发表机构
The Chinese University of Hong Kong（香港中文大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究BTL模型下偏好诱导中未知参数向量的估计，推导了无约束MLE的有限样本误差界，揭示了Fisher信息几何决定估计难度，并给出达到极小极大速率的充分条件，为参数化效用诱导提供统一非渐近理论。
AI 中文摘要
我们研究了Bradley-Terry-Luce (BTL)模型下的偏好诱导问题，其中真实的部分价值向量未知，需要通过诱导的偏好信息作为参数进行估计。所选成对查询的集合是非均匀的、确定性的，并且在一组备选方案上是任意的，前提是它满足联合可辨识性条件。我们专注于理解典型的最大似然估计量(MLE)何时是有限的，并在可行集上没有显式紧性约束或外部正则化器的情况下具有尖锐的误差界。为此，我们在标准有界动态范围条件下推导了极小极大下界，并发现经典Cramér-Rao下界中的相同Fisher信息几何支撑了该估计问题的有限样本难度。通过将似然得分方程的非渐近展开与不动点定位论证相结合，我们确定了一个依赖于设计的样本量阈值，超过该阈值，无约束的典型MLE以高概率存在且唯一。相同的展开将估计误差分解为线性随机项、显式二阶偏差和高阶余项。精细分析给出了充分的样本量条件，在这些条件下，典型MLE达到极小极大速率，直至对数因子和常数因子。这些结果为参数化效用诱导提供了统一的非渐近理论，并揭示了推理何时仅由响应数据决定，而非由外部正则化决定。初步数值结果与理论发现一致。
英文摘要
We study preference elicitation under the Bradley-Terry-Luce (BTL) model where the true partworth vector is unknown and has to be estimated as a parameter with elicited preference information. The set of selected pairwise queries is non-uniform, deterministic, and arbitrary over a collection of alternatives, provided that it satisfies a joint identifiability condition. We focus on understanding when the canonical maximum likelihood estimator (MLE) is finite and admits sharp error bounds without explicit compactness constraints on the feasible set or external regularizers. To this end, we derive minimax lower bounds under the standard bounded dynamic range condition, and find that the same Fisher-information geometry in the classic Cramér-Rao lower bounds underpins the finite-sample difficulty of the estimation problem. By combining a non-asymptotic expansion of the likelihood score equation with a fixed-point localization argument, we identify a design-dependent sample size threshold above which the unconstrained canonical MLE exists and is unique with high probability. The same expansion yields a decomposition of the estimation error into a linear stochastic term, an explicit second-order bias, and a higher-order remainder. A refined analysis gives sufficient sample size conditions under which the canonical MLE attains the minimax rates up to logarithmic and constant factors. These results provide a unified non-asymptotic theory for parametric utility elicitation and reveal when the inference is determined by response data alone rather than by external regularization. Preliminary numerical results are consistent with the theoretical findings.
