WEAK CONVERGENCE OF AN EMPIRICAL MONOTONIC DEPENDENCEFUNCTION UNDER DEPENDENCE
T. Bednarski T. Ledwina
Abstract: The weak convergence of a consistent estimator of a monotonic dependence
function of two random variables and is studied. The estimator is treated as a random
element of and of where stands for the Lebesgue measure. Its
asymptotic distribution is derived for the two spaces in the following cases: independence of
and , distributions contiguous to independence, and dependence of and .
Except for the case of independence the asymptotic distributions depend strongly on the
marginals of and . Therefore, the asymptotic distribution of rank counterpart of the
estimator is also considered. The obtained results extend the possibility of practical
applications of the measure of monotonic dependence and its consistent estimator.
and 
is studied. The estimator is treated as a random
element of ![D[0,1]](files/4.1/HTML/4.1.7.abs2x.png)
and of ![L ([0,1],c),
2](files/4.1/HTML/4.1.7.abs3x.png)
where 
stands for the Lebesgue measure. Its
asymptotic distribution is derived for the two spaces in the following cases: independence of
and 
, distributions contiguous to independence, and dependence of 
and 
.
Except for the case of independence the asymptotic distributions depend strongly on the
marginals of 
and 
. Therefore, the asymptotic distribution of rank counterpart of the
estimator is also considered. The obtained results extend the possibility of practical
applications of the measure of monotonic dependence and its consistent estimator.