On exactness and unbiasedness of the simple confidence bands for a continuous distribution function

摘要

We define a class of confidence bands for distribution functions, named simple confidence bands. The class of bands includes the common step bands and continuous bands, some of which may perform better than the smoothed bands not belonging to the class, e.g. the kernel smoothed bands. It is shown that under some trivial assumptions, the simple bands with exact coverage for continuous distribution functions are all step bands. The unbiasedness problem of the step bands is also investigated. It is proved that most of two-sided step bands are biased and one-sided step bands are unbiased.