The Glivenko-Cantelli theorem based on data with randomly imputed missing values
Glivenko-Cantelli-type results will be derived for the performance of the empirical distribution function under imputation of the missing data in the univariate case, when there are no auxiliary covariates. According to these results, both random imputation and the adjusted random imputation produce reliable estimates of the unknown distribution function. They are reliable in the sense that the corresponding empirical distribution functions stay (w.p.1) "close" to F(t), uniformly over t. At the same time, it is also shown that if r(n)[less-than-or-equals, slant]n is the number of nonmissing observations, then one cannot improve on the performance of the empirical distribution function Fr(n) by generating artificial data, in order to increase the sample size from r(n) to n (using either one of the above two imputation procedures).
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Volume (Year): 55 (2001)
Issue (Month): 4 (December)
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- Arenal-Gutiérrez, Eusebio & Matrán, Carlos & Cuesta-Albertos, Juan A., 1996. "Unconditional Glivenko-Cantelli-type theorems and weak laws of large numbers for bootstrap," Statistics & Probability Letters, Elsevier, vol. 26(4), pages 365-375, March.
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