Template:Loglogistic distribution characteristics: Difference between revisions

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(Created page with '====Distribution Characteristics==== For <math>\sigma >1</math> : :• <math>f(T)</math> decreases monotonically and is convex. Mode and mean do not exist. For <math>\sigma…')
 
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For  <math>\sigma >1</math> :
For  <math>\sigma >1</math> :


: <math>f(T)</math>  decreases monotonically and is convex. Mode and mean do not exist.
:* <math>f(t)</math>  decreases monotonically and is convex. Mode and mean do not exist.


For  <math>\sigma =1</math> :
For  <math>\sigma =1</math> :


: <math>f(T)</math>  decreases monotonically and is convex. Mode and mean do not exist. As  <math>T\to 0</math> ,  <math>f(T)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>  
:* <math>f(t)</math>  decreases monotonically and is convex. Mode and mean do not exist. As  <math>t\to 0</math> ,  <math>f(t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>  
:As  <math>T\to 0</math>  ,  <math>\lambda (T)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>  
:* As  <math>t\to 0</math>  ,  <math>\lambda (t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.</math>  


For  <math>0<\sigma <1</math> :
For  <math>0<\sigma <1</math> :


:The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
:* The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
:The  <math>pdf</math>  starts at zero, increases to its mode, and decreases thereafter.
:* The  <math>pdf</math>  starts at zero, increases to its mode, and decreases thereafter.
:As  <math>\mu </math>  increases, while  <math>\sigma </math>  is kept the same, the  <math>pdf</math>  gets stretched out to the right and its height decreases, while maintaining its shape.
:* As  <math>\mu </math>  increases, while  <math>\sigma </math>  is kept the same, the  <math>pdf</math>  gets stretched out to the right and its height decreases, while maintaining its shape.
:As  <math>\mu </math>  decreases,while  <math>\sigma </math>  is kept the same, the  ..  gets pushed in towards the left and its height increases.
:* As  <math>\mu </math>  decreases,while  <math>\sigma </math>  is kept the same, the  ..  gets pushed in towards the left and its height increases.
: <math>\lambda (T)</math>  increases till  <math>T={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}</math>  and decreases thereafter.  <math>\lambda (T)</math>  is concave at first, then becomes convex.
:* <math>\lambda (t)</math>  increases till  <math>t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}</math>  and decreases thereafter.  <math>\lambda (t)</math>  is concave at first, then becomes convex.


[[Image:ldaLLD10.1.gif|thumb|center|400px| ]]
[[Image:ldaLLD10.1.gif|thumb|center|400px| ]]

Revision as of 01:52, 15 February 2012

Distribution Characteristics

For [math]\displaystyle{ \sigma \gt 1 }[/math] :

  • [math]\displaystyle{ f(t) }[/math] decreases monotonically and is convex. Mode and mean do not exist.

For [math]\displaystyle{ \sigma =1 }[/math] :

  • [math]\displaystyle{ f(t) }[/math] decreases monotonically and is convex. Mode and mean do not exist. As [math]\displaystyle{ t\to 0 }[/math] , [math]\displaystyle{ f(t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}. }[/math]
  • As [math]\displaystyle{ t\to 0 }[/math] , [math]\displaystyle{ \lambda (t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}. }[/math]

For [math]\displaystyle{ 0\lt \sigma \lt 1 }[/math] :

  • The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
  • The [math]\displaystyle{ pdf }[/math] starts at zero, increases to its mode, and decreases thereafter.
  • As [math]\displaystyle{ \mu }[/math] increases, while [math]\displaystyle{ \sigma }[/math] is kept the same, the [math]\displaystyle{ pdf }[/math] gets stretched out to the right and its height decreases, while maintaining its shape.
  • As [math]\displaystyle{ \mu }[/math] decreases,while [math]\displaystyle{ \sigma }[/math] is kept the same, the .. gets pushed in towards the left and its height increases.
  • [math]\displaystyle{ \lambda (t) }[/math] increases till [math]\displaystyle{ t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}} }[/math] and decreases thereafter. [math]\displaystyle{ \lambda (t) }[/math] is concave at first, then becomes convex.
LdaLLD10.1.gif