[R] Parameters of Beta distribution

[David Winsemius]

On Oct 8, 2009, at 12:53 PM, Albyn Jones wrote:

> Maithili
>
> I find it really hard to believe that a beta distribution would be a  
> reasonable probability model for loss data.  There would have to be  
> an upper bound on the size of losses.   What is the process that  
> generates the data.  Is there any natural upper bound?  Why is there  
> a lower bound greater than zero?

In insurance situation there is typically a cap on the covered losses  
and there is also typically an amount below which it would not make  
sense to offer a policy. So a minimum and a maximum are sensible  
assumptions about loss distributions in may real modeling situations.

-- 
David.
>
> That said, the MLE's would be the min and max, but those will  
> underestimate the range of a beta.  It is an elementary exercise to  
> see why with the uniform[0,B] (ie beta(1,1)), for which the expected  
> value of the max of a sample of size n is B*n/(n+1).  If you have a  
> lot of data, this may not bother you.  For an arbitrary beta  
> distribution you would have 4 parameters to estimate...    probably  
> a Bayes estimator would be easiest.
>
> I'll put this one away for an exercise my next math stats course...
>
> albyn
>
> Quoting Maithili Shiva <maithili_shiva at yahoo.com>:
>
>> Dear Albyn,
>> Thanks for your reply.
>> Yes "A" and "B" are unknown. I was just thinking to assign -
>> A = min(amounts) and B = max(amounts).
>> The actual loss data I am dealing with is large. I am trying to fit  
>> some statistical distributions to this data. I already have done  
>> with R code pertaining to many other distributions like Normal,  
>> Weibull, Pareto, Generalized extreme Value distribution etc. So  
>> just want to know how to estimate the parameters if I need to check  
>> whether the Beta distribution fits the loss data.
>> Is it possible for you to guide me how to estimate A and B or can I  
>> assume A = min(amounts) and B = max(Amounts)
>> Regards
>> Maithili
>>   --- On Wed, 7/10/09, Albyn Jones <jones at reed.edu> wrote:
>>
>>
>> From: Albyn Jones <jones at reed.edu>
>> Subject: Re: [R] Parameters of Beta distribution
>> To: JLucke at ria.buffalo.edu
>> Cc: "Maithili Shiva" <maithili_shiva at yahoo.com>, r-help at r- 
>> project.org, r-help-bounces at r-project.org
>> Date: Wednesday, 7 October, 2009, 3:30 PM
>>
>>
>> Are A and B known?  That is, are there known upper and lower bounds
>> for this credit loss data?  If not, you need to think about how to
>> estimate those bounds.  Why do you believe the data have a beta  
>> distribution?
>>
>> albyn
>>
>>
>> On Wed, Oct 07, 2009 at 09:03:31AM -0400, JLucke at ria.buffalo.edu  
>> wrote:
>>> Rescale your data x to  (x-A)/(B-A).
>>>
>>> Maithili Shiva <maithili_shiva at yahoo.com>
>>> Sent by: r-help-bounces at r-project.org
>>> 10/07/2009 08:39 AM
>>>
>>> To
>>> r-help at r-project.org
>>> cc
>>>
>>> Subject
>>> [R] Parameters of Beta distribution
>>>
>>> Supose I have a data pertaining to credit loss as
>>> amounts <-
>>> c(46839.50,31177.12,35696.69,21192.57,29200.91,42049.64,42422.19,
>>> 44976.18, 32135.36,47936.57,27322.91,37359.09,43179.60, 48381.02,
>>> 45872.38, 28057.30,44643.83,36156.33,16037.62, 45432.28)
>>> I am trying to fit Beta distribution (two parameters distribution  
>>> but
>>> where lower bound and upper bounds are NOT  0 and 1 respectively).  
>>> For
>>> this I need to estimate the two parameters of Beta distribution. I  
>>> found
>>> some code in VGAM pacakge but it deals with standard Beta  
>>> distribution
>>> i.e. lower bound (say A) = 0 and upper bound (say B) = 1.
>>> How do I estimate the parameters of the Beta distribution for  
>>> above data
>>> where A and B are not 0's?
>>> Please guide.
>>> Thanking you in advance
>>> Maithili

David Winsemius, MD
Heritage Laboratories
West Hartford, CT