Multivariate Boundaries of a Self Representing Stratum of Large Units in Agricultural Survey Design

In business surveys in general, and in multipurpose agricultural surveys in particular, the prob- lem of designing a sample from a list frame usually consists of two different aspects. The first is concerned with the choice of a rule for stratifying the population when several size variables are available and the second is devoted to sample size determination and sample allocation to a given set of strata. The main property that is required of the sample design is that it delivers a specified level of precision for a set of variables of interest using as few sampling units as possible. This article examines how this can be achieved via a basic partition into two strata, one completely enumerated and the other sampled, defined in such a way as to achieve both these objectives. The procedure was used to design the Italian Milk Products Monthly Survey on the basis of a set of auxiliary variables obtained from an annual census of the same target population. Given the combinatorial optimization nature of the problem, we use stochastic relaxation theory, and in particular, we use simulated annealing because of its flexibility. Our results indicate that in this situation the multivariate partition obtained by using this random search strategy is a suit- able solution as it permits identification of boundaries of any shape. Furthermore, numerical comparisons between sampling designs obtained by using these procedures and some simple extensions of univariate stratification rules are made. The gain from using the proposed strategy is nontrivial as it achieves the required precision using a sample size that is notably smaller than that required by simple extensions to univariate stratification rules.