Semi-parametric order-based generalized multivariate regression

In this paper, we consider a generalized multivariate regression problem where the responses are some functions of linear transformations of predictors. We assume that these functions are strictly monotonic, but their form and parameters are unknown. We propose a semi-parametric estimator based on the ordering of the responses which is invariant to the functional form of the transformation function as long as it is strictly monotonic. We prove that our estimator, which maximizes the rank similarity between 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/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.