[R] median and joint distribution

[Roger Koenker]

For distribution functions F and G we have the (Frechet) bounds:

	max{0, F(x)+G(y) -1} <= H(x,y) <= min{F(x),F(y)}

where H is the joint df of (X,Y) having marginals F and G.  If
X and Y are comonotonic (Schmeidler (Econometrica, 1989)), that is
if there is a random variable Z, such that X = f(Z), and Y=g(Z)
for monotone f and g, then the upper Frechet bound holds and the
quantile function of X+Y is the sum of the quantile functions of
X and Y.  For "multiplicative linkage" take logs, presuming, of
course, that I'm not misinterpreting this phrase.  One can think
of comonotone X and Y as "perfectly concordant" in the language
of rank correlation.



url:	www.econ.uiuc.edu/~roger/my.html	Roger Koenker
email	rkoenker at uiuc.edu			Department of Economics
vox: 	217-333-4558				University of Illinois
fax:   	217-244-6678				Champaign, IL 61820

On Thu, 24 Jul 2003, Salvatore Barbaro wrote:

> Dear R-"helpers"!
>
> May I kindly ask the pure statistics-experts to help me for a
> purpose which first part is not directly concerned with R.
> Consider two distribution functions, say f and g. For both, the
> median is smaller than a half. Now, the multiplicative or additive
> linkage of both distribution leads to a new distribution function,
> say h, whereas the median of h is greater than a half. Does
> anybody know under which circumstances such a construction of h is
> possible (my intuition is that it depends on the correlation of f
> and g) or can anybody advice a helpful literature. Furthermore,
> does anybody know whether or how such a construction can be done
> with R. Thanks in advance.
>
> s.
>
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