The randomness of the occurrences of earthquakes, together with our limited ability to detect and measure earthquakes, combine to present challenges for the testing of scientific hypothesis about earthquakes. This dissertation examines implications of these challenges and presents methods for addressing them. In contrast to physical systems characterized by a dominating length scale, the relevant scales of earthquakes span many orders of magnitude. Our limited observations of the smallest of these scales, in the form of small, undetected earthquakes, severely impacts our ability to faithfully model observable seismicity because, as we show, small earthquakes contribute significantly to observed seismicity. Using the Epidemic-Type Aftershock Sequence model, a time-dependent model of triggered seismicity, we introduce a formalism that distinguishes between the detection threshold and a smaller size above which earthquakes may trigger others, and place constraints on its size. We derive equations that relate observed clustering parameters obtained from different thresholds. We show that parameters are biased and discuss the failure of the maximum likelihood estimator. As an example of the power of simulation-based null hypothesis testing, we investigate a recent claim of a scaling law in the distribution of the spatial distances between successive earthquakes. Motivated by the debate on the relevance of critical phenomena to earthquakes and by the suggested contradiction of aftershock zone scaling, we analyze other regions and generate synthetic data using a realistic model that explicitly includes mainshock rupture length scales. We show that the proposed law does not hold. Earthquake catalogs contain a wide variety of uncertainties. We quantify magnitude uncertainties and find they are more broadly distributed than a Gaussian distribution. We show their severe impact on short term forecasts by proving that the deviations of a noisy forecast from an exact forecast