Assessing the Effectiveness of Bootstrapped Confidence Intervals in Small Sample Statistical Estimation Problems

Posted: Jan 13, 2008

• Grace Wright
• Levi Lee
• Olivia Adams

Abstract

Statistical inference in small sample contexts presents significant challenges for researchers across numerous disciplines. Traditional asymptotic methods often fail to provide accurate coverage probabilities when sample sizes are severely limited, a common scenario in fields such as medical research with rare diseases, engineering with expensive prototype testing, and social sciences with hard-to-reach populations. The bootstrap method, introduced by Bradley Efron in 1979, has revolutionized statistical practice by providing a computationally intensive but distribution-free approach to interval estimation. However, the conventional wisdom regarding bootstrap methodology has primarily been established for moderate to large sample sizes, leaving a substantial gap in our understanding of its behavior under extreme sample size constraints. This research addresses critical limitations in existing literature by systematically evaluating bootstrap performance in sample sizes ranging from n=5 to n=30, with particular emphasis on the challenging region below n=15. We develop a novel hybrid bootstrap methodology that intelligently combines percentile-t approaches with bias-corrected and accelerated techniques, optimized specifically for small-sample applications. Our investigation is guided by three research questions that have received limited attention in statistical literature: How do various bootstrap variants perform in terms of coverage probability and interval width under extreme sample size constraints? What is the impact of distributional characteristics on bootstrap performance in small samples? Can we develop a diagnostic framework to guide method selection based on observable sample characteristics? The theoretical foundation of this work builds upon but significantly extends the original bootstrap concept by incorporating adaptive resampling strategies and moment-based diagnostics. Our approach challenges the conventional threshold-based recommendations for bootstrap application and instead proposes a continuum-based framework where method selection depends on multiple sample characteristics rather than a simple sample size cutoff.