Bootstrap Confidence Intervals in Mixtures of Discrete Distributions

The problem of building bootstrap con¯dence intervals for small probabilitieswith count data is considered. The true probability distribution generating the in-dependent observations is supposed to be a mixture of a given family of power seriesdistributions. The mixing distribution is estimated by nonparametric maximum like-lihood and the corresponding mixture is used for resampling. We build percentile¡tand Efron percentile bootstrap con¯dence intervals for the probabilities and we provetheir consistency in probability. The theoretical results are supported by simulationexperiments for Poisson and Geometric mixtures. We compare percentile¡t andEfron percentile bootstrap intervals with other eight bootstrap or asymptotic theorybased intervals. It appears that Efron percentile bootstrap interval outperforms thecompetitors in terms of coverage probability and length.