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.

Due to its generality and flexibility, the state-space model has become one of the most popular models in modern time domain analysis for the description and prediction of time series data. The model is often used to characterize processes that can be conceptualized as "signal plus noise," where the realized series is viewed as the manifestation of a latent signal that has been corrupted by observation noise. In the state-space framework, parameter estimation is generally accomplished by maximizing the innovations Gaussian log-likelihood. The maximum likelihood estimator (MLE) is efficient when the normality assumption is satisfied. However, in the presence of contamination, the MLE suffers from a lack of robustness. Basu, Harris, Hjort, and Jones (1998) introduced a discrepancy measure (BHHJ) with a non-negative tuning parameter that regulates the trade-off between robustness and efficiency. In this manuscript, we propose a new parameter estimation procedure based on the BHHJ discrepancy for fitting state-space models. As the tuning parameter is increased, the estimation procedure becomes more robust but less efficient. We investigate the performance of the procedure in an illustrative simulation study. In addition, we propose a numerical method to approximate the asymptotic variance of the estimator, and we provide an approach for choosing an appropriate tuning parameter in practice. We justify these procedures theoretically and investigate their efficacy in simulation studies. Based on the proposed parameter estimation procedure, we then develop a new model selection criterion in the state-space framework. The traditional Akaike information criterion (AIC), where the goodness-of-fit is assessed by the empirical log-likelihood, is not robust to outliers. Our new criterion is comprised of a goodness-of-fit term based on the empirical BHHJ discrepancy, and a penalty term based on both the tuning parameter and the dimension of the candidate model. We present a comprehensive simulation study to investigate the performance of the new criterion. In instances where the time series data is contaminated, our proposed model selection criterion is shown to perform favorably relative to AIC. Lastly, using the BHHJ discrepancy based on the chosen tuning parameter, we propose two versions of an influence diagnostic in the state-space framework. Specifically, our diagnostics help to identify cases that influence the recovery of the latent signal, thereby providing initial guidance and insight for further exploration. We illustrate the behavior of these measures in a simulation study.