Bounds on Expectations of Linear Systematic Statistics Based on Dependent Samples

David summarized distribution-free bounds for E(Xk:n), the expected value of the kth order statistic, and for the expected value of certain linear combinations of the order statistics, when sampling n i.i.d. observations from a population with expectation μ and variance σ². Here the problem of finding distribution-free bounds for the expectations of linear systematic statistics is considered in the case in which the observations Xᵢ, i = 1,2, ⋯, n, satisfy only E(Xᵢ) = μ and Var(Xᵢ) = σ². The observations may be dependent and have different distributions. Bounds are obtained for the expectations of the kth order statistic, the trimmed mean, the range, and quasi-ranges, the spacings and Downton's estimator of σ. The sharpness of these bounds is considered. In contrast with the i.i.d. case all the bounds obtained are shown to be sharp.