In the use of smoothing methods in data analysis, an important question is often: which observed features are \really there?", as opposed to being spurious sampling artifacts. An approach is described, based on scale space ideas that were originally developed in computer vision literature. Assessment of SIgnicant ZERo crossings of derivatives, results in the SiZer map, a graphical device for display of signicance of features, with respect to both location and scale. Here \scale" means \level of resolution", i.e. \bandwidth".

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association.An extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered. The approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles, both of which have been extensively studied in the literature. These geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other L estimates of multivariate location, and they lead to a directional notion of central and extreme points in a multidimensional setup. Such quantiles can be defined as meaningful and natural objects even in infinite-dimensional Hilbert and Banach spaces, and they yield an effective generalization of quantile regression in multiresponse linear model problems. Desirable equivariance properties are shown to hold for these multivariate quantiles, and issues related to their computation for data in finite-dimensional spaces are discussed. nl/2 consistency and asymptotic normality of sample geometric quantiles estimating the corresponding population quantiles are established after deriving a Bahadur-type linear expansion. The sampling variation of geometric quantiles is carefully investigated, and estimates for dispersion matrices, which may be used in developing confidence ellipsoids, are constructed. In course of this development of sampling distributions and related statistical properties, we observe several interesting facts, some of which are quite counterintuitive. In particular, many of the intriguing properties of spatial medians documented in the literature appear to be inherited by geometric quantiles. KEY WORDS: Bahadur representation; Geometric quantiles; L estimation in multidimension; Multiresponse quantile regression; nl/2-consistent estimate; Spatial median; Trimmed multivariate mean.Probal Chaudhuri is Member for many stimulating comments and two anonymous referees, an anonymous associate editor, and Myles Hollander for several helpful suggestions.

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