From Peter.Ruckdeschel at uni-bayreuth.de  Tue Feb  3 15:50:09 2004
From: Peter.Ruckdeschel at uni-bayreuth.de (Peter Ruckdeschel)
Date: Tue Feb  3 15:50:50 2004
Subject: [Rd] request for comments --- package "distr" --- S4 Classes
	for Distributions
In-Reply-To: <401FBCE7.4000006@uni-bayreuth.de>
References: <401FBCE7.4000006@uni-bayreuth.de>
Message-ID: <401FB521.6060804@uni-bayreuth.de>

On Tue, 3 Feb 2004, Prof Brian D Ripley wrote:

> On Tue, 3 Feb 2004, Duncan Murdoch wrote:
>
>>On Tue, 03 Feb 2004 09:45:52 +0000, Matthias Kohl
>><Matthias.Kohl@uni-bayreuth.de> wrote:
>>
>>    
>>
>>>I think the most common example is the Cantor distribution.
>>>      
>>>
>>That's the most common 1-dimensional singular distribution, but higher
>>dimensional distributions are much more commonly singular.  For
>>example, mixed continuous-discrete distributions, and other
>>distributions whose support is of lower dimension than the sample
>>space, e.g. X ~ N(0,1), Y=X.
>>    
>>
>
>The most common 1d singular distribution is probably a lifetime with an
>atom at zero.
>
>I think the question was about a continuous but not absolutely continuous
>distribution, and indeed the Cantor distribution is the standard example
>in theory courses.
>  
>
I do not mean to give statistical advice to any of the cited contributors,
who will all be familiar with this, but I might add that by iterated use
of the Lebesgue decomposition,
 
cf       http://mathworld.wolfram.com/LebesgueDecomposition.html

you may decompose any measure on the 1d Borel sets uniquely
into a discrete, an absolutely coninuous and a singular part.

Using this nomenclatura, Prof. Ripley's lifetime example would have
non-trivial discrete and  absolutely continuous parts  but only a 
trivial singular part.

In dimension d>1 things may become more complicated, though, as you 
might want to
distinguish the dimensions of sets on which [ Lebesgue^d and discrete] - 
singular parts
throw their mass....



[Rd] request for comments --- package "distr" --- S4 Classes for Distributions
Peter Ruckdeschel