Abstract
The linearisation approach to approximating variance of complex non-linear statistics is a well-established procedure. The basis of this approach is to reduce non-linear statistics to a linear form, justified on the basis of asymptotic properties of large populations and samples. For diverse cross-sectional measures of inequality such linearised forms are available, though the derivations involved can be complex. Replication methods based on repeated resampling of the parent sample provide an alternative approach to variance estimation of complex statistics from complex samples. These procedures can be computationally demanding but tend to be straightforward technically. Perhaps the simplest and the best established among these is the Jackknife Repeated Replication (JRR) method. Recently the JRR method has been shown to produce comparable variance for cross-sectional poverty measures (Verma and Betti in J Appl Stat 38(8):1549–1576, 2011); and it has also been extended to estimate the variance of longitudinal poverty measures for which Taylor approximation is not currently available, or at least cannot be easily derived. This paper extends the JRR methodology further to the estimation of variance of differences and averages of inequality measures. It illustrates the application of JRR methodology using data from four waves of the EU-SILC for Spain. For cross-sectional measures design effect can be decomposed into the effect of clustering and stratification, and that of weighting under both methodologies. For differences and averages of these poverty measures JRR method is applied to compute variance and three separate components of the design effect—effect of clustering and stratification, effect of weighting, and an additional effect due to correlation of different cross-sections from panel data—combining these the overall design effect can be estimated.
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Notes
A brief definition of the indicators is the following.
Mean equivalised income: constructed using the equivalised income, defined as the total disposable household income divided by equivalent household size (constructed using the modified-OECD scale which gives a weight of 1.0 to the first adult in a household, 0.5 to each subsequent member aged 14 and over, and 0.3 to each child aged under 14), is ascribed to each member of the household.
Inequality of income distribution Gini coefficient: it is defined as the relationship of cumulative shares of the population arranged according to the level of equivalised disposable income, to the cumulative share of the equivalised total disposable income received by that population.
Inequality of income distribution S80/S20 income quintile share ratio: ratio of the shares of equivalised income of the top and the bottom 20% of the population.
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Betti, G., Gagliardi, F. Extension of JRR Method for Variance Estimation of Net Changes in Inequality Measures. Soc Indic Res 137, 45–60 (2018). https://doi.org/10.1007/s11205-017-1590-8
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DOI: https://doi.org/10.1007/s11205-017-1590-8