Semi-parametric Order-based Generalized Multivariate Regression
This addresses regression problems where responses are monotonic functions of predictors, offering a method that avoids specifying the functional form, though it appears incremental relative to existing semi-parametric approaches.
The paper tackles generalized multivariate regression with monotonic response functions by proposing a semi-parametric algorithm based on response ordering that is invariant to the transformation function's form. It proves the algorithm is a consistent estimator with squared error decaying at rate o(1/√n), and demonstrates performance through simulations and comparisons with traditional methods.
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.