Selection procedures for sparse data

The problem of selecting the best of k populations is studied for data which are incomplete as some of the values have been deleted randomly. This situation is met in extreme value analysis where only data exceeding a threshold are observable. For increasing sample size we study the case where the probability that a value is observed tends to zero, but the sparse condition is satisfied, so that the mean number of observable values in each population is bounded away from zero and infinity as the sample size tends to infinity. The incomplete data are described by thinned point processes which are approximated by Poisson point processes. Under weak assumptions and after suitable transformations these processes converge to a Poisson point process. Optimal selection rules for the limit model are used to construct asymptotically optimal selection rules for the original sequence of models. The results are applied to extreme value data for high thresholds data.