Rate robustness for semiparametric estimation of a causal effect on a concentration index

M. Ghasempour1 X. de Luna

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and P.E. Gustafsson

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Abstract

A concentration index, a standardised covariance between a health variable and relative income ranks, is often used to quantify income-related health inequalities. There is a lack of formal approach to study the effect of an exposure, e.g., education, on such measures of inequality. In this paper we contribute by filling this gap and developing the necessary theory and method. Thus, we define a counterfactual concentration index for different levels of an exposure. We give conditions for the identification of this complex estimand, and then deduce its efficient influence function. This allows us to propose estimators, which are regular asymptotic linear under certain conditions. In particular, we show that these estimators are $\sqrt{n}$ -consistent and asymptotically normal, as well as locally efficient. The implementation of the estimators is based on the fit of several nuisance functions. The estimators proposed have rate robustness properties allowing for convergence rates slower than $\sqrt{n}$ -rate for some of the nuisance function fits. The relevance of the asymptotic results for finite samples is studied with simulation experiments. We also present a case study of the effect of education on income-related health inequalities for a Swedish cohort.

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Department of Statistics, USBE, University of Umeå, Umeå, Sweden [mohammad.ghasempour@umu.se]
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Department of Statistics, USBE, University of Umeå, Umeå, Sweden [xavier.deluna@umu.se]
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Department of Epidemiology and Global Health, University of Umeå, Umeå, Sweden [per.gustafsson@umu.se]

Keywords: Efficient influence function – Gini index – Rate robustness – Record linked data – Semiparametric efficiency bound