Due to technological limitations and peculiarities of Nature, classes of astronomical distance indicators are applicable only in specific distance ranges. The Cosmic Distance Ladder is the framework by which we link together distance indicators, climbing from one rung to the next, in order to measure physical distance on an absolute scale. The object of this dissertation is one category of distance indicators, called RR Lyrae pulsating variable stars, which has commanded substantial scientific study for more than a century.

RR Lyrae stars are low mass (M ~ 0.7 M_sol), old (age > 10¹⁰ yr) Population II objects that are found mixed in with any stellar population of requisite age. They are unstable to radial harmonic oscillations (pulsations) because of their specific mass, metallicity content, and interior composition. It has been empirically determined, and theoretically justified, that the pulsation periods of individual RR Lyrae stars are correlated with their intrinsic luminosity; hereafter referred to as the RR Lyrae period--luminosity relation. Thus, if one can measure the period of a star (a relatively straightforward task given sufficient observations), then one can use that star as a standard candle and infer its distance.

The work in this dissertation is aimed at improving our understanding of the period--luminosity relation of RR Lyrae stars, and particularly at improving the precision of RR Lyrae distance measurements. By leveraging (and advancing) new observational facilities, gathering an abundance of new classical observations, and developing new statistical methods to combine a wealth of multi-wavelength data, this goal has been accomplished. In this dissertation I describe the involved methodology and report distances to a calibration sample of 134 RR Lyrae stars with a median fractional distance error of 0.66 per cent.

In the following chapters I describe the arc of this research. First, I present an instrumentation development project that contributed to a new simultaneous multi-band imaging camera which is well-suited to study RR Lyrae stars and accumulate the invaluable near-infrared photometry necessary for highly-precise distance measurements. Then, I present a series of RR Lyrae period--luminosity relation studies that iteratively combine more and more data (increasing both in calibration sample size and number of wavebands) while simultaneously developing the necessary statistical models and computational methods. Finally, as an application of the results of these earlier investigations, I combine catalog data with new, longer-wavelength observations of the Large Magellanic Cloud (LMC) to measure the three-dimensional shape of the distribution of RR Lyrae stars in the LMC and derive a new distance measurement to the LMC of 50.2482 ± 0.0546 (statistical) ± 0.4628 (systematic) kpc, which is a fractional distance error of 1.03 per cent.