Assessing the Effectiveness of Monte Carlo Simulation Techniques in High-Dimensional Statistical Estimation Problems

Posted: Jan 15, 2020

• Maria Wilson
• Michael Miller
• Lucas Thomas

Abstract

Monte Carlo simulation techniques have long been established as fundamental tools in statistical estimation, providing powerful methods for approximating complex integrals, solving high-dimensional problems, and performing Bayesian inference. However, the application of these techniques to genuinely high-dimensional statistical estimation problems presents unique challenges that remain inadequately addressed in the current literature. The curse of dimensionality manifests in Monte Carlo methods through exponentially increasing variance, deteriorating mixing properties in Markov Chain Monte Carlo (MCMC) algorithms, and computational bottlenecks that render many traditional approaches impractical. This research addresses a critical gap in understanding how Monte Carlo simulation techniques perform under extreme dimensionality conditions. While numerous studies have examined Monte Carlo methods in moderate dimensions, few have systematically investigated their behavior in dimensions exceeding 100, where traditional assumptions about sample efficiency and convergence rates break down. Our work bridges this gap by providing a comprehensive assessment of both classical and contemporary Monte Carlo variants across a spectrum of high-dimensional statistical estimation problems. The novelty of our approach lies in the development and evaluation of a hybrid methodology that combines Hamiltonian Monte Carlo with dimension-adaptive sparse grid techniques. This integration represents a significant departure from conventional approaches, as it leverages the geometric properties of Hamiltonian dynamics while simultaneously addressing the dimensionality challenge through adaptive sparse representations. Our methodology enables more efficient exploration of high-dimensional parameter spaces while maintaining computational feasibility.