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When testing hypotheses about which of two competing models is better, say A and B, the difference is often not significant. An alternative, complementary approach, is to measure how often model A is better than model B regardless of how slight or large the difference. The hypothesis concerns whether or not the percentage of time that model A is better than model B is larger than 50%. One generalized test statistic that can be used is the power-divergence test, which encompasses many familiar goodness-of-fit test statistics, such as the loglikelihood-ratio and PearsonX²tests. Theoretical results justify using thedistribution for the entire family of test statistics, wherekis the number of categories. However, these results assume that the underlying data are independent and identically distributed, which is often violated. Empirical results demonstrate that the reduction to two categories (i.e., model A is better than model B versus model B is better than A) results in a test that is reasonably robust to even severe departures from temporal independence, as well as contemporaneous correlation. The test is demonstrated on two different example verification sets: 6-h forecasts of eddy dissipation rate (m^2/3s⁻¹) from two versions of the Graphical Turbulence Guidance model and for 12-h forecasts of 2-m temperature (°C) and 10-m wind speed (m s⁻¹) from two versions of the High-Resolution Rapid Refresh model. The novelty of this paper is in demonstrating the utility of the power-divergence statistic in the face of temporally dependent data, as well as the emphasis on testing for the “frequency-of-better” alongside more traditional measures.

Studies of the ages, abundances, and motions of individual stars in the Milky Way provide one of the best ways to study the evolution of disc galaxies over cosmic time. The formation of the Milky Way’s barred inner region in particular is a crucial piece of the puzzle of disc galaxy evolution. Using data from APOGEE and Gaia, we present maps of the kinematics, elemental abundances, and age of the Milky Way bulge and disc that show the barred structure of the inner Milky Way in unprecedented detail. The kinematic maps allow a direct, purely kinematic determination of the bar’s pattern speed of $41\pm 3\, \mathrm{km\, s}^{-1}\, \mathrm{kpc}^{-1}$ and of its shape and radial profile. We find the bar’s age, metallicity, and abundance ratios to be the same as those of the oldest stars in the disc that are formed in its turbulent beginnings, while stars in the bulge outside of the bar are younger and more metal-rich. This implies that the bar likely formed ${\approx}8\, \mathrm{Gyr}$ ago, when the decrease in turbulence in the gas disc allowed a thin disc to form that quickly became bar-unstable. The bar’s formation therefore stands as a crucial epoch in the evolution of the Milky Way, a picture that is in line with the evolutionary path that emerges from observations of the gas kinematics in external disc galaxies over the last ${\approx}10\, \mathrm{Gyr}$.