Optimal Linear Estimation of Bounds of Random Variables

The problem of estimating the bounds of random variables has been previously discussed. Here we discuss optimality of estimates when the data is censored so that only the r largest or smallest of the observations is available for estimating a bound. For fixed r we find the linear function of the censored data which is the optimal estimator of a bound in the sense that, when the sample size is large, the estimator has smallest mean squared error among all such linear estimators. Provided r is not close to one, these estimators are almost optimal when the entire sample is available since, for example, when estimating an upper bound and the sample size is large, the largest few observations carry most of the information about the bound. This fact is illustrated in one case.