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：我们提出了一种面向类别或有序响应的前向充分降维方法，通过将梯度外积与最小平均方差估计量扩展到多项广义线性模型中实现。该方向的前期研究将前向回归推广至二元响应，并以成对方式应用于多项数据，其效率低于我们的方法。与其他基于前向回归的充分降维方法类似，我们的方法避免了对逆回归替代方法所需的相对严格的分布条件。我们证明了所提估计量的一致性并推导出其收敛速率。通过重复应用现有前向回归算法，我们开发了适用于本方法的算法。同时提出一种基于聚类的调参过程来估计调节参数。模拟实验与应用案例验证了估计量及相关算法的有效性。