Optimal estimation of slope vector in high-dimensional linear transformation models

In a linear transformation model, there exists an unknown monotone, typically nonlinear, transformation function such that the transformed response variable is related to the predictor variables via a linear regression model. This paper presents CENet, a new method for estimating the slope vector and simultaneously performing variable selection in the high-dimensional sparse linear transformation model. CENet is the solution to a convex optimization problem which can be computed efficiently from an algorithm with guaranteed convergence to the global optimum. It is shown that when the joint distribution of the predictors and errors is elliptical, under some regularity conditions, CENet attains the same optimal rate of convergence as the best regression method in the high-dimensional sparse linear regression model. The empirical performance of CENet is shown on both simulated and real datasets. The connection of CENet with existing nonlinear regression/multivariate methods is also discussed.