1. Introduction
Eric Graf and Yves Tillé
Deville (2000) proposed that the precision of non-linear statistics in sampling designs be estimated using the generalized linearization method, which relies on the concept of influence function proposed by Hampel (1974) in the field of robust statistics. Osier (2009) applied these theories to estimate the variance of complex statistics such as the Laeken indicators (Eurostat 2005) in the European Statistics on Income and Living Conditions (EU-SILC) survey. Goga, Deville and Ruiz-Gazen (2009) extend the theory of Deville (2000) to two-sample surveys. Verma and Betti (2011) provide a comprehensive list of traditional poverty indicators and associated linearized variables, and they also compare the performance of the linearization technique with that of the jackknife repeated replication method. In this article, we will limit ourselves to poverty indicators published in the EU-SILC survey and focus on the way to estimate the income density function at various points in the distribution.
In Section 2, we review the required theoretical foundations, the expressions for the poverty and inequality indicators being studied, and the linearized variables of those indicators. Some linearized variables are dependent on the density function of the variable of interest, which is usually estimated using the Gaussian kernel estimation method. Two alternative methods are presented in Section 3. The R-language simulations are described and discussed in Section 4. We show that Gaussian kernel estimation can generate strong bias in the estimated variance of indicators when an estimate of the income density function is being used. We also show that the other two density estimation methods proposed in Section 3 reduce the observed bias, and this is discussed in the findings in the last part of this article.
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