request for comments --- package "distr" --- S4 Classes for Distributions
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....