Vidyashankar Anand N, Collamore Jeffrey F
Department of Statistics, George Mason University, Fairfax, VA 22030, USA.
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark.
Entropy (Basel). 2021 Mar 24;23(4):386. doi: 10.3390/e23040386.
Hellinger distance has been widely used to derive objective functions that are alternatives to maximum likelihood methods. While the asymptotic distributions of these estimators have been well investigated, the probabilities of rare events induced by them are largely unknown. In this article, we analyze these rare event probabilities using large deviation theory under a potential model misspecification, in both one and higher dimensions. We show that these probabilities decay exponentially, characterizing their decay via a "rate function" which is expressed as a convex conjugate of a limiting cumulant generating function. In the analysis of the lower bound, in particular, certain geometric considerations arise that facilitate an explicit representation, also in the case when the limiting generating function is nondifferentiable. Our analysis involves the modulus of continuity properties of the affinity, which may be of independent interest.
赫林格距离已被广泛用于推导作为最大似然方法替代方案的目标函数。虽然这些估计量的渐近分布已得到充分研究，但由它们引起的罕见事件的概率在很大程度上尚不清楚。在本文中，我们在潜在模型误设的情况下，使用大偏差理论在一维和更高维度上分析这些罕见事件概率。我们表明这些概率呈指数衰减，通过一个“速率函数”来表征它们的衰减，该速率函数表示为极限累积量生成函数的凸共轭。特别是在分析下界时，会出现某些几何考虑因素，这有助于在极限生成函数不可微的情况下也能得到明确的表示。我们的分析涉及亲和性的连续性模性质，这可能具有独立的研究价值。
