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A Tutorial on Asymptotic Properties for Biostatisticians with Applications to COVID-19 Data

Elvis Han Cui

TL;DR

A roadmap of general procedures for deriving asymptotic properties under fixed designs and the observations need not to be iid is built and their applications in many statistical applications are provided.

Abstract

Asymptotic properties of statistical estimators play a significant role both in practice and in theory. However, many asymptotic results in statistics rely heavily on the independent and identically distributed (iid) assumption, which is not realistic when we have fixed designs. In this article, we build a roadmap of general procedures for deriving asymptotic properties under fixed designs and the observations need not to be iid. We further provide their applications in many statistical applications. Finally, we apply our results to Poisson regression using a COVID-19 dataset as an illustration to demonstrate the power of these results in practice.

A Tutorial on Asymptotic Properties for Biostatisticians with Applications to COVID-19 Data

TL;DR

A roadmap of general procedures for deriving asymptotic properties under fixed designs and the observations need not to be iid is built and their applications in many statistical applications are provided.

Abstract

Asymptotic properties of statistical estimators play a significant role both in practice and in theory. However, many asymptotic results in statistics rely heavily on the independent and identically distributed (iid) assumption, which is not realistic when we have fixed designs. In this article, we build a roadmap of general procedures for deriving asymptotic properties under fixed designs and the observations need not to be iid. We further provide their applications in many statistical applications. Finally, we apply our results to Poisson regression using a COVID-19 dataset as an illustration to demonstrate the power of these results in practice.
Paper Structure (9 sections, 5 theorems, 47 equations)

This paper contains 9 sections, 5 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Consistency
  3. Weak Consistency
  4. Strong Consistency
  5. Asymptotic Normality
  6. Classical CLT
  7. Cramer's Conditions
  8. Generalized Linear Models (GLM) with Fixed Design
  9. Applications to COVID-19 Data

Key Result

Theorem 2.1

Suppose that $X_1,\cdots,X_n$ are independent and identically distributed (iid) with finite means and variances, $var(X_i)=\sigma^2$. Then as $n$ goes to infinity where $\epsilon$ is a small positive number.

Theorems & Definitions (15)

  • Theorem 2.1: Khinchin's WLLN
  • Example 2.1: Hoeffding's inequality for boosting algorithm
  • Example 2.2: Independent but not identically distributed sequence
  • Example 2.3: Weak consistency of dependent variables
  • Theorem 2.2: Feller-Kolmogorov WLLN feller1967introductiondabrowska2019
  • proof
  • Example 2.4: St Petersburg paradox
  • Example 2.5: Exponential spacings and order statistic
  • Example 2.6: Pareto's distribution
  • Theorem 2.3: Glivenko-Cantelli dabrowska2019wellner2013weak
  • ...and 5 more