The multi-variate sampling problem

Abstract Sampling theory deals primarily with the problem of estimating for a given finite population the mean per element of some particular variate Y. It is thus characteristic of the theory that it is uni-variate. Most sample surveys, however, are multi-variate. In some surveys, the number of variates is even very large; e.g. in the sample surveys carried out each year in Sweden in order to estimate acreages under different crops, numbers of different kinds of cattle, amount of milk produced etc., the number of variates is more than 40. Every survey statistician has encountered situations where the (uni-variate) sampling theory alone does not provide a sufficient guide to the problem of constructing a good sample design for a proposed (multi-variate) survey. In this paper a short general discussion will first be given of the multi-variate sampling problem; thereafter, a special case (the problem of optimum allocation III stratified sampling) will be dealt with in more detail.