Empirical measures for incomplete data with applications. (English) Zbl 1326.62206

Summary: Methods are proposed to construct empirical measures when there are missing terms among the components of a random vector. Furthermore, Vapnik-Chevonenkis type exponential bounds are obtained on the uniform deviations of these estimators, from the true probabilities. These results can then be used to deal with classical problems such as statistical classification, via empirical risk minimization, when there are missing covariates among the data. Another application involves the uniform estimation of a distribution function.


62N99 Survival analysis and censored data
60G50 Sums of independent random variables; random walks
62H30 Classification and discrimination; cluster analysis (statistical aspects)
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[1] Bennett, G. (1962). Probability inequalities for the sum of independent random variables., J. Amer. Statist Assoc. , 57:33-45. · Zbl 0104.11905
[2] Cheng, P. E. and Chu, C. K. (1996). Kernel estimation of distribution functions and quantiles with missing data., Statist. Sinica , 6:63-78. · Zbl 0839.62038
[3] Devroye, L. (1982). Bounds on the uniform deviation of empirical meaures., Journal of Multivariate Analysis , 12:72-79. · Zbl 0492.60006
[4] Devroye, L., Györfi, L., and Lugosi, G. (1996)., A Probabilistic Theory of Pattern Recognition . Springer-Verlag, New York. · Zbl 0853.68150
[5] Dudley, R. (1978). Central limit theorems for empirical measures., Ann. Probab. , 6:899-929. · Zbl 0404.60016
[6] Little, R. J. A. and Rubin, D. B. (2002)., Statistical Analysis With Missing Data . Wiley, New York. · Zbl 1011.62004
[7] Massart, P. (1990). The tight constant in the Devoretzky-Kiefer-Wolfowitz inequality., Ann. Probab. , 18:1269-1283. · Zbl 0713.62021
[8] Pollard, D. (1984)., Convergence of Stochastic Processes . Springer-Verlag, New York. · Zbl 0544.60045
[9] Talagrand, M. (1994). Sharper bounds for gaussian and empirical processes., Ann. Probab. , 22:28-76. · Zbl 0798.60051
[10] van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes, with Applications to Statistics . Springer-Verlag, New York. · Zbl 0862.60002
[11] Vapnik, V. N. and Chervonenkis, A. Y. (1971). On the uniform convergence of relative frequencies of events to their probabilities., Theory Probab. Appl. , 16:264-280. · Zbl 0247.60005
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