  In this paper, we consider a generalized multivariate regression problem
where the responses are monotonic functions of linear transformations of
predictors. We propose a semi-parametric algorithm based on the ordering of the
responses which is invariant to the functional form of the transformation
function. We prove that our algorithm, which maximizes the rank correlation of
responses and linear transformations of predictors, is a consistent estimator
of the true coefficient matrix. We also identify the rate of convergence and
show that the squared estimation error decays with a rate of $o(1/\sqrt{n})$.
We then propose a greedy algorithm to maximize the highly non-smooth objective
function of our model and examine its performance through extensive
simulations. Finally, we compare our algorithm with traditional multivariate
regression algorithms over synthetic and real data.
