Nonparametrically consistent depth-based classifiers

We introduce a class of depth-based classification procedures that are of a nearest-neighbor nature. Depth, after symmetrization, indeed provides the center-outward ordering that is necessary and sufficient to define nearest neighbors. Like all their depth-based competitors, the resulting classifiers are affine-invariant, hence in particular are insensitive to unit changes. Unlike the former, however, the latter achieve Bayes consistency under virtually any absolutely continuous distributions - a concept we call nonparametric consistency, to stress the difference with the stronger universal consistency of the standard $k$NN classifiers. We investigate the finite-sample performances of the proposed classifiers through simulations and show that they outperform affine-invariant nearest-neighbor classifiers obtained through an obvious standardization construction. We illustrate the practical value of our classifiers on two real data examples. Finally, we shortly discuss the possible uses of our depth-based neighbors in other inference problems.Comment: Published at http://dx.doi.org/10.3150/13-BEJ561 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Binary populations and stellar dynamics in young clusters

We first summarize work that has been done on the effects of binaries on theoretical population synthesis of stars and stellar phenomena. Next, we highlight the influence of stellar dynamics in young clusters by discussing a few candidate UFOs (unconventionally formed objects) like intermediate mass black holes, Eta Carinae, Zeta Puppis, Gamma Velorum and WR 140.Comment: Contributed paper IAU 250: Massive Stars as Cosmic Engine

Tyler shape depth

In many problems from multivariate analysis, the parameter of interest is a shape matrix, that is, a normalized version of the corresponding scatter or dispersion matrix. In this paper, we propose a depth concept for shape matrices that involves data points only through their directions from the center of the distribution. We use the terminology Tyler shape depth since the resulting estimator of shape, namely the deepest shape matrix, is the median-based counterpart of the M-estimator of shape of Tyler (1987). Beyond estimation, shape depth, like its Tyler antecedent, also allows hypothesis testing on shape. Its main benefit, however, lies in the ranking of shape matrices it provides, whose practical relevance is illustrated in principal component analysis and in shape-based outlier detection. We study the invariance, quasi-concavity and continuity properties of Tyler shape depth, the topological and boundedness properties of the corresponding depth regions, existence of a deepest shape matrix and prove Fisher consistency in the elliptical case. Finally, we derive a Glivenko-Cantelli-type result and establish almost sure consistency of the deepest shape matrix estimator.Comment: 28 pages, 5 figure

The formation and evolution of very massive stars in dense stellar systems

The early evolution of dense stellar systems is governed by massive single star and binary evolution. Core collapse of dense massive star clusters can lead to the formation of very massive objects through stellar collisions ($M\geq$ 1000 \msun). Stellar wind mass loss determines the evolution and final fate of these objects, and decides upon whether they form black holes (with stellar or intermediate mass) or explode as pair instability supernovae, leaving no remnant. We present a computationaly inexpensive evolutionary scheme for very massive stars that can readily be implemented in an N-body code. Using our new N-body code 'Youngbody' which includes a detailed treatment of massive stars as well as this new scheme for very massive stars, we discuss the formation of intermediate mass and stellar mass black holes in young starburst regions. A more detailed account of these results can be found in Belkus et al. 2007.Comment: 2 pages, 2 figures. To appear in conference proceedings for IAUS246, 200