In 21 cm cosmology, we are seeking to detect the highly redshifted spin-flip line of HI to constrain the properties of the Epoch of Reionization and to better understand the astrophysical and cosmological processes of the early Universe. The sensitivity requirements for such a detection are exacting, since astrophysical foregrounds overshadow the 21 cm signal by roughly five orders of magnitude. These foregrounds are, in principle, separable from the 21 cm signal, as they are naturally sequestered in Fourier space owing to their spectral smoothness. However, real-world factors such as calibration errors or array non-redundancy can taint this separability, which have the effect of proliferating foreground power into otherwise signal-dominated regions. A foreground avoidance strategy can be used; however, extreme calibration and instrumental requirements are needed to minimize this foreground leakage.

Radio astronomy is synonymous with big data: large data volumes need to be reduced, much of which is corrupted by radio-frequency interference, systematics and other non-thermal effects. Traditional outlier rejection algorithms are ad-hoc, often require manual input, and can be intricate and costly; these can miss anomalous effects and cause overflagging. Furthermore, commonly used calibration methods assume an underlying Gaussian noise distribution for visibilities, which can lead to sub-optimal results due to radio-frequency interference contamination and array non-redundancy. With the advent of the Square Kilometre Array and other next-generation radio telescopes, the flagging requirements for such traditional processes will be insurmountable. Automated methods that employ robust statistics will be able to adequately reduce these immense streams of interferometric data to produce uncorrupted estimates, with little to no manual input. This thesis focuses on robust and alternative statistical techniques that better deal with non-normal or contaminated radio data, with the aim of improving 21 cm power spectrum results.

The thesis outline is as follows: I start by reviewing the physics of the Epoch of Reionization and the 21 cm signal. I subsequently introduce the fundamentals of radio interferometry, and present the Hydrogen Epoch of Reionization Array (HERA) experiment and its data reduction pipeline. Following this theoretical and experimental underpinning, HERA data is scrutinized and its normality is probed, thus supporting the use of robust statistics for any interferometric data reduction. Robust multivariate location estimators are employed to obtain new visibility results, which are compared to those from the standard pipeline in both the visibility and power spectrum domains. In addition, robust multivariate outlier detection, high-pass filtering before robust averaging and alternative inpainting methods are also discussed. Shifting the focus to calibration, I present a generalized approach to redundant calibration with non-Gaussian maximum likelihood estimation. As an extension of this work, redundant calibration solutions are compared by degenerate translation, and a unified redundant calibration solver across days is developed that provides optimal gain estimates for whole multi-day datasets. Lastly, I use multiresolution analysis with wavelets to better characterize the 21cm signal, as it is non-stationary across redshifts due to the light-cone effect. I also show how the wavelet transform can be used for error detection by inspecting frequency-delay scaleograms.

Through this research, I demonstrate that robust statistics in radio interferometric data reduction and calibration are crucial for obtaining uncontaminated estimates, and eliminate any reliance on abstruse and ad hoc flagging procedures. While these robust methods are computationally more expensive, they can be accelerated with machine learning libraries and hardware. These robust techniques will become a cornerstone of radio astronomy on account of their ability to reduce large amounts of data with confidence and with little intervention.