Bounds on Expectations of Linear Systematic Statistics Based on Dependent Samples

Abstract

David summarized distribution-free bounds for $E(X_{k:n})$, the expected value of the $k$th 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 $\mu$ and variance $\sigma^2$. Here the problem of finding distribution-free bounds for the expectations of linear systematic statistics is considered in the case in which the observations $X_i, i = 1,2, \cdots, n$, satisfy only $E(X_i) = \mu$ and $\operatorname{Var}(X_i) = \sigma^2$. The observations may be dependent and have different distributions. Bounds are obtained for the expectations of the $k$th order statistic, the trimmed mean, the range, and quasi-ranges, the spacings and Downton's estimator of $\sigma$. The sharpness of these bounds is considered. In contrast with the i.i.d. case all the bounds obtained are shown to be sharp.