Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yifan Yang, Xiaoyu Zhou, Ming Wang

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.

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

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, -estimation, and profile estimation. Our findings contribute a rigorous foundation for the applications of Taylor expansions in statistics.

Paper Structure

This paper contains 2 sections, 8 theorems, 37 equations.

Table of Contents

  1. PRELIMINARY
  2. PROOFS OF MAIN RESULTS

Key Result

Lemma 1

Let $\mathcal{U} \subseteq \mathcal{E} = \mathbb{R} ^ {p}$, $\mathcal{F} = \mathbb{R}$, and $m \geq 1$ be an integer. Let $f: \mathcal{U} \to \mathcal{F}$ be a function. Suppose the following three conditions hold: (1) $\mathcal{U}$ is an open set. (2) $f(\cdot)$ is $m$-th continuously differentiabl where $u_{\ell}$ and $y_{\ell}$ are the $\ell$-th components of $\bm{u}$ and $\bm{y}$, and the oper

Theorems & Definitions (22)

  • Lemma 1: Taylor's Theorems for Deterministic Functions and Deterministic Vectors
  • proof : Proof of Lemma \ref{['Lem:dfdv']}
  • Definition 1: Multifunction
  • Definition 2: Measurable Multifunction
  • Definition 3: Distance Function for a Nonempty Set and Distance Function Associated with a Multifunction
  • Definition 4: Quasi-Continuous Function
  • Definition 5: Carathéodory Function and Quasi-Carathéodory Function
  • Definition 6: Closed Value of a Function
  • Lemma 2: Filippov Implicit Function Theorems for Carathéodory and Quasi-Carathéodory Functions
  • proof : Proof of Lemma \ref{['Lem:Filippov']}
  • ...and 12 more