This dissertation focuses on resampling-type methods for complex structured data collected over time. In particular, a notoriously difficult problem from time series is considered regarding resampling inference in the frequency domain. Despite many developments over the past 25 years, existing bootstrap methods struggle with complicated variance structures that arise in such inference. The first project starts with a more basic form of resampling, called subsampling, which has been largely ignored in efforts to develop bootstrap for frequency domain inference. We show that, non-trivially, subsampling solves the general variance estimation problem in the frequency domain under far weaker conditions than any existing bootstrap. We then link subsampling to the current state-of-the-art bootstrap methods and show that subsampling is key to broadly expanding the application of such bootstraps. The subsampling work in the complicated context of frequency domain inference for time series also suggests a larger and broader potential for alternative statistics and bootstrap schemes for dependent data. With the frequency domain, in particular, all existing bootstraps involve resampling periodogram ordinates, which commonly encounter problems because the periodogram ordinates exhibit are not independent. Essentially, the bootstrap principle can break down, so that existing resampling plans cannot entirely re-create spectral statistics and finite-sample performance suffers for distributional approximation. Using a sub-data scale perspective, though, it is possible to re-imagine bootstraps and statistics in a way that breaks with the past. In this spirit, the second project combines resampling techniques, including empirical likelihood, to formulate a fundamentally new bootstrap method for time series in the frequency domain. The third project focuses on another important topic in the frequency domain, involving the interval estimation of spectral densities, where current methods cannot produce meaningful results in practice. In the latter project, we re-innovate a current frequency domain bootstrap method and combine its strength with empirical likelihood to develop a novel hybrid inference method. The hybrid method is proven to be valid under a wide range of time processes and also demonstrates promising numerical performance for interval estimation
