I'm currently interested in Positive Quadrant Dependent (PQD) random variables and its properties, in contrast to just positive correlated random variables. Since PQD implies positive correlation, what are the properties of random variables from PQD-family, which random variables from Non-PQD-family with positive correlation do not fulfill?
I guess the answer is included in the question
PQD is a weak form of the association properties due to mathematical abstrations
definitely the question I would ask is wether the question refers to its mathematical relevance
I personally would prefer to seek for a modelling purpose
Thank you Paul Doukhan for your answer. But it cannot be the case that the inherited property of PQD family is "only" positive correlation. Otherwise, it would be such an "uninteresting" object.
I research a pair of finance parameters, which are well-known to be positively correlated, but PQD structure is still unknown. I want to know whether testing PQD structure will be worth the time. Therefore, I first look at the potential research impact, when PQD structure can be significantly identified.
The property can be rewritten in terms of copulas, see page 2 here :
http://www.karlin.mff.cuni.cz/~omelka/Soubory/papers/pqd_2010.pdf
Thank you Frédéric Ouimet for your answer and the link. However, from a short glance, I fail to see why a r.v pair (X,Y) from Non-PQD-family with positive correlation cannot be rewritten in terms of copulas. Can you assist?
Copulas are notions related to distribution functions (or more generally probabilities) whereas correlation is a notion related to expectation.
PQD is a notion relating survival probabilities. So PQD and copulas are linked directly, but not positive correlation and copulas necessarily.
The PQD definition can be rewritten with expectations as follows. If 1_A(x) is the indicator function of the set A (meaning 1_A(x) = 1 if x is in A and =0 otherwise), then X and Y are PQD if and only if
E[1_{X > x} 1_{Y > y}] - E[1_{X > x}] E[1_{Y > y}] >= 0 for all x,y
if and only if
Covariance(1_{X > x}, 1_{Y > y}) >= 0 for all x,y.
In other words,
X and Y are PQD if and only if
1_{X > x} and 1_{Y > y} are non-negatively correlated for all x,y
In the reverse direction, I just wrote the formal proof that X,Y being PQD and non-negative implies X,Y being non-negatively correlated.
See the uploaded pdf here : http://docdro.id/f4IPq1a
If we assume that X and Y are non-negative, the reverse implication is probably false, meaning that X and Y being non-negatively correlated doesn't imply that X and Y are PQD (left as an exercise to the reader !)
To summarize everything above :
1) X,Y are PQD if and only if 1_{X > x} and 1_{Y > y} are non-negatively correlated for all x,y
2) X,Y being PQD and non-negative implies that X,Y are non-negatively correlated
3) Even if X,Y are non-negative, then X,Y being non-negatively correlated doesn't imply that X,Y are PQD
So PQD is a STRONGER notion of dependence than non-negative correlation, not weaker.
It would be greatly appreciated if you could recommend my answers (if you find them useful). Thank you.
Thanks a lot, the proof for PQD => positive correlation is rather trivial when one uses Höffding's covariance identity, it does not have to be positive r.v.
I learnt a lot from your answers. However, for the main purpose of the question, I probably do not have any particular reason to prove the existence of PQD structure of two particular finance parameters.
This is the first time I've ever heard about Hoeffding's covariance identity and PQD, so that's good to know.
I wrote the above proof on the fly.
I find this post interesting but not sure if an answer to the original question posed by Christopher has appeared.
Give an example of a pair of random variables (X, Y) that has a positive correlation but they are not necessarily PQD.
By Hoeffding's covariance identity, this means one has to find a joint cdf F(x,y) that violates PQD over (small) subset of 2-d Euclidean space but the integral/sum of F(x,y)-F_1(x)F_2(y) is positive (where F_1(x) and F_2(y) are the marginal cdf's of X and Y, respectively).
The following paper provides several generalizations and examples (mostly with discrete distributions):
A Multivariate Extension of Hoeffding's Lemma
https://www.jstor.org/stable/2243993
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