Exploring the Relationship Between Statistical Bias and Estimator Consistency in Finite Sample Analysis

Posted: Sep 22, 2015

• Victoria Martin
• Henry Thompson
• Riley Perez

Abstract

The conventional statistical paradigm has long maintained a clear distinction between bias and consistency as distinct properties of estimators. Bias refers to the systematic deviation of an estimator's expected value from the true parameter value, while consistency describes the convergence of an estimator to the true parameter as sample size increases indefinitely. Traditional statistical education and practice often emphasize unbiasedness as a desirable property, with consistent but biased estimators receiving less attention in applied settings. However, this perspective fails to account for the complex interplay between these properties in finite samples, where most real-world statistical analysis occurs. This research challenges the conventional separation of bias and consistency by demonstrating their intricate relationship in finite sample contexts. We investigate how bias influences the path to consistency and how consistency requirements constrain the permissible forms of bias. The motivation for this work stems from observed phenomena in applied statistics where intentionally biased estimators, such as ridge regression or James-Stein estimators, often outperform their unbiased counterparts in finite samples while maintaining asymptotic consistency. Our primary research questions address fundamental gaps in current understanding: How does the magnitude and direction of bias affect the rate of convergence to consistency? Under what conditions do biased estimators achieve superior finite-sample performance while maintaining consistency? Can we develop a unified framework that quantifies the bias-consistency trade-off across different estimator classes and sample sizes?