CMA Research Report
SRR92-011
Peter Hall and Andrew T.A. Wood
On the effect of smoothing on the coverage accuracy of confidence bands for
a distribution function
Abstract:
We derive formal Edgeworth expansions for the distribution of smoothed
empirical processes, and apply these to obtain formal Edgeworth expansions
of the coverage error of smoothed confidence bands for distribution
functions. If the smooth bands are obtained by smoothing the empricial
distribution using the kernel method with window size h, then our
expansions suggest that the resulting confidence bands have coverage error
of size n−1h + h2, relative to the
precisely-known coverage of the unsmoothed bands. We also develop bands for
distribution functions based on weighted and unweighted empirical
likelihood, and show that our discussion of smoothing and coverage error
applies to appropriately weighted empricial likelihood bands as well. In
practice, the window size need not be taken much larger than order
n−1, since the jumps in the empirical distribution are of
this size. Therefore, the coverage error should typically be only
O(n−2) — a very low level of error indeed.
AMS Classification:
62F25, 62E20
Date:
1992
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