In this paper, we consider a generalized multivariate regression problemwhere the responses are monotonic functions of linear transformations ofpredictors. We propose a semi-parametric algorithm based on the ordering of theresponses which is invariant to the functional form of the transformationfunction. We prove that our algorithm, which maximizes the rank correlation ofresponses and linear transformations of predictors, is a consistent estimatorof the true coefficient matrix. We also identify the rate of convergence andshow 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 objectivefunction of our model and examine its performance through extensivesimulations. Finally, we compare our algorithm with traditional multivariateregression algorithms over synthetic and real data.
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