论文标题
Cramér-rao的下限是由普遍的csiszár差异引起的
Cramér-Rao Lower Bounds Arising from Generalized Csiszár Divergences
论文作者
论文摘要
我们研究了概率分布的几何形状，相对于一般的csiszár$ f $ diverences。该家族的成员是相对$α$ - 肠道，它也是信息理论中相对熵的rényi类似物，在统计中被称为对数或投影能力差异。我们运用Eguchi的理论来得出Fisher信息度量标准以及由这些广义差异函数引起的双重仿射连接。这使我们能够到达Cramér-Rao不平等的更广泛适用的版本，该版本为估算器的差异提供了下限，用于护送潜在的参数概率分布。然后，我们将指数和混合器模型的Amari-Nagaoka的双重平坦结构扩展到了上述广义度量的其他分布。我们表明，这些配方使我们找到了护送模型的无偏和有效估计器。最后，我们将我们的工作与从非信息几何框架得出的广义Cramér-rao不平等现象进行了比较。
We study the geometry of probability distributions with respect to a generalized family of Csiszár $f$-divergences. A member of this family is the relative $α$-entropy which is also a Rényi analog of relative entropy in information theory and known as logarithmic or projective power divergence in statistics. We apply Eguchi's theory to derive the Fisher information metric and the dual affine connections arising from these generalized divergence functions. This enables us to arrive at a more widely applicable version of the Cramér-Rao inequality, which provides a lower bound for the variance of an estimator for an escort of the underlying parametric probability distribution. We then extend the Amari-Nagaoka's dually flat structure of the exponential and mixer models to other distributions with respect to the aforementioned generalized metric. We show that these formulations lead us to find unbiased and efficient estimators for the escort model. Finally, we compare our work with prior results on generalized Cramér-Rao inequalities that were derived from non-information-geometric frameworks.
