Best Invariant and Minimax Estimation of Quantiles in Finite Populations

We study estimation of finite population quantiles, with emphasis on estimators that are invariant under monotone transformations of the data, and suitable invariant loss functions. We discuss non-randomized and randomized estimators, best invariant and minimax estimators and sampling strategies relative to different classes. The combination of natural invariance of the kind discussed here, and finite population sampling appears to be novel, and leads to interesting statistical and combinatorial aspects.