Taylor's Theorem and Mean Value Theorem for Random Functions and Random Variables

TL;DR

This paper develops rigorous multivariate Taylor and mean value theorems for random functions and variables, addressing measurability issues often overlooked in statistical applications like likelihood and M-estimation.

Contribution

It introduces new Taylor and mean value theorems for stochastic functions, ensuring measurability of intermediate points in statistical contexts.

Findings

Provides conditions for measurability of intermediate points

Demonstrates applicability to maximum likelihood and M-estimation

Establishes a rigorous foundation for Taylor expansions in statistics

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.