Minimum distance estimation of Milky Way model parameters and related inference

We propose a method to estimate the location of the Sun in the disk of the Milky Way using a method based on the Hellinger distance and construct confidence sets on our estimate of the unknown location using a bootstrap-based method. Assuming the Galactic disk to be two-dimensional, the sought solar location then reduces to the radial distance separating the Sun from the Galactic center and the angular separation of the Galactic center to Sun line, from a pre-fixed line on the disk. On astronomical scales, the unknown solar location is equivalent to the location of us earthlings who observe the velocities of a sample of stars in the neighborhood of the Sun. This unknown location is estimated by undertaking pairwise comparisons of the estimated density of the observed set of velocities of the sampled stars, with the density estimated using synthetic stellar velocity data sets generated at chosen locations in the Milky Way disk. The synthetic data sets are generated at a number of locations that we choose from within a constructed grid, at four different base astrophysical models of the Galaxy. Thus, we work with one observed stellar velocity data and four distinct sets of simulated data comprising a number of synthetic velocity data vectors, each generated at a chosen location. For a given base astrophysical model that gives rise to one such simulated data set, the chosen location within our constructed grid at which the estimated density of the generated synthetic data best matches the density of the observed data is used as an estimate for the location at which the observed data was realized. In other words, the chosen location corresponding to the highest match offers an estimate of the solar coordinates in the Milky Way disk. The “match” between the pair of estimated densities is parameterized by the affinity measure based on the familiar Hellinger distance. We perform a novel cross-validation procedure to establish a desirable “consistency” property of the proposed method.