We examine the non-asymptotic properties of robust density ratio estimation (DRE) in contaminated settings. Weighted DRE is the most promising among existing methods, exhibiting doubly strong robustness from an asymptotic perspective. This study demonstrates that Weighted DRE achieves sparse consistency even under heavy contamination within a non-asymptotic framework. This method addresses two significant challenges in density ratio estimation and robust estimation. For density ratio estimation, we provide the non-asymptotic properties of estimating unbounded density ratios under the assumption that the weighted density ratio function is bounded. For robust estimation, we introduce a non-asymptotic framework for doubly strong robustness under heavy contamination, assuming that at least one of the following conditions holds: (i) contamination ratios are small, and (ii) outliers have small weighted values. This work provides the first non-asymptotic analysis of strong robustness under heavy contamination.

翻译：本文研究了污染环境下鲁棒密度比估计（DRE）的非渐近性质。在现有方法中，加权DRE最具前景，从渐近视角展现出双重强鲁棒性。本研究表明，在非渐近框架下，即使存在重度污染，加权DRE仍能实现稀疏一致性。该方法解决了密度比估计与鲁棒估计中的两个关键挑战。针对密度比估计，我们在加权密度比函数有界的假设下，给出了无界密度比估计的非渐近性质。针对鲁棒估计，我们提出了重度污染下双重强鲁棒性的非渐近框架，其假设至少满足以下条件之一：（i）污染比例较小，且（ii）异常值具有较小的加权值。本研究首次对重度污染下的强鲁棒性进行了非渐近分析。