发表机构
CREST/ENSAE Paris（巴黎高等统计分析与经济研究所/ENSAE巴黎）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究利用经验过程理论和随机锥几何，揭示了泊松回归中MLE存在的相变阈值，并给出了保证最优风险所需的样本量指数阈值，同时揭示了中间状态并缩小了差距。
AI 中文摘要
我们研究在设定良好的泊松回归中系数β的最大似然估计。利用经验过程理论和随机锥几何的工具，我们证明最大似然估计（MLE）存在的概率在阈值n > d处表现出尖锐的相变。然后，我们确定一个关于样本量n的、以β的范数为指数的阈值，以保证以高概率达到渐近阶d/n的过剩风险。我们揭示了泊松回归中存在一个中间状态，当n大于d但小于该指数阈值时，MLE存在但未达到最优速率d/n。我们通过提供当n > d^{1+ε}时MLE与β之间距离的上界，将两个状态之间的差距缩小到d^{1+ε}项以内，其中ε∈(0,1)。在此过程中，我们提供了关于次伽马随机向量和次伽马随机矩阵的两个众所周知的PAC-Bayes不等式的推广，这些推广具有独立的意义，并且我们广泛使用它们来证明本文的主要结果。
英文摘要
We study the maximum likelihood estimation of the coefficient β in well-specified Poisson regression. Using tools from empirical process theory and random conic geometry, we show that the probability of existence of the maximum likelihood estimator (MLE) exhibits a sharp phase transition at the threshold n > d. We then determine a minimum threshold exponential in the norm of β on the sample size n to guarantee with high probability an excess risk of the asymptotic order d/n. We reveal the existence of an intermediate regime in Poisson regression, when n is larger than d but smaller than this exponential threshold, where the MLE exists but does not achieve the optimal rate d/n. We close the gap between the two regimes up to a d^{1+ε} term with ε \in (0, 1) by providing an upper bound on the distance between the MLE and β whenever n > d^{1+ε}. Along the way, we provide two generalizations of well-known PAC-Bayes inequalities regarding sub-Gamma random vectors and sub-Gamma random matrices that are of independent interest and that we use extensively to prove the main results of the present paper.
