Title: On the Precise Asymptotics of Universal Inference Show Chinese title

Title: 关于通用推断的精确渐近性

Abstract: In statistical inference, confidence set procedures are typically evaluated based on their validity and width properties. Even when procedures achieve rate-optimal widths, confidence sets can still be excessively wide in practice due to elusive constants, leading to extreme conservativeness, where the empirical coverage probability of nominal $1-\alpha$ level confidence sets approaches one. This manuscript studies this gap between validity and conservativeness, using universal inference (Wasserman et al., 2020) with a regular parametric model under model misspecification as a running example. We identify the source of asymptotic conservativeness and propose a general remedy based on studentization and bias correction. The resulting method attains exact asymptotic coverage at the nominal $1-\alpha$ level, even under model misspecification, provided that the product of the estimation errors of two unknowns is negligible, exhibiting an intriguing resemblance to double robustness in semiparametric theory. Show Chinese abstract

Abstract: 在统计推断中，置信集过程通常根据其有效性和宽度特性进行评估。 即使过程达到最优率的宽度，由于难以捉摸的常数，在实践中置信集可能仍然过于宽泛，导致极端保守性，其中名义上的$1-\alpha$水平置信集的经验覆盖概率接近于一。 本文以模型误指下的常规参数模型中的通用推断（Wasserman等，2020）为一个持续的例子，研究有效性和保守性之间的差距。 我们确定了渐近保守性的来源，并提出了一种基于学生化和偏差校正的一般补救方法。 在两个未知数的估计误差的乘积可以忽略不计的情况下，该方法即使在模型误指下也能在名义上的$1-\alpha$水平上达到精确的渐近覆盖，表现出与半参数理论中双重稳健性相似的有趣相似性。