揭示逻辑回归中似然比统计量在小样本下的精确行为，无需设计假设。
Beyond Modern Asymptotics for Log-Likelihood Ratios in Logistic Regression
- 给出似然比统计量在所有设计和参数下的统一上界，不依赖正则性条件。
- 当 $n\gtrsim d+\log(1/δ)$ 时，上界为 $d + \log(1/δ)$，接近经典威尔克斯尺度。
- 低维情形（$d=1,2$）有特殊行为，与传统渐近理论不同，适合高维统计研究者。
我们刻画了二分类逻辑回归中对数似然比统计量的有限样本行为，覆盖所有固定的设计向量集合和目标参数。当 $n\geq d\geq 3$ 时，其最坏情况 $(1-δ)$ 分位数（至多差一个绝对常数）为 $d\log(e n/d) + \log(1/δ)$，这是威尔克斯 $χ^2_d$ 现象的非渐近类比，且无需对设计施加正则性假设。低维情形表现出异常行为：$d=2$ 时最坏分位数为 $\log\log\log n + \log(1/δ)$，$d=1$ 时为 $\log(1/δ)$，与 $n$ 无关。当设计为独立同分布高斯时，恢复经典威尔克斯尺度。在 $n\gtrsim d+\log(1/δ)$ 下，证明了紧上界 $d + \log(1/δ)$。不同于现有渐近结果，本界对目标参数一致，该参数可依赖于 $n$、$d$、$δ$。
原文摘要 · Abstract (English)
We characterize the finite sample behavior of the log-likelihood ratio statistic in binary logistic regression, uniformly over both the design and the target parameter. For $n\geq d\geq 3$, we determine, up to universal constants, its worst case $(1-δ)$ quantile over all fixed collections of design vectors and all target parameters: \[ d\log\left(\frac{e n}{d}\right)+\log\left(\frac{1}δ\right). \] This is a nonasymptotic analogue of the Wilks $χ^2_d$ phenomenon and requires no regularity assumptions on the design. The low dimensional cases exhibit unusual behavior. The worst case quantile in dimension $d=2$ is sharply of order \[ \log\log\log n+\log\left(\frac{1}δ\right). \] The worst case quantile in dimension $d=1$ is of order $\log(1/δ)$, with no dependence on $n$. Finally, i.i.d. Gaussian design vectors recover the classical Wilks scale. In the regime $n\gtrsim d+\log(1/δ)$, we prove the sharp bound \[ d+\log\left(\frac{1}δ\right). \] Unlike existing asymptotic results, our bounds are uniform over the target parameter, which may depend on $n$, $d$, and $δ$.
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