Employing Modern Statistics To Explore the Universe with Type Ia Supernovae

The Large Synoptic Survey Telescope (LSST) anticipates observing hundreds of thousands of well-measured Type Ia supernovae (SNe Ia). These stellar remnant explosions are exceptional in that they have a standardizeable light curve which allows for an accurate measurement of their luminosity. The standard nature of SNe Ia allow us to measure relative distances in the Universe with better than 6\% precision in distance. With distance estimates in hand to large sets of galaxies through Type Ia Supernova (SN Ia) measurements, we can measure the expansion history of the Universe or create flow models of how galaxies (matter) near the Milky Way are moving. \ud \ud In this new regime of large datasets, weaknesses and limitations of the current techniques for estimating cosmological parameters and modeling local flows are becoming apparent. As statistical errors are reduced systematic uncertainties ranging from calibration to survey design and cadence to host galaxy contamination are dominating the error budget and limiting our ability to make improvements on cosmological measurements. Similarly, recent comparisons of flow models reveal systematic inconsistencies between different approaches. \ud \ud For my dissertation I have employed modern statistical methods to improve flow models in the local Universe by accounting for the non-uniform distribution of data across the sky and demonstrated how Approximate Bayesian Computation can tackle complicated likelihood functions in supernova cosmology. I also present the first results of a new near-infrared SN~Ia survey called "SweetSpot" whose focus is on improving our ability to standardize the total luminosity of SNe Ia.\u