Iterated Bootstrap‐ t Confidence Intervals for Density Functions

.  Conventional bootstrap‐ t intervals for density functions based on kernel density estimators exhibit poor coverages due to failure of the bootstrap to estimate the bias correctly. The problem can be resolved by either estimating the bias explicitly or undersmoothing the kernel density estimate to undermine its bias asymptotically. The resulting bias‐corrected intervals have an optimal coverage error of order arbitrarily close to second order for a sufficiently smooth density function. We investigated the effects on coverage error of both bias‐corrected intervals when the nominal coverage level is calibrated by the iterated bootstrap. In either case, an asymptotic reduction of coverage error is possible provided that the bias terms are handled using an extra round of smoothed bootstrapping. Under appropriate smoothness conditions, the optimal coverage error of the iterated bootstrap‐ t intervals has order arbitrarily close to third order. Examples of both simulated and real data are reported to illustrate the iterated bootstrap procedures.