Envelopes in multivariate regression models with nonlinearity and heteroscedasticity

Summary Envelopes have been proposed in recent years as a nascent methodology for sufficient dimension reduction and efficient parameter estimation in multivariate linear models. We extend the classical definition of envelopes in Cook et al. (2010) to incorporate a nonlinear conditional mean function and a heteroscedastic error. Given any two random vectors ${X}\in\mathbb{R}^{p}$ and ${Y}\in\mathbb{R}^{r}$, we propose two new model-free envelopes, called the martingale difference divergence envelope and the central mean envelope, and study their relationships to the standard envelope in the context of response reduction in multivariate linear models. The martingale difference divergence envelope effectively captures the nonlinearity in the conditional mean without imposing any parametric structure or requiring any tuning in estimation. Heteroscedasticity, or nonconstant conditional covariance of ${Y}\mid{X}$, is further detected by the central mean envelope based on a slicing scheme for the data. We reveal the nested structure of different envelopes: (i) the central mean envelope contains the martingale difference divergence envelope, with equality when ${Y}\mid{X}$ has a constant conditional covariance; and (ii) the martingale difference divergence envelope contains the standard envelope, with equality when ${Y}\mid{X}$ has a linear conditional mean. We develop an estimation procedure that first obtains the martingale difference divergence envelope and then estimates the additional envelope components in the central mean envelope. We establish consistency in envelope estimation of the martingale difference divergence envelope and central mean envelope without stringent model assumptions. Simulations and real-data analysis demonstrate the advantages of the martingale difference divergence envelope and the central mean envelope over the standard envelope in dimension reduction.