R help - Jul 2003 - median and joint distribution

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.

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.
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
>

Dear Salvatore,

Assuming that you mean "convolution" when you write
"additive linkage", the answer is that there is no general 
answer. It will depend heavily on the joint distribution
of the two random variables.

Just to give a simple example, let X~f, Y~g, and
P(X=0.4)=P(Y=0.4)=1. Then, X and Y are independent, their
medians are <0.5, but there sum has a median >0.5.

Its different for "multiplicative", since in general from
X~f, Y~f, P(X>=0.5)<=0.5, and P(Y>=0.5)<=0.5 it follows
that P(XY>=0.5) <= P(X>=p or Y>=q) if p*q=0.5. Thus, if there
are numbers p and q with that property, such that
P(X>=p) + P(Y>=q) <= 0.5, then the median of XY will be <=0.5.

You might argue that there is a relationship between additive
and multiplicative scale through a log-transformation (note that
the median is stable under monotone transformations). However,
I assume there is no obvious formulation of the above statement
on the additive scale.

There is no way of carrying convolutions out in R directly; you'd
need to do numerical integration to do that, e.g. using 
integrate().

HTH

Thomas

---

Thomas Hotz
Research Associate in Medical Statistics
University of Leicester
United Kingdom

Department of Epidemiology and Public Health
22-28 Princess Road West
Leicester
LE1 6TP
Tel +44 116 252-5410
Fax +44 116 252-5423

Division of Medicine for the Elderly
Department of Medicine
The Glenfield Hospital
Leicester
LE3 9QP
Tel +44 116 256-3643
Fax +44 116 232-2976

> -----Original Message-----
> From: Salvatore Barbaro [mailto:sbarbar at gwdg.de]
> Sent: 24 July 2003 12:56
> To: r-help at stat.math.ethz.ch
> Subject: [R] median and joint distribution
> 
> 
> 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.
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
>