Exploring the Role of Bootstrapping in Estimating Standard Errors for Complex Statistical Estimators

Posted: Aug 16, 2008

• Kylie Rogers
• Lauren Parker
• Layla Evans

Abstract

The estimation of standard errors represents a fundamental challenge in statistical inference, particularly for complex estimators where closed-form variance expressions are unavailable or rely on unrealistic assumptions. Bootstrapping, introduced by Bradley Efron in 1979, has emerged as a powerful resampling technique for assessing the variability of statistical estimators without stringent distributional assumptions. However, despite decades of research and application, significant gaps remain in our understanding of bootstrap performance for estimators with non-standard properties, including those with non-smooth objective functions, high-dimensional parameters, or complex dependence structures. Traditional approaches to standard error estimation often depend on asymptotic approximations derived from central limit theorems or delta method applications. While these methods perform adequately in large samples under regular conditions, their reliability diminishes in finite samples, with non-standard estimators, or when model assumptions are violated. The bootstrap offers an attractive alternative by empirically approximating the sampling distribution through resampling, but its application to complex estimators presents unique challenges that have not been systematically addressed in the existing literature. This research addresses several critical limitations in current bootstrap methodology. First, we investigate the performance of bootstrap methods for estimators defined through non-smooth estimating equations, where conventional bootstrap theory based on smooth function models breaks down. Second, we examine bootstrap behavior in high-dimensional settings where the number of parameters grows with sample size, creating scenarios where standard bootstrap consistency may fail. Third, we develop novel adaptations of bootstrap methods for dependent data structures where simple random resampling destroys essential correlation patterns. Our contribution is threefold. Methodologically, we introduce an integrated bootstrap framework that combines multiple resampling strategies to address different aspects of estimator complexity. Theoretically, we establish new conditions for bootstrap consistency in non-standard settings and identify boundary