奇异回归问题中的一个极限定理
A Limit Theorem in Singular Regression Problem
摘要：
在统计问题中，一组参数化的概率分布被用来估计真实的概率分布。如果真实分布下的Fisher信息矩阵是奇异的，那么我们从随机样本中能估计出真实分布的多少是未知的。本文研究了一类奇异回归问题，证明了一个极限定理，该定理表明了奇异回归问题与两个双有理不变量、一个实对数正则阈值和一个奇异波动之间的关系。所得到的定理有一个重要的应用统计，因为它使我们能够估计泛化误差的训练误差，没有任何知识的真实概率分布。
英文摘要：
In statistical problems, a set of parameterized probability distributions is used to estimate the true probability distribution. If Fisher information matrix at the true distribution is singular, then it has been left unknown what we can estimate about the true distribution from random samples. In this paper, we study a singular regression problem and prove a limit theorem which shows the relation between the singular regression problem and two birational invariants, a real log canonical threshold and a singular fluctuation. The obtained theorem has an important application to statistics, because it enables us to estimate the generalization error from the training error without any knowledge of the true probability distribution.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
