Small area estimation methods under cut-off sampling
Section 1. Introduction
Haziza, Chauvet and Deville (2010) describe cut-off sampling as a technique in which a set of units is deliberately excluded from possible selection in the sample. For the Organisation for Economic Co-operation and Development (OECD), it is a sampling procedure in which a threshold is established such that all units above or below the threshold are excluded from selection in a sample. According to Särndal, Swensson and Wretman (1992, pages 531-533), this sampling technique is typically used when the distribution of the study variable is highly skewed and there is no reliable frame covering the small elements. Benedetti, Bee and Espa (2010) recognizes the advantage of cut-off sampling in terms of survey reduction cost. This procedure is often used in business surveys, where small firms are deliberately excluded from the sample due to difficulty of getting information from them. The cost of obtaining and keeping a reliable frame for the whole population does not compensate the subsequent gain in accuracy.
The monthly survey of manufacturing performed by Statistics Canada is an example of cut-off sampling (Benedetti et al., 2010). In Spain, the monthly survey of industrial production index (IPI) performed by the Spanish National Statistical Institute (in Spanish, INE) collects data from firms that produce a significant volume of products according to the annual industrial survey of products (in Spanish EIAP), see INE (2018). Related surveys, e.g., the index of industrial prices (IIP) and the index of business turnover (IBT) also use one form of cut-off sampling. Since the inclusion probabilities for the excluded units are zero, this procedure leads to biased design-based estimators, see e.g., Särndal et al. (1992) or Haziza et al. (2010) among others. To reduce the cut-off sampling bias, Haziza et al. (2010) propose to use auxiliary information either at the design or at the estimation stage; concretely, they propose to use balanced sampling and/or calibration.
In this work, we restrict ourselves to the estimation stage and study how cut-off sampling affects the estimation of domain (or area) parameters. We analyze some of the calibration methods proposed by Haziza et al. (2010) to reduce this problem. For domains with small sample size (small domains or areas), even in absence of cut-off sampling, calibration estimators might be inefficient. To improve efficiency, we consider small area estimation methods. For estimation of linear parameters, we consider the empirical best linear unbiased predictor (EBLUP) and, for general non-linear parameters, we consider the empirical best/Bayes predictor (EBP). We apply the methods studied in this work to the estimation of the total sales of certain tobacco product in Spanish provinces.
In the absence of cut-off sampling, the considered model-based estimators are approximately optimal when the model holds for all the population units. However, since no model holds exactly, we wish to study whether model-based estimators still perform better than basic design-based estimators (which do not depend on models) and calibration estimators under the sampling replication mechanism; i.e., without model assumptions and when cut-off sampling is present.
The article is organized as follows. Section 2 describes the theoretical set-up. The following four sections describe the considered estimation methods, namely the basic direct estimators (Section 3), different approaches to calibration (Section 4), the EBLUP for estimation of linear parameters (Section 5) and the EBP for estimation of more general parameters in small domains (Section 6). Section 7 describes a bootstrap procedure for estimating the mean squared error of the proposed small area estimators. Section 8 compares, through simulation experiments, the performance of several small area estimators under cut-off sampling. Section 9 describes the application and, finally, Section 10 draws some conclusions.
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