This document describes statistical techniques for analyzing rare circuit failure events caused by process variations in high-dimensional variation spaces. It introduces the subset simulation (SUS) and scaled-sigma sampling (SSS) methods. SUS estimates the rare failure probability by calculating conditional probabilities in subsets of the variation space. SSS scales the sampling distribution to make rare failures less rare, allowing more efficient estimation. Both methods were tested on 45nm CMOS circuits with hundreds of random variables and shown to accurately estimate failure rates, outperforming existing techniques challenged by high dimensions.