1 Introduction

Anne Massiani

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SILC (Statistics on Income and Living Conditions) is an annual European survey designed to obtain indicators that are comparable from one country to another on poverty, social exclusion and living conditions within the population. For a detailed description of this survey, see Clemenceau and Museux (2006). In accordance with the recommendations of Eurostat (Eurostat 2003), the survey is conducted in Switzerland based on a four-panel rotation scheme; the first panel was surveyed in 2007. Each panel lasts four years, and every year one panel is replaced. When complete, a sample selected for a panel consists of approximately 3,600 households. This article will focus on cross-sectional estimation, for the population present in a year $t,$ of the indicators selected by Eurostat on poverty and living conditions, such as the at-risk-of-poverty rate and the quintile share ratio. See Osier (2009) for a description of these indicators and Ardilly and Lavallée (2007) for a description of the cross-sectional approach in the context of the SILC survey. Under this cross-sectional approach, the change over time in the composition of the households in the panels selected is taken into account using the indirect sampling theory developed by Lavallée (2002).

In this study, the formulas presented for estimating the variance of indicators take account of the great complexity of SILC-Switzerland. The results obtained, while developed specifically for the survey conducted in Switzerland, are likely to interest the other participating countries since they use similar methods. The main factors taken into account in estimating the variance of the indicators are the non-linearity of the estimators, total non-response at different survey stages, indirect sampling and calibration. One solution for estimating the variance of non-linear indicators is to use linearization techniques (see Deville 1999). Formulas specifically adapted to the indicators selected by Eurostat have been developed by Osier (2009). An alternative formula for the quintile share ratio is available in Langel and Tillé (2011). Once linearization techniques are applied, there is still the problem of estimating the variance of a total in a complex survey design. One difficulty is the presence of non-response after weight sharing. Lavallée (2002) proposes an estimator of the variance of a total that takes this into account. However, this estimator is generally not unbiased, even when the response probabilities are known. In this study, we propose an adaptation that corrects this bias.

Section 2 briefly describes how the survey is conducted and its sampling design. Section 3 describes the weighting used. Section 4 describes how linearization techniques are applied to these estimators to obtain an approximation of their variance. Section 5 is devoted to the problem of estimating the variance of a total where there is non-response after weight sharing. The final formula used for variance estimation is given in Section 6. Finally, a numerical application is provided in Section 7, followed by the conclusions of this study.

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