Statistical inference for richness measures

Richness indices are distributional statistics used to measure the incomes, earnings or wealth of the rich. This article uses a linearization method to derive the sampling variances for recently introduced distributionally sensitive richness measures when estimated from survey data. The results are derived for two cases: (1) when the richness line is known and (2) when it has to be estimated from the sample. The proposed approach enables easy consideration of the effects of a complex sampling design. Monte Carlo results suggest that the proposed approach allows for reliable inference in case of 'concave' richness indices, but that it is not satisfactory in case of 'convex' richness measures. The standard bootstrap methods give similar results for 'concave' measures, but they are also unreliable for 'convex' indices. The performance of the bootstrap inference can be improved in some cases using a semi-parametric approach. The variance formulae are illustrated with a comparison of wealth richness in Canada, Sweden, the United Kingdom and the United States.