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Kernel Smoothing to Improve Bootstrap Confidence Intervals
Author(s) -
Polansky Alan M.,
Schucany William. R.
Publication year - 1997
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/1467-9868.00099
Subject(s) - smoothing , mathematics , resampling , confidence interval , kernel smoother , kernel density estimation , percentile , statistics , kernel (algebra) , variable kernel density estimation , empirical distribution function , cdf based nonparametric confidence interval , bootstrapping (finance) , mean squared error , kernel method , econometrics , computer science , artificial intelligence , combinatorics , estimator , radial basis function kernel , support vector machine
Some studies of the bootstrap have assessed the effect of smoothing the estimated distribution that is resampled, a process usually known as the smoothed bootstrap. Generally, the smoothed distribution for resampling is a kernel estimate and is often rescaled to retain certain characteristics of the empirical distribution. Typically the effect of such smoothing has been measured in terms of the mean‐squared error of bootstrap point estimates. The reports of these previous investigations have not been encouraging about the efficacy of smoothing. In this paper the effect of resampling a kernel‐smoothed distribution is evaluated through expansions for the coverage of bootstrap percentile confidence intervals. It is shown that, under the smooth function model, proper bandwidth selection can accomplish a first‐order correction for the one‐sided percentile method. With the objective of reducing the coverage error the appropriate bandwidth for one‐sided intervals converges at a rate of n −1/4 , rather than the familiar n −1/5 for kernel density estimation. Applications of this same approach to bootstrap t and two‐sided intervals yield optimal bandwidths of order n −1/2 . These bandwidths depend on moments of the smooth function model and not on derivatives of the underlying density of the data. The relationship of this smoothing method to both the accelerated bias correction and the bootstrap t methods provides some insight into the connections between three quite distinct approximate confidence intervals.