This article presents novel sequential methods of sample coordination appropriate for a repeated survey, with a stratified design and simple random sampling without replacement (SRSWOR) selection within each stratum, when the composition or definition of strata changes. Such changes could be the result of updating the frame for births, deaths, or the modification of the industry classification system. Given that a sample has already been selected according to a first (before the frame updates) SRSWOR design, our general aim is to select a minimum number of new units for the second (after the updates) survey while preserving the first-order inclusion probabilities of units in the second SRSWOR design. Sequential methods presently in use can attain a large expected overlap, but do not control the overlap on each pair of selected samples. In this article we present a set of new methods for maximizing the expected overlap, which can handle realistic situations when strata and the associated sample sizes are large. These methods include one that not only maximizes the expected overlap but, for any initially selected sample, maximizes its overlap with the second sample; its superior performance is illustrated with numerical examples.

In survey sampling there is often a need to coordinate the selection of pairs of samples drawn from two overlapping populations so as to maximize or minimize their expected overlap, subject to constraints on the marginal probabilities determined by the respective designs. For instance, maximizing the expected overlap between repeated samples can stabilize the resulting estimates of change and reduce the costs of first contacts; minimizing the expected overlap can avoid overburdening respondents with multiple surveys. We focus on the important special case in which both samples are selected by simple random sampling without replacement (SRSWOR) conducted independently within each stratum. Optimizing the expected sample overlap can be formulated as a linear programming problem known as a transportation problem (TP). We show that by appropriately grouping and ordering the possible samples in each survey, one can reduce the initial TP to a much smaller TP amenable to solution by an algorithm known as the Northwest Corner Rule (NWCR). The proposed NWCR method proceeds in two easily implemented steps: first selecting the numbers of births (new units) and deaths (deleted units) by a random selection from a hypergeometric distribution, and then selecting the births and deaths by SRSWOR. We formally prove properties of the NWCR solutions, including a minimal variance property of the minimal overlap solution. In a simulation study, the NWCR method compares favorably with a popular method based on assignment of permanent random numbers to each sampling unit.

Convergence properties of weighted sums of functions in D([0, 1]; E) (E a Banach space) are investigated. We show that convergence in the Skorokhod J1-topology of a sequence (xn) in D([0, 1]; E) does not imply convergence of a sequence (xn) of averages. Convergence in the J1-topology of a sequence (xn) of averages is shown, under the growth condition ∥ xn ∥ ∞ = o(n), to be equivalent to the convergence of (xn) in the uniform topology. Convergence of a sequence (xn,) is shown to imply convergence of the sequence (xn) of averages in the M1 and M2 topologies. The strong law of large numbers in D[0, 1] is considered and an example is constructed to show that different definitions of the strong law of large numbers are nonequivalent.

ABSTRACT Weview,the problem,of maximizing,or minimizing the expected overlap of two,surveys as a transportation problem,(TP) and give simple selection algorithms for solving it. We compare our method,with the sequential SRSWOR for positive and negative coordination, the method of collocated samples and the PRN method with full-stratum rotation. KEY WORDS: SRSWOR, transportation problem, NWCR, expected overlap, hypergeometricdistribution, p.p.s. selection. RÉSUMÉ Onregarde,le problème ,de la ,maximisation ou ,de la ,minimisation ,du chevauchement, espéré entre deux enquêtes comme ,un problème de transport. On donne des algorithmes de sélection simples pour solutionner ce problème. On compare notre méthode avec le sondage aléatoire simple séquentiel pour les coordinations positive et négative, la méthode des échantillons colloqués et la méthode des nombres,aléatoires permanents,avec rotation complète dans les strates. MOTS CLÉS : Sondage aléatoire simple, problème de transport, règle du coin nord-ouest, chevauchement espéré, loi hypergéométrique, sélection ppt. ,,,,,,,,,,,,,