We test the hypothesis that accelerating moment release (AMR) is a precursor to large earthquakes, using data from California, Nevada, and Sumatra. Spurious cases of AMR can arise from data fitting because the time period, area, and sometimes magnitude range analyzed before each main shock are often optimized to produce the strongest AMR signal. Optimizing the search criteria can identify apparent AMR even if no robust signal exists. For both 1950-2006 California-Nevada M ??? 6.5 earthquakes and the 2004 M9.3 Sumatra earthquake, we can find two contradictory patterns in the pre-main shock earthquakes by data fitting: AMR and decelerating moment release. We compare the apparent AMR found in the real data to the apparent AMR found in four types of synthetic catalogs with no inherent AMR. When spatiotemporal clustering is included in the simulations, similar AMR signals are found by data fitting in both the real and synthetic data sets even though the synthetic data sets contain no real AMR. These tests demonstrate that apparent AMR may arise from a combination of data fitting and normal foreshock and aftershock activity. In principle, data-fitting artifacts could be avoided if the free parameters were determined from scaling relationships between the duration and spatial extent of the AMR pattern and the magnitude of the earthquake that follows it. However, we demonstrate that previously proposed scaling relationships are unstable, statistical artifacts caused by the use of a minimum magnitude for the earthquake catalog that scales with the main shock magnitude. Some recent AMR studies have used spatial regions based on hypothetical stress loading patterns, rather than circles, to select the data. We show that previous tests were biased and that unbiased tests do not find this change to the method to be an improvement. The use of declustered catalogs has also been proposed to eliminate the effect of clustering but we demonstrate that this does not increase the statistical significance of AMR. Given the ease with which data fitting can find desired patterns in seismicity, future studies of AMR-like observations must include complete tests against synthetic catalogs that include spatiotemporal clustering.