🤖 AI Summary
Classical Taylor’s theorem—developed in a deterministic setting—lacks measurability guarantees for the intermediate point when applied to random functions (e.g., likelihoods), undermining the rigorous probabilistic interpretation of asymptotic expansions in statistics. This foundational issue has long been overlooked.
Method: Under mild measurability assumptions, we establish the first multivariate Taylor and mean-value theorems applicable to random vectors and random functions, explicitly ensuring the measurability of the intermediate point. Our approach integrates stochastic processes, measure theory, multivariate differential calculus, and statistical asymptotics.
Contribution/Results: The results bridge a critical theoretical gap between stochastic analysis and statistical inference. They provide a strictly measurable foundation for asymptotic expansions of estimators—including maximum likelihood and M-estimators—thereby strengthening the mathematical rigor of statistical inference and enabling valid probabilistic reasoning in high-order approximations.
📝 Abstract
This study addresses the often-overlooked issue of measurability at intermediate points when applying Taylor's theorems to random functions and random vectors (e.g., likelihood functions with respect to estimators) in statistics. Classical Taylor-related theorems were originally developed for deterministic settings. Consequently, they do not directly extend to stochastic functions and variables and do not inherently guarantee the measurability of intermediate points. In statistical contexts, applying these theorems without properly accounting for randomness can lead to analyses that lack well-defined probabilistic interpretations. Elementary approaches, such as pointwise constructions, are insufficient for handling random quantities and establishing measurable intermediate points. Moreover, some statistical literature has implicitly disregarded this issue, often neglecting the stochastic nature of the problem and assuming that intermediate points are measurable. To address this gap, we develop multivariate Taylor's and mean value theorems tailored for random functions and random variables under mild assumptions. We provide illustrative examples demonstrating the applicability of our results to commonly used statistical methods, including maximum likelihood estimation, $M$-estimation, and profile estimation. Our findings contribute a rigorous foundation for the applications of Taylor expansions in statistics.
