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Estimating the Support of a HighDimensional Distribution
, 1999
"... Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified between 0 and 1. We propo ..."
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Cited by 766 (29 self)
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Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified between 0 and 1. We propose a method to approach this problem by trying to estimate a function f which is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. The expansion coefficients are found by solving a quadratic programming problem, which we do by carrying out sequential optimization over pairs of input patterns. We also provide a preliminary theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabelled d...
Smooth Discrimination Analysis
 Ann. Statist
, 1998
"... Discriminant analysis for two data sets in IR d with probability densities f and g can be based on the estimation of the set G = fx : f(x) g(x)g. We consider applications where it is appropriate to assume that the region G has a smooth boundary. In particular, this assumption makes sense if di ..."
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Cited by 154 (3 self)
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Discriminant analysis for two data sets in IR d with probability densities f and g can be based on the estimation of the set G = fx : f(x) g(x)g. We consider applications where it is appropriate to assume that the region G has a smooth boundary. In particular, this assumption makes sense if discriminant analysis is used as a data analytic tool. We discuss optimal rates for estimation of G. 1991 AMS: primary 62G05 , secondary 62G20 Keywords and phrases: discrimination analysis, minimax rates, Bayes risk Short title: Smooth discrimination analysis This research was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 373 "Quantifikation und Simulation okonomischer Prozesse", HumboldtUniversitat zu Berlin 1 Introduction Assume that one observes two independent samples X = (X 1 ; : : : ; X n ) and Y = (Y 1 ; : : : ; Ym ) of IR d valued i.i.d. observations with densities f or g, respectively. The densities f and g are unknown. An additional random variabl...
Kernel estimation of density level sets
 J. Multivariate Anal
, 2006
"... Abstract. Let f be a multivariate density and fn be a kernel estimate of f drawn from the nsample X1, · · ·,Xn of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the tlevel set {f ≥ t} and its plug ..."
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Cited by 24 (2 self)
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Abstract. Let f be a multivariate density and fn be a kernel estimate of f drawn from the nsample X1, · · ·,Xn of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the tlevel set {f ≥ t} and its plugin estimator {fn ≥ t}. As a corollary, we obtain the exact rate of convergence of a plugin type estimate of the density level set corresponding to a fixed probability for the law induced by f.
The excess mass approach and the analysis of multimodality
 Proc. 18th Annual Conference of the GfKl
, 1996
"... Summary: The excess mass approach is a general approach to statistical analysis. It can be used to formulate a probabilistic model for clustering and can be applied to the analysis of multimodality. Intuitively, a mode is present where an excess of probability mass is concentrated. This intuitive i ..."
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Cited by 1 (0 self)
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Summary: The excess mass approach is a general approach to statistical analysis. It can be used to formulate a probabilistic model for clustering and can be applied to the analysis of multimodality. Intuitively, a mode is present where an excess of probability mass is concentrated. This intuitive idea can be formalized directly by means of the excess mass functional. There is no need for intervening steps like initial density estimation. The excess mass measures the local difference of a given distribution to a reference model, usually the uniform distribution. The excess mass defines a functional which can be estimated efficiently from the data and can be used to test for multimodality. 1. The problem of multimodality We want to find the number of modes of a distribution in R k, based on a sample of n independent observations. There are many approaches to this problem. Any approach has to face an inherent difficulty of the modalityproblem: the functional which associates the number of modes to a distribution is only semicontinuous. In any neighbourhood (with respect to the testing topology) of a given
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"... Nonparametric estimation of regression level sets using kernel plugin estimator ..."
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Nonparametric estimation of regression level sets using kernel plugin estimator