Smooth Discrimination Analysis

Discriminant analysis for two data sets in IRd with probability densities f and g can be based on the estimation of the set G = {x : f(x) I g(x)}. 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.

[1]  Shun-ichi Amari,et al.  A Theory of Pattern Recognition , 1968 .

[2]  R. Dudley.  Metric Entropy of Some Classes of Sets with Differentiable Boundaries , 1974 .

[3]  J. Bretagnolle,et al.  Estimation des densités: risque minimax , 1978 .

[4]  J. Marron.  Optimal Rates of Convergence to Bayes Risk in Nonparametric Discrimination , 1983 .

[5]  P. Assouad.  Deux remarques sur l'estimation , 1983 .

[6]  K. Alexander,et al.  Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm , 1984 .

[7]  J. Hartigan.  Estimation of a Convex Density Contour in Two Dimensions , 1987 .

[8]  Andrew R. Barron,et al.  Complexity Regularization with Application to Artificial Neural Networks , 1991 .

[9]  A. Barron.  Approximation and Estimation Bounds for Artificial Neural Networks , 1991, COLT '91.

[10]  G. Sawitzki,et al.  Excess Mass Estimates and Tests for Multimodality , 1991 .

[11]  E. Mammen.  Nonparametric regression under qualitative smoothness assumptions , 1991 .

[12]  P. Massart,et al.  Rates of convergence for minimum contrast estimators , 1993 .

[13]  A. Tsybakov,et al.  Minimax theory of image reconstruction , 1993 .

[14]  M. Rudemo,et al.  Approximating the distribution of maximum likelihood contour estimators in two-region images , 1994 .

[15]  W. Polonik.  Measuring Mass Concentrations and Estimating Density Contour Clusters-An Excess Mass Approach , 1995 .

[16]  W. Wong,et al.  Probability inequalities for likelihood ratios and convergence rates of sieve MLEs , 1995 .

[17]  Vladimir Vapnik,et al.  The Nature of Statistical Learning , 1995 .

[18]  E. Mammen,et al.  Asymptotical minimax recovery of sets with smooth boundaries , 1995 .

[19]  László Györfi,et al.  A Probabilistic Theory of Pattern Recognition , 1996, Stochastic Modelling and Applied Probability.

[20]  A. Tsybakov.  On nonparametric estimation of density level sets , 1997 .

[21]  P. Massart,et al.  Minimum contrast estimators on sieves: exponential bounds and rates of convergence , 1998 .

[22]  P. Massart,et al.  Risk bounds for model selection via penalization , 1999 .

[23]  S. Geer.  Applications of empirical process theory , 2000 .

[24]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.