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Mathematically bad behavioral distributions

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I'm studying asymptotic theory and have found that most of the distributions shown in textbooks have "good" properties like differentiability and integrability.
Edgeworth expansion, for example, apparently gives a good approximation for a density function, but the theory needs strong assumptions.

In some areas, "bad" behavioral distributions play a great role. Cauchy distribution is used in mathematical science since it has a fat tail.

I would like to know important distributions with troublesome properties and how researchers are approaching them.

Thank you in advance!

probability-theory statistics probability-distributions asymptotics

share|cite|improve this question

asked Mar 23 at 18:58

ParuruParuru

83

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  You can get a fat tail without giving up CLT benefits the way Cauchy does; it's enough to use Student's $t$ with multiple degrees of freedom.
  $endgroup$
  – J.G.
  Mar 23 at 19:08
  

  
  
  
  


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  Yes, certainly t distribution may work well in this case. I mentioned this just as an example. What I want to know is how to approach bad distributions like Cauchy. I suppose that CLT cannot be applied to such distributions, but is there any similar method to analyze them? I would appreciate it if you could give some examples. I'm also glad if you could introduce some useful articles.
  $endgroup$
  – Paruru
  Mar 23 at 19:22
  

  
  
  
  


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  It depends what you mean by "bad". It's one thing to lack a finite mean and variance; it's quite another for the CDF to not be differentiable. And incidentally, although the Cauchy's sample mean doesn't become approximately Gaussian as sample size $toinfty$, the sample median does (see e.g. this application of the delta method).
  $endgroup$
  – J.G.
  Mar 23 at 19:28
  

  
  
  
  

add a comment | 

0

1

$begingroup$

I'm studying asymptotic theory and have found that most of the distributions shown in textbooks have "good" properties like differentiability and integrability.
Edgeworth expansion, for example, apparently gives a good approximation for a density function, but the theory needs strong assumptions.

In some areas, "bad" behavioral distributions play a great role. Cauchy distribution is used in mathematical science since it has a fat tail.

I would like to know important distributions with troublesome properties and how researchers are approaching them.

Thank you in advance!

probability-theory statistics probability-distributions asymptotics

share|cite|improve this question

asked Mar 23 at 18:58

ParuruParuru

83

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can get a fat tail without giving up CLT benefits the way Cauchy does; it's enough to use Student's $t$ with multiple degrees of freedom.
  $endgroup$
  – J.G.
  Mar 23 at 19:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, certainly t distribution may work well in this case. I mentioned this just as an example. What I want to know is how to approach bad distributions like Cauchy. I suppose that CLT cannot be applied to such distributions, but is there any similar method to analyze them? I would appreciate it if you could give some examples. I'm also glad if you could introduce some useful articles.
  $endgroup$
  – Paruru
  Mar 23 at 19:22
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It depends what you mean by "bad". It's one thing to lack a finite mean and variance; it's quite another for the CDF to not be differentiable. And incidentally, although the Cauchy's sample mean doesn't become approximately Gaussian as sample size $toinfty$, the sample median does (see e.g. this application of the delta method).
  $endgroup$
  – J.G.
  Mar 23 at 19:28
  

  
  
  
  

add a comment | 

$begingroup$

I'm studying asymptotic theory and have found that most of the distributions shown in textbooks have "good" properties like differentiability and integrability.
Edgeworth expansion, for example, apparently gives a good approximation for a density function, but the theory needs strong assumptions.

In some areas, "bad" behavioral distributions play a great role. Cauchy distribution is used in mathematical science since it has a fat tail.

I would like to know important distributions with troublesome properties and how researchers are approaching them.

Thank you in advance!

probability-theory statistics probability-distributions asymptotics

share|cite|improve this question

asked Mar 23 at 18:58

ParuruParuru

83

$endgroup$

share|cite|improve this question

asked Mar 23 at 18:58

ParuruParuru

83

share|cite|improve this question

asked Mar 23 at 18:58

ParuruParuru

83

asked Mar 23 at 18:58

ParuruParuru

83

asked Mar 23 at 18:58

ParuruParuru

83