Let $X$ be a random variable which takes values from $Omega = (0,1)^m$ with a probability distribution $p(x)$. Assume $p$ is a BV function with non zero total variation and $p(x)>0forall xinOmega$. There is a discrete random variable $Y$ which takes values from ${0,1}$ and depends on $X$ with $P(Y=1/X=x) = eta(x)$. Assume that $eta$ is also a BV function with non zero total variation and no removable discontinuities.

Binary Regression Problem

Given no other information except $n$ samples of the random variable pair $(X,Y)$ drawn *iid*, that is $(x_1,y_1),(x_2,y_2),ldots(x_n,y_n)$ one need to give a method for computing $tilde{eta}_n$, an estimate of $eta$ such that $$limlimits_{ntoinfty}|tilde{eta}_n-eta|_{L^2(Omega)} = 0$$

This problem I believe is open (needs confirmation as I have little knowledge on statistics literature). There are methods like $k$-nearest neighbours method, which solves when $eta$ belongs to a narrower class of absolutely continuous functions. There seem to be some methods when $eta$ is a Lipschitz continous function, but with known Lipschitz constant.

For $eta$ belongs to class of all BV functions with nonzero total variation (a wider class), I believe I have come up with a method and proof of convergence. Is this sufficiently interesting to be considered for a mathematics journal or it should be considered for a (mathematical)statistics journal? How interesting is this problem for mathematicians and/or statisticians?