Preliminary Regional Forecasts for the Outcome of an Estimation Problem

Sets bounded by random variables that are functions of the first k, k < n, members of a random sample of n members from a normal distribution are studied. These sets have the property that, before the random sample is drawn, the probability is 1 − α, 0 < a < 1, that they contain a point whose coordinates are specified functions of all of the n members of the random sample. Usually, these specified functions will be estimates of the distribution parameters. After numerical values are obtained for the k members of the subsample, the set is fixed and affords a regional forecast along with an index of the reliance to be placed on the forecast for the eventual parameter estimate which will be based on all n members of the random sample. Such sets and indices are called “forecast sets” and “forecast coefficients,” respectively.