Electronic Journal of Statistics

Empirical measures for incomplete data with applications

Shojaeddin Chenouri, Majid Mojirsheibani, and Zahra Montazeri
Source: Electron. J. Statist. Volume 3 (2009), 1021-1038.

Abstract

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.

First Page: Show Hide
Primary Subjects: 60G50, 62G15
Secondary Subjects: 62H30
Full-text: Open access
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.ejs/1255440399
Digital Object Identifier: doi:10.1214/09-EJS420
Mathematical Reviews number (MathSciNet): MR2557127

References

Bennett, G. (1962). Probability inequalities for the sum of independent random variables., J. Amer. Statist Assoc., 57:33–45.
Cheng, P. E. and Chu, C. K. (1996). Kernel estimation of distribution functions and quantiles with missing data., Statist. Sinica, 6:63–78.
Mathematical Reviews (MathSciNet): MR1379049
Zentralblatt MATH: 0839.62038
Devroye, L. (1982). Bounds on the uniform deviation of empirical meaures., Journal of Multivariate Analysis, 12:72–79.
Mathematical Reviews (MathSciNet): MR650929
Zentralblatt MATH: 0492.60006
Digital Object Identifier: doi:10.1016/0047-259X(82)90083-5
Devroye, L., Györfi, L., and Lugosi, G. (1996)., A Probabilistic Theory of Pattern Recognition. Springer-Verlag, New York.
Mathematical Reviews (MathSciNet): MR1383093
Zentralblatt MATH: 0853.68150
Dudley, R. (1978). Central limit theorems for empirical measures., Ann. Probab., 6:899–929.
Mathematical Reviews (MathSciNet): MR512411
Digital Object Identifier: doi:10.1214/aop/1176995384
Project Euclid: euclid.aop/1176995384
Little, R. J. A. and Rubin, D. B. (2002)., Statistical Analysis With Missing Data. Wiley, New York.
Mathematical Reviews (MathSciNet): MR1925014
Massart, P. (1990). The tight constant in the Devoretzky-Kiefer-Wolfowitz inequality., Ann. Probab., 18:1269–1283.
Mathematical Reviews (MathSciNet): MR1062069
Zentralblatt MATH: 0713.62021
Digital Object Identifier: doi:10.1214/aop/1176990746
Project Euclid: euclid.aop/1176990746
Pollard, D. (1984)., Convergence of Stochastic Processes. Springer-Verlag, New York.
Mathematical Reviews (MathSciNet): MR762984
Zentralblatt MATH: 0544.60045
Talagrand, M. (1994). Sharper bounds for gaussian and empirical processes., Ann. Probab., 22:28–76.
Mathematical Reviews (MathSciNet): MR1258865
Zentralblatt MATH: 0798.60051
Digital Object Identifier: doi:10.1214/aop/1176988847
Project Euclid: euclid.aop/1176988847
van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes, with Applications to Statistics. Springer-Verlag, New York.
Mathematical Reviews (MathSciNet): MR1385671
Zentralblatt MATH: 0862.60002
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.

2013 © Institute of Mathematical Statistics

Electronic Journal of Statistics

Electronic Journal of Statistics

Turn MathJax Off
What is MathJax?