We present a forward sufficient dimension reduction method for categorical or ordinal responses by extending the outer product of gradients and minimum average variance estimator to multinomial generalized linear model. Previous work in this direction extend forward regression to binary responses, and are applied in a pairwise manner to multinomial data, which is less efficient than our approach. Like other forward regression-based sufficient dimension reduction methods, our approach avoids the relatively stringent distributional requirements necessary for inverse regression alternatives. We show consistency of our proposed estimator and derive its convergence rate. We develop an algorithm for our methods based on repeated applications of available algorithms for forward regression. We also propose a clustering-based tuning procedure to estimate the tuning parameters. The effectiveness of our estimator and related algorithms is demonstrated via simulations and applications.

翻译：我们通过将梯度和最低平均差异估计值的外产产品扩展至多等通用线性模型,为绝对或正统反应提出了一个充分的远方减少维度方法。以前在这方面开展的工作将前向回归扩展至二元反应,并以双向方式应用于多重数据,这比我们的方法效率低。与其他远方基于回归的足够维度减少方法一样,我们的方法避免了反向回归替代方法所需的相对严格的分配要求。我们显示了我们提议的估计值的一致性,并得出了它的趋同率。我们根据对前向回归反复使用现有算法的方法开发了一种算法。我们还提议了一个基于集群的调控程序来估计调控参数。我们的估计值和相关算法的有效性通过模拟和应用得到证明。