" /> is the gamma-function; the coefficients of variation, skewness and excess are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737014.png" />, which facilitates their tabulation and the compilation of auxiliary tables which can be used to obtain parameter estimates. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737015.png" />, the Weibull distribution is unimodal with mode equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737016.png" />, while the risk function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737017.png" /> is non-decreasing. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737018.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737019.png" /> monotonically decreases. It is possible to construct so-called Weibull probability paper ([[#References|[3]]]). On this paper, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737020.png" /> becomes a straight line; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737021.png" />, the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737022.png" /> is concave, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737023.png" /> it is convex. Estimates of the parameters of the Weibull distribution by the quantile method yields equations which are much simpler than those obtained by the maximum-likelihood method. The simultaneous asymptotic efficiency of the estimates of the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737025.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737026.png" />) by the quantile method is maximal (and equal to 0.64) if level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737027.png" /> and level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737028.png" /> quantiles are employed. The distribution function (*) is well-approximated by the distribution function of the log-normal distribution,
+
\right )  ^ {p} \right \} , & t > \mu , \\
 +
0, & t \leq  \mu , \\
 +
\end{array} \right.
  
<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/w/w097/w097370/w09737029.png" /></td> </tr></table>
+
$$
  
(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737030.png" /> is the distribution function of the standardized normal distribution and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097370/w09737033.png" />):
+
where $  p $
 +
is a parameter of the shape of the distribution curve,  $  \sigma $
 +
is a scale parameter and $  \mu $
 +
is a shift parameter. The family of distributions (*) was named after W. Weibull [[#References|[1]]], who was the first to use it in the approximation of extremal data on the tensile strength of steel during fatigue testing and to propose methods for estimating the parameters of the distribution (*). Weibull's distribution belongs to the limit distributions of the third kind for the extremal terms of a series of order statistics. It is extensively used to describe the laws governing the breakdown of such things as ball bearings, vacuum instruments and electronic components. The [[Exponential distribution|exponential distribution]] ( $  p = 1 $)
 +
and the [[Rayleigh distribution|Rayleigh distribution]] ( $  p = 2 $)
 +
are special cases of the Weibull distribution. The distribution functions (*) do not belong to the Pearson family. There are auxiliary tables, from which the Weibull distribution functions may be calculated ([[#References|[2]]]). If  $  \mu = 0 $,
 +
the  $  q $-level quantile is  $  \sigma [ -  \mathop{\rm ln} ( 1 - q)]  ^ {1/p} $,
  
<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/w/w097/w097370/w09737034.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} X _ {w}  ^ {k}  = \sigma  ^ {k} \Gamma \left ( 1+
 +
\frac{k}{p}
 +
\right ) ,\ \
 +
k = 1, 2 \dots
 +
$$
  
<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/w/w097/w097370/w09737035.png" /></td> </tr></table>
+
$$
 +
{\mathsf D} X _ {w}  = \sigma  ^ {2} \left [ \Gamma \left ( 1+
 +
\frac{2}{p}
 +
\right ) - \Gamma  ^ {2} \left ( 1+
 +
\frac{1}{p}
 +
\right ) \right ] ,
 +
$$
 +
 
 +
where  $  \Gamma ( x) $
 +
is the gamma-function; the coefficients of variation, skewness and excess are independent of  $  \sigma $,
 +
which facilitates their tabulation and the compilation of auxiliary tables which can be used to obtain parameter estimates. If  $  p \geq  1 $,
 +
the Weibull distribution is unimodal with mode equal to  $  \sigma ( p - 1)  ^ {1/p} $,
 +
while the risk function  $  \lambda ( t) = pt ^ {p- 1 } / \sigma  ^ {p} $
 +
is non-decreasing. If  $  p < 1 $,
 +
the function  $  \lambda ( t) $
 +
monotonically decreases. It is possible to construct so-called Weibull probability paper ([[#References|[3]]]). On this paper,  $  {F _ {w} } ( t, p, \sigma , 0) $
 +
becomes a straight line; if  $  \mu > 0 $,
 +
the graph of  $  {F _ {w} } ( t, p, \sigma , \mu ) $
 +
is concave, and if  $  \mu < 0 $
 +
it is convex. Estimates of the parameters of the Weibull distribution by the quantile method yields equations which are much simpler than those obtained by the maximum-likelihood method. The simultaneous asymptotic efficiency of the estimates of the parameters  $  p $
 +
and  $  \sigma $ (for  $  \mu = 0 $)
 +
by the quantile method is maximal (and equal to 0.64) if level  $  - 0.24 $
 +
and level  $  - 0.93 $
 +
quantiles are employed. The distribution function (*) is well-approximated by the distribution function of the log-normal distribution,
 +
 
 +
$$
 +
\Phi \left (
 +
\frac{ \mathop{\rm ln} ( t- b)- a }{c}
 +
\right )
 +
$$
 +
 
 +
(where  $  \Phi ( x) $
 +
is the distribution function of the standardized normal distribution and  $  - \infty < b < \infty $,
 +
$  - \infty < a < \infty $,
 +
$  c > 0 $):
 +
 
 +
$$
 +
\inf  _ {p, \sigma ,a,c } \
 +
\sup  _ { t } \
 +
\left | F _ {w} ( t, p, \sigma , 0)-
 +
\Phi \left (
 +
\frac{ \mathop{\rm ln}  t- a }{c}
 +
\right ) \right | =
 +
$$
 +
 
 +
$$
 +
= \
 +
\inf  _ { a,c }  \sup  _ { t }  \left | F _ {w} ( t, 1, 1, 0) - \Phi \left (
 +
\frac{ \mathop{\rm ln}  t- a }{c}
 +
\right ) \right |  =  0.038 .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Weibull,  "A statistical theory of the strength of materials" , Stockholm  (1939)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Gnedenko,  Yu.K. Belyaev,  A.D. Solov'ev,  "Mathematical methods of reliability theory" , Acad. Press  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Johnson,  "The statistical treatment of fatigue experiments" , Amsterdam  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Weibull,  "A statistical theory of the strength of materials" , Stockholm  (1939)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Gnedenko,  Yu.K. Belyaev,  A.D. Solov'ev,  "Mathematical methods of reliability theory" , Acad. Press  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Johnson,  "The statistical treatment of fatigue experiments" , Amsterdam  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</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. Continuous univariate distributions" , Wiley  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Galambos,  "The theory of extreme order statistics" , Wiley  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,  S. Kotz,  "Distributions in statistics. Continuous univariate distributions" , Wiley  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Galambos,  "The theory of extreme order statistics" , Wiley  (1987)</TD></TR></table>

Revision as of 05:59, 13 June 2022


A particular probability distribution of random variables $ X _ {w} $, characterized by the distribution function

$$ \tag{* } F _ {w} ( t, p, \sigma , \mu ) = \ \left \{ \begin{array}{ll} 1- \mathop{\rm exp} \left \{ - \left ( \frac{t- \mu } \sigma \right ) ^ {p} \right \} , & t > \mu , \\ 0, & t \leq \mu , \\ \end{array} \right. $$

where $ p $ is a parameter of the shape of the distribution curve, $ \sigma $ is a scale parameter and $ \mu $ is a shift parameter. The family of distributions (*) was named after W. Weibull [1], who was the first to use it in the approximation of extremal data on the tensile strength of steel during fatigue testing and to propose methods for estimating the parameters of the distribution (*). Weibull's distribution belongs to the limit distributions of the third kind for the extremal terms of a series of order statistics. It is extensively used to describe the laws governing the breakdown of such things as ball bearings, vacuum instruments and electronic components. The exponential distribution ( $ p = 1 $) and the Rayleigh distribution ( $ p = 2 $) are special cases of the Weibull distribution. The distribution functions (*) do not belong to the Pearson family. There are auxiliary tables, from which the Weibull distribution functions may be calculated ([2]). If $ \mu = 0 $, the $ q $-level quantile is $ \sigma [ - \mathop{\rm ln} ( 1 - q)] ^ {1/p} $,

$$ {\mathsf E} X _ {w} ^ {k} = \sigma ^ {k} \Gamma \left ( 1+ \frac{k}{p} \right ) ,\ \ k = 1, 2 \dots $$

$$ {\mathsf D} X _ {w} = \sigma ^ {2} \left [ \Gamma \left ( 1+ \frac{2}{p} \right ) - \Gamma ^ {2} \left ( 1+ \frac{1}{p} \right ) \right ] , $$

where $ \Gamma ( x) $ is the gamma-function; the coefficients of variation, skewness and excess are independent of $ \sigma $, which facilitates their tabulation and the compilation of auxiliary tables which can be used to obtain parameter estimates. If $ p \geq 1 $, the Weibull distribution is unimodal with mode equal to $ \sigma ( p - 1) ^ {1/p} $, while the risk function $ \lambda ( t) = pt ^ {p- 1 } / \sigma ^ {p} $ is non-decreasing. If $ p < 1 $, the function $ \lambda ( t) $ monotonically decreases. It is possible to construct so-called Weibull probability paper ([3]). On this paper, $ {F _ {w} } ( t, p, \sigma , 0) $ becomes a straight line; if $ \mu > 0 $, the graph of $ {F _ {w} } ( t, p, \sigma , \mu ) $ is concave, and if $ \mu < 0 $ it is convex. Estimates of the parameters of the Weibull distribution by the quantile method yields equations which are much simpler than those obtained by the maximum-likelihood method. The simultaneous asymptotic efficiency of the estimates of the parameters $ p $ and $ \sigma $ (for $ \mu = 0 $) by the quantile method is maximal (and equal to 0.64) if level $ - 0.24 $ and level $ - 0.93 $ quantiles are employed. The distribution function (*) is well-approximated by the distribution function of the log-normal distribution,

$$ \Phi \left ( \frac{ \mathop{\rm ln} ( t- b)- a }{c} \right ) $$

(where $ \Phi ( x) $ is the distribution function of the standardized normal distribution and $ - \infty < b < \infty $, $ - \infty < a < \infty $, $ c > 0 $):

$$ \inf _ {p, \sigma ,a,c } \ \sup _ { t } \ \left | F _ {w} ( t, p, \sigma , 0)- \Phi \left ( \frac{ \mathop{\rm ln} t- a }{c} \right ) \right | = $$

$$ = \ \inf _ { a,c } \sup _ { t } \left | F _ {w} ( t, 1, 1, 0) - \Phi \left ( \frac{ \mathop{\rm ln} t- a }{c} \right ) \right | = 0.038 . $$

References

[1] W. Weibull, "A statistical theory of the strength of materials" , Stockholm (1939)
[2] B.V. Gnedenko, Yu.K. Belyaev, A.D. Solov'ev, "Mathematical methods of reliability theory" , Acad. Press (1969) (Translated from Russian)
[3] L. Johnson, "The statistical treatment of fatigue experiments" , Amsterdam (1964)
[4] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)

Comments

References

[a1] N.L. Johnson, S. Kotz, "Distributions in statistics. Continuous univariate distributions" , Wiley (1970)
[a2] J. Galambos, "The theory of extreme order statistics" , Wiley (1987)
How to Cite This Entry:
Weibull distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weibull_distribution&oldid=18906
This article was adapted from an original article by Yu.K. BelyaevE.V. Chepurin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article