Abstract
Let X be a random variable taking values in a function space \(\mathcal{F}\), and let Y be a discrete random label with values 0 and 1. We investigate asymptotic properties of the moving window classification rule based on independent copies of the pair (X,Y). Contrary to the finite dimensional case, it is shown that the moving window classifier is not universally consistent in the sense that its probability of error may not converge to the Bayes risk for some distributions of (X,Y). Sufficient conditions both on the space \(\mathcal{F}\) and the distribution of X are then given to ensure consistency.
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References
Abraham C., Cornillon P.A., Matzner-Løber E., Molinari N. (2003). Unsupervised curve clustering using B-splines. Scandinavian Journal of Statistics 30:581–595
Akaike H. (1954). An approximation to the density function. Annals of the Institute of Statistical Mathematics 6:127–132
Antoniadis A., Sapatinas T. (2003). Wavelet methods for continuous-time prediction using representations of autoregressive processes in Hilbert spaces. Journal of Multivariate Analysis 87:133–158
Biau G., Bunea F., Wegkamp M.H. (2005). Functional classification in Hilbert spaces. IEEE Transactions on Information Theory 51:2163–2172
Bosq D. (2000). Linear processes in function spaces—theory and applications. Lecture Notes in Mathematics. Springer–Verlag, New York
Boucheron S., Bousquet O., Lugosi G. (2005). Theory of classification: A survey of some recent advances. ESAIM: Probability and Statistics 9:323–375
Ciesielski Z. (1961). Hölder conditions for the realizations of Gaussian processes. Transactions of the American Mathematical Society 99:403–413
Cover T.M., Hart P.E. (1965). Nearest neighbor pattern classification. The Annals of Mathematical Statistics 36:1049–1051
Devroye L., Krzyżak A. (1989). An equivalence theorem for L 1 convergence of the kernel regression estimate. Journal of Statistical Planning and Inference 23:71–82
Devroye L., Györfi L., Lugosi G. (1996). A probabilistic theory of pattern recognition. Springer–Verlag, New York
Diabo-Niang, S., Rhomari, N. (2001). Nonparametric regression estimation when the regressor takes its values in a metric space. University Paris VI, Technical Report, http://www.lsta.upmc.fr/
Ferraty F., Vieu P. (2000). Dimension fractale et estimation de la régression dans des espaces vectoriels semi-normés. Comptes Rendus de l’Académie des Sciences de Paris 330:139–142
Ferraty F., Vieu P. (2002). The functional nonparametric model and application to spectrometric data. Computational Statistics 17:545–564
Ferraty F., Vieu P. (2003). Curves discrimination: a nonparametric functional approach. Computational Statistics and Data Analysis 44:161–173
Hall P., Poskitt D., Presnell B. (2001). A functional data-analytic approach to signal discrimination. Technometrics 43:1–9
Kneip A., Gasser T. (1992). Statistical tools to analyse data representing a sample of curves. The Annals of Statistics 20:1266–1305
Kolmogorov A.N., Tihomirov V.M. (1961). ɛ-entropy and ɛ-capacity of sets in functional spaces. American Mathematical Society Translations, Series 2, 17:277–364
Kulkarni S.R., Posner S.E. (1995). Rates of convergence of nearest neighbor estimation under arbitrary sampling. IEEE Transactions on Information Theory 41:1028–1039
Mattila P. (1980). Differentiation of measures on uniform spaces. Lecture Notes in Mathematics, vol. 794, pp. 261–283. Springer–Verlag, Berlin
McDiarmid C. (1989). On the method of bounded differences. Surveys in Combinatorics 1989 (pp. 148–188). Cambridge University Press, Cambridge
Nadaraya E.A. (1964). On estimating regression. Theory of Probability and its Applications 9:141–142
Nadaraya E.A. (1970). Remarks on nonparametric estimates for density functions and regression curves. Theory of Probability and its Applications 15:134–137
Parzen E. (1962). On estimation of a probability density function and mode. The Annals of Mathematical Statistics 33: 1065–1076
Preiss D., Tiser J. (1982). Differentiation of measures on Hilbert spaces. Lecture Notes in Mathematics, vol. 945, pp. 194–207. Springer–Verlag, Berlin
Ramsay J.O., Silverman B.W. (1997). Functional data analysis. Springer–Verlag, New York
Rice J.A., Silverman B.W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society, Series B 53:233–243
Rosenblatt M. (1956). Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistics 27:832–837
Rudin W. (1987). Real and complex analysis (3rd ed). McGraw-Hill, New York
Tiser J. (1988). Differentiation theorem for Gaussian measures on Hilbert space. Transactions of the American Mathematical Society 308:655–666
Van der Vaart A.W., Wellner J.A. (1996). Weak convergence and empirical processes – with applications to statistics. Springer–Verlag, New York
Watson G.S. (1964). Smooth regression analysis. Sankhyā, Series A 26:359–372
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Abraham, C., Biau, G. & Cadre, B. On the Kernel Rule for Function Classification. Ann Inst Stat Math 58, 619–633 (2006). https://doi.org/10.1007/s10463-006-0032-1
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DOI: https://doi.org/10.1007/s10463-006-0032-1