Robust parameter estimation and pivotal inference is crucial for credible statistical conclusions. This thesis addresses these issues in three contexts: long-memory parameter estimation robust to low frequency nonstationary contamination, long-memory properties of financial time series, and inference on structural changes in a joint segmented trend with heterogeneous noise. Chapter 1 considers robust estimation of the long-memory parameter allowing for a wide collection of contamination processes, in particular low frequency nonstationary processes such as random level shifts. We propose a robust modified local-Whittle estimator and show it has the usual asymptotic distribution. We also provide modifications to further account for short-memory dynamics and additive noise. The proposed estimator provides substantial efficiency gains compared to existing methods in the presence of contaminations, without sacrificing efficiency when these are absent. Chapter 2 applies the modified local-Whittle estimator to various volatilities series for stock indices and exchange rates to robustly estimate the long-memory parameter. Our findings suggest that all series are a combination of long and short-memory processes and random level shifts, with the magnitude of each component varying across series. Our results contrast with the view that long-memory is the dominant feature. Chapter 3 is concerned with pivotal inference about structural changes in a joint segmented trend with heterogeneous noise. We provide tests for changes in the slope and the variance of the noise valid when both may be present, each allowed to occur at different dates. We suggest procedures for four testing problems.