Orthogonal Decomposition of Finite Population Statistics and Its Applications to Distributional Asymptotics

We study orthogonal decomposition of symmetric statistics based on samples drawn without replacement from finite populations. Under very mild smoothness conditions the first k terms of the decomposition provide stochastic expansion with remainder O(N−k/2). Assuming that the linear part of the decomposition is nondegenerate we establish one term Edgeworth expansion of the distribution function of a general symmetric statistic. Several applications are discussed: second order asymptotics of the jackknife histogram, consistency of the jackknife estimator of variance, Efron–Stein inequality.