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Smoothed bootstrap confidence intervals with discrete data. (English) Zbl 0908.62052

Summary: The smoothed bootstrap paradigm involves replacing the empirical distribution function \(F_{n}\) with a smoothed version. Thus far, this idea has been mainly considered for continuous data arising from a distribution F with density f. In terms of mean squared error, the results indicate no global preference for smoothing in the continuous case. We depart from these investigations in that we consider (1) discrete data and (2) assess the merits of smoothed bootstraps through confidence intervals. We present a class of nonparametric smoothed bootstraps for ordered categorical data, and report on simulation studies comparing various proposed resampling methods to standard methods, including unsmoothed bootstraps. To serve as an illustration using real data, one Monte Carlo study involves samples from two large populations obtained from toxicological research. Some versions of the smoothed bootstrap yield a worthwhile small-sample improvement in coverage for this application to a ratio estimator with integer-valued data. The potential gains from the smoothing methodology are evident, as are the dangers from oversmoothing.

MSC:

62G09 Nonparametric statistical resampling methods
62G15 Nonparametric tolerance and confidence regions
62G07 Density estimation
65C05 Monte Carlo methods

Software:

bootstrap
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Full Text: DOI

References:

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