A Roadmap to Asymptotic Properties with Applications to COVID-19 Data

A good estimator should, at least in the asymptotic sense, be close to the true quantity that it wishes to estimate and we should be able to give uncertainty measure based on a finite sample size. An estimator with well-behaved asymptotic properties can help clinicians in many ways such as reducing the number of patients needed in a trial, cutting down the budget for toxicology studies and providing insightful findings for late phase trials. Suggested by Sr. Fisher [1], generations of statisticians have worked on the so-called "consistency" and "asymptotic normality" of estimators. The former is based on different versions of law of large numbers (LLN) and the later is based on various types of central limit theorems (CLT) [2]. In addition to these two main tools, statisticians also apply other important but less well-known results in probability theory and other mathematical fields. To name a few, extreme value theory for distributions of maxima and minima [3], convex analysis for checking the optimality of a statistical design [4], asymptotic relative efficiency (ARE) of an estimator [5], concentration inequalities for finite sample properties and selection consistency [6] and other non-normal limits, robustness and simultaneous confidence bands of common statistical estimators [7, 8]. Despite of different properties, consistency and asymptotic normality are still the most celebrated and important properties of statistical estimators in either academia or industry. Hence, in the following, we present a roadmap to consistency and asymptotic normality. Then we provide their applications in toxicology studies and clinical trials using a COVID-19 dataset.