1 Introduction

Pierre Lavallée and Sébastien Labelle-Blanchet

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Business surveys differ in a number of ways from social surveys. One is that data associated with a business frame are highly skewed, whereas those associated with social surveys are much more homogenous. The sampling of businesses takes place by transforming operating structures into standardized units known as statistical units. These are represented as a hierarchy, or series of levels that allow subsequent integration of the various data items available at different levels within the organization. The number of levels within the hierarchy differs between statistical agencies. For example, the Canadian Business Register has four levels: enterprise, company, establishment and location. The statistical enterprise corresponds to the legal unit in most cases. The statistical establishment is, in most cases, equivalent to a profit centre and provides data on the value of output, the cost of inputs and labour. These data are sufficient to compute value added (profit, salary and wages). In this paper, we will consider only two levels of the hierarchy: enterprise and establishment, as defined by the Canadian Business Register. For more information, one can consult Statistics Canada (2010).

For selecting the sample, stratification is often performed at the establishment level. This allows to control for geographical (e.g., by stratifying by province), industrial (e.g., by stratifying by the industrial activity), and size representativeness (e.g., by stratifying by revenue classes or by the number of employees). Controlling for such representativeness is not possible if stratification is performed at the enterprise level. However, in addition to establishment-based statistics, statistics at the enterprise level are often required. Therefore, to achieve these two goals, we perform the sample selection at the establishment level, and once the sample of establishments is obtained, we extend the sample to the complete set of establishments belonging to the enterprises of the initially selected establishments. Note that selecting enterprises through the selection of establishments is like selecting clusters through their components. This procedure allows producing estimates at the establishment and the enterprise levels, as well as some reduction in collection costs by selecting clusters of establishments.

One way to consider the production of enterprise-level estimates using a sample of establishments is by viewing the sampling frame and the target population separately. The former is a set of establishments, while the latter is a set of enterprises corresponding to clusters of establishments. When the sampling frame is not the same as the target population, but still related to the latter, we are in a situation of Indirect Sampling (see Lavallée 2002 (French version), 2007 (English version)). More formally, we wish to produce an estimate for a target population $U^B,$ using a sampling frame $U^A,$ which is somehow linked to $U^B.$ We select a sample $s^A$ from $U^A$ to produce an estimate for $U^B$ using the existing links between the two populations. To produce unbiased estimates of quantities of interest (e.g., totals or means) for the target population $U^B$ using $s^A,$ we obtain estimation weights using the Generalised Weight Share Method (GWSM).

Although the theory of the GWSM is widely developed (see Lavallée 2002, 2007), its application for business surveys presents some difficulties. Indeed, while yielding unbiased estimates, the GWSM tends to lead to large variances. This lack of precision is due to the skewness of the population, i.e., a small number of establishments cover a major portion of the economy.

The purpose of this paper is to suggest some adjustments to the estimation weights to reduce the variance of the estimates in the context of skewed populations, while keeping the method unbiased. After a brief overview of Indirect Sampling and the GWSM in Section 2, we present the problem with skewed population in Section 3. We describe the proposed adjustments to be done to the GWSM in Section 4. In Section 5, the estimates produced with these adjustments are compared to those from the original GWSM using a small numerical example, and using real data that come from the Statistics Canada's Business Register. A brief conclusion is presented in Section 6.

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