LAUSR

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 ∈ Rp and Y ∈ Rr, we propose two new model-free envelopes – called the Martingale Difference Divergence Envelope (MDDE) and 20 the Central Mean Envelope (CME) – and study their relationships with the standard envelope in the context of response reduction in the multivariate linear models. The MDDE effectively captures the nonlinearity in the conditional mean without imposing any parametric structure or requiring any tuning in estimation. Heteroscedasticity, or the non-constant conditional covariance of Y | X , is further detected by the CME based on a slicing scheme for the data. We reveal the nested structure 25 of different envelopes: (1) the CME contains the MDDE, with equality when Y | X has a constant conditional covariance; and (2) the MDDE contains the standard envelope, with equality when Y | X has a linear conditional mean. We further develop an estimation procedure that obtains the MDDE first and then estimates the additional envelope components in the CME. We establish consistency in envelope estimation of MDDE and CME without stringent model assumptions. Simulations and real data 30 analysis demonstrate the advantages of MDDE and CME over standard envelope in dimension reduction.