标题： 基于密度幂的散度的统一表示可简化为M估计 显示英文标题

标题： A Unified Representation of Density-Power-Based Divergences Reducible to M-Estimation

摘要： 基于密度幂的散度已知能在异常值情况下提供稳健的推断过程，其扩展已被广泛研究。 成功散度的一个特征是估计问题可以简化为M估计。 在本文中，我们定义了一种基于范数的Bregman密度幂散度（NB-DPD）——一种在Bregman散度框架内具有功能灵活性的基于密度幂的散度，可以简化为M估计。 我们证明，通过指定函数$\phi_\gamma$，NB-DPD 可以简化为已知的散度，如密度幂散度和$\gamma$-散度。 此外，通过检查对应于现有散度的函数$\phi_\gamma$的组合，我们证明可以推导出一种连接这些现有散度的新散度。 最后，我们证明，红退特性（稳健性的一个关键指标）仅适用于$\gamma$-散度。 显示英文摘要

摘要： Density-power-based divergences are known to provide robust inference procedures against outliers, and their extensions have been widely studied. A characteristic of successful divergences is that the estimation problem can be reduced to M-estimation. In this paper, we define a norm-based Bregman density power divergence (NB-DPD) -- density-power-based divergence with functional flexibility within the framework of Bregman divergences that can be reduced to M-estimation. We show that, by specifying the function $\phi_\gamma$, NB-DPD reduces to well-known divergences, such as the density power divergence and the $\gamma$-divergence. Furthermore, by examining the combinations of functions $\phi_\gamma$ corresponding to existing divergences, we show that a new divergence connecting these existing divergences can be derived. Finally, we show that the redescending property, one of the key indicators of robustness, holds only for the $\gamma$-divergence.