We present stellar velocity dispersion profiles for seven Milky Way dwarf spheroidal (dSph) satellite galaxies. We have measured 8394 line-of-sight velocities (+/- 2.5 km/s) for 6804 stars from high-resolution spectra obtained at the Magellan and MMT telescopes. We combine these new data with previously published velocities to obtain the largest available kinematic samples, which include more than 5500 dSph members. All the measured dSphs have stellar velocity dispersion of order 10 km/s that remains approximately constant with distance from the dSph center, out to and in some cases beyond the radius at which the mean surface brightness falls to the background level. Assuming dSphs reside within dark matter halos characterized by the NFW density profile, we obtain reasonable fits to the empirical velocity dispersion profiles. These fits imply that, among the seven dSphs, M_vir ~ 10^[8-9] M_sun. The mass enclosed at a radius of 600 pc, the region common to all data sets, ranges from (2-7) x 10^7 M_sun .Comment: Accepted for publication in the Astrophysical Journal Letter

We develop an algorithm for estimating parameters of a distribution sampled with contamination. We employ a statistical technique known as "expectation maximization" (EM). Given models for both member and contaminant populations, the EM algorithm iteratively evaluates the membership probability of each discrete data point, then uses those probabilities to update parameter estimates for member and contaminant distributions. The EM approach has wide applicability to the analysis of astronomical data. Here we tailor an EM algorithm to operate on spectroscopic samples obtained with the Michigan-MIKE Fiber System (MMFS) as part of our Magellan survey of stellar radial velocities in nearby dwarf spheroidal (dSph) galaxies. These samples, to be presented in a companion paper, contain discrete measurements of line-of-sight velocity, projected position, and pseudo-equivalent width of the Mg-triplet feature, for ∼ 1000 − 2500 stars per dSph, including some fraction of contamination by foreground Milky Way stars. The EM algorithm uses all of the available data to quantify dSph and contaminant distributions. For distributions (e.g., velocity and Mg-index of dSph stars) assumed to be Gaussian, the EM algorithm returns maximum-likelihood estimates of the mean and variance, as well as the probability that each star is a dSph member. These probabilities can serve as weights in subsequent analyses. Applied to our MMFS data, the EM algorithm identifies more than 5000 stars as probable dSph members. We test the performance of the EM algorithm on simulated data sets that represent a range of sample size, level of contamination, and amount of overlap between dSph and contaminant velocity distributions. The simulations establish that for samples ranging from large (N ∼ 3000, characteristic of the MMFS samples) to small (N ∼ 30, resembling new samples for extremely faint dSphs), the EM algorithm distinguishes members from contaminants and returns accurate parameter estimates much more reliably than conventional methods of contaminant removal (e.g., sigma clipping).

This paper deals with the consistency of the nonparametric least squares estimator of a convex regression function when the predictor is multidimensional. We characterize and discuss the computation of such an estimator via the solution of certain quadratic and linear programs. Mild sufficient conditions for the consistency of this estimator and its subdifferentials in fixed and stochastic design regression settings are provided.

We consider the problem of estimating an unknown θ ∈ R n from noisy observations under the constraint that θ belongs to certain convex polyhedral cones in R n . Under this setting, we prove bounds for the risk of the least squares estimator (LSE). The obtained risk bound behaves differently depending on the true sequence θ which highlights the adaptive behavior of θ. As special cases of our general result, we derive risk bounds for the LSE in univariate isotonic and convex regression. We study the risk bound in isotonic regression in greater detail: we show that the isotonic LSE converges at a whole range of rates from log n/n (when θ is constant) to n −2/3 (when θ is uniformly increasing in a certain sense). We argue that the bound presents a benchmark for the risk of any estimator in isotonic regression by proving nonasymptotic local minimax lower bounds. We prove an analogue of our bound for model misspecification where the true θ is not necessarily nondecreasing.

Extensive photometric studies of the globular clusters located toward the center of the Milky Way have been historically neglected. The presence of patchy differential reddening in front of these clusters has proven to be a significant obstacle to their detailed study. We present here a well defined and reasonably homogeneous photometric database for 25 of the brightest Galactic globular clusters located in the direction of the inner Galaxy. These data were obtained in the B, V, and I bands using the Magellan 6.5 m Telescope and the Hubble Space Telescope. A new technique is extensively used in this paper to map the differential reddening in the individual cluster fields, and to produce cleaner, dereddened color-magnitude diagrams for all the clusters in the database. Subsequent papers will detail the astrophysical analysis of the cluster populations, and the properties of the obscuring material along the clusters' lines of sight.

We study the nonparametric least squares estimator (LSE) of a multivariate convex regression function. The LSE, given as the solution to a quadratic program with O(n 2 ) linear constraints (n being the sample size), is difficult to compute for large problems. Exploiting problem specific structure, we propose a scalable algorithmic framework based on the augmented Lagrangian method to compute the LSE. We develop a novel approach to obtain smooth convex approximations to the fitted (piecewise affine) convex LSE and provide formal bounds on the quality of approximation. When the number of samples is not too large compared to the dimension of the predictor, we propose a regularization scheme -Lipschitz convex regression -where we constrain the norm of the subgradients, and study the rates of convergence of the obtained LSE. Our algorithmic framework is simple and flexible and can be easily adapted to * 1 arXiv:1509.08165v1 [stat.CO] 28 Sep 2015 handle variants: estimation of a non-decreasing/non-increasing convex/concave (with or without a Lipschitz bound) function. We perform numerical studies illustrating the scalability of the proposed algorithm.

We consider the problem of nonparametric estimation of a convex regression function φ 0 . We study the risk of the least squares estimator (LSE) under the natural squared error loss. We show that the risk is always bounded from above by n −4/5 modulo logarithmic factors while being much smaller when φ 0 is well-approximable by a piecewise affine convex function with not too many affine pieces (in which case, the risk is at most 1/n up to logarithmic factors). On the other hand, when φ 0 has curvature, we show that no estimator can have risk smaller than a constant multiple of n −4/5 in a very strong sense by proving a "local" minimax lower bound. We also study the case of model misspecification where we show that the LSE exhibits the same global behavior provided the loss is measured from the closest convex projection of the true regression function. In the process of deriving our risk bounds, we prove new results for the metric entropy of local neighborhoods of the space of univariate convex functions. These results, which may be of independent interest, demonstrate the non-uniform nature of the space of univariate convex functions in sharp contrast to classical function spaces based on smoothness constraints.

We present multifiber echelle radial velocity results for 551 stars in the Sextans dwarf spheroidal galaxy and identify 294 stars as probable Sextans members. The projected velocity dispersion profile of the binned data remains flat to a maximum angular radius of $30^{\prime}$. We introduce a nonparametric technique for estimating the projected velocity dispersion surface, and use this to search for kinematic substructure. Our data do not confirm previous reports of a kinematically distinct stellar population at the Sextans center. Instead we detect a region near the Sextans core radius that is kinematically colder than the overall Sextans sample with 95% confidence.Comment: accepted for publication in ApJ Letters; 4 figures (2 color