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Trimmed Mean for Partially Observed Functional Data

Yixiao Wang

TL;DR

This work addresses the analysis of partially observed functional data by extending the concept of integrated functional depth to partially observed settings via POIFD. It defines an $\ abla$-trimmed mean based on POIFD, and proves strong consistency of both the depth and the trimmed estimator under suitable regularity conditions. Theoretical results cover fixed and weighted observation schemes, ensuring almost-sure convergence of the empirical trimmed mean to its population counterpart. A comprehensive simulation study demonstrates that the trimmed mean offers superior accuracy and robustness compared with the ordinary mean across various contamination and missingness scenarios, highlighting its practical utility in incomplete FDA contexts.

Abstract

In recent years, partially observable functional data has gained significant attention in practical applications and has become the focus of increasing interest in the literature. In this thesis, we build upon the concept of data integration depth for partially observable functions, as proposed by Elias et al. (2023), and the trimmed-mean estimator method along with its consistency proof introduced by Fraiman and Muniz (2001) for completely observable functions. We introduce the concept of trimmed mean specifically for partially observable functional data. Additionally, we address several theoretical and practical issues, including a proof of the strong consistency of the proposed trimmed mean, and we provide a simulation study. The results demonstrate that our estimator outperforms the ordinary mean in terms of accuracy and robustness when applied to partially observable functional data.

Trimmed Mean for Partially Observed Functional Data

TL;DR

This work addresses the analysis of partially observed functional data by extending the concept of integrated functional depth to partially observed settings via POIFD. It defines an -trimmed mean based on POIFD, and proves strong consistency of both the depth and the trimmed estimator under suitable regularity conditions. Theoretical results cover fixed and weighted observation schemes, ensuring almost-sure convergence of the empirical trimmed mean to its population counterpart. A comprehensive simulation study demonstrates that the trimmed mean offers superior accuracy and robustness compared with the ordinary mean across various contamination and missingness scenarios, highlighting its practical utility in incomplete FDA contexts.

Abstract

In recent years, partially observable functional data has gained significant attention in practical applications and has become the focus of increasing interest in the literature. In this thesis, we build upon the concept of data integration depth for partially observable functions, as proposed by Elias et al. (2023), and the trimmed-mean estimator method along with its consistency proof introduced by Fraiman and Muniz (2001) for completely observable functions. We introduce the concept of trimmed mean specifically for partially observable functional data. Additionally, we address several theoretical and practical issues, including a proof of the strong consistency of the proposed trimmed mean, and we provide a simulation study. The results demonstrate that our estimator outperforms the ordinary mean in terms of accuracy and robustness when applied to partially observable functional data.
Paper Structure (23 sections, 8 theorems, 69 equations, 4 figures, 4 tables)

This paper contains 23 sections, 8 theorems, 69 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background and Literature Review
  3. Organization of the Paper
  4. Functional Data Depth
  5. Data Depth
  6. Integrated Functional Depth
  7. Partially Observed Integrated Functional Depth (POIFD)
  8. Basic Setting
  9. Population Form
  10. Sample Form
  11. The Trimmed Mean for Partially Observed Functional Data and Related Theorems
  12. Definition
  13. Strong Consistency Results
  14. The Case of a Fixed Weight Function
  15. The Case of Weighted Functions for Partially Observed Functions
  16. ...and 8 more sections

Key Result

Theorem 1

Under the conditions H1 and H2, if the weight function is $\omega(t)$ and $\sup_{t \in [0,1]} |\omega(t)| \leq C$ for some constant $C$, let where $F_{n,t}(x(t))$ and $F_{t}(x(t))$ are the empirical and population distribution functions, respectively. We have and

Figures (4)

  • Figure 1: Illustration of partially observable functions.
  • Figure 2: Symmetric complete contamination model
  • Figure 3: Asymmetric complete contamination model
  • Figure 4: Partial contamination model

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 2 more