Sufficient Reductions in Regressions With Exponential Family Inverse Predictors
针对预测变量在给定响应变量下服从多元指数族分布的回归问题，提出识别和估计充分降维的方法，可处理连续、分类或混合型预测变量，并给出最小充分降维及其最大似然估计。
We develop methodology for identifying and estimating sufficient reductions in regressions with predictors that, given the response, follow a multivariate exponential family distribution. This setup includes regressions where predictors are all continuous, all categorical, or mixtures of categorical and continuous. We derive the minimal sufficient reduction of the predictors and its maximum likelihood estimator by modeling the conditional distribution of the predictors given the response. Whereas nearly all extant estimators of sufficient reductions are linear and only partly capture the sufficient reduction, our method is not limited to linear reductions. It also provides the exact form of the sufficient reduction, which is exhaustive, its maximum likelihood (ML) estimates via an iterated reweighted least-square (IRLS) estimation algorithm, and asymptotic tests for the dimension of the regression. Supplementary materials for this article are available online.