2.8 Clark-Hemming-Ulph class of poverty measures (svychu)

Clark, Hemming, and Ulph (1981Clark, Stephen, Richard Hemming, and David Ulph. 1981. “On Indices for the Measurement of Poverty.” The Economic Journal 91 (362). John Wiley; Sons. doi:10.2307/2232600.) proposes two classes of distribution-sensitive poverty measures. Yet, the poverty measurement literature focuses on the second class4 See Atkinson (1987Atkinson, Anthony B. 1987. “On the Measurement of Poverty.” Econometrica 55 (4). John Wiley; Sons. doi:10.2307/1911028.) and Verma and Betti (2011Verma, Vijay, and Gianni Betti. 2011. “Taylor Linearization Sampling Errors and Design Effects for Poverty Measures and Other Complex Statistics.” Journal of Applied Statistics 38 (8). Taylor; Francis Group. doi:10.1080/02664763.2010.515674.), for instance., expressed as

\[ CHU_\alpha = \begin{cases} \frac{1}{\alpha N} \sum_{i \in U} \big[ 1-(y_i/\theta)^\alpha \big] \cdot \delta ( y_i \leqslant \theta ) , & \alpha \leqslant 1 , \alpha \neq 0 \\ 1 - \bigg( \prod_{i \in U} y_i^{\delta ( y_i \leqslant \theta )} \bigg)^{1/N} \bigg/ \theta , & \alpha = 0 \end{cases} \]

As an special case, \(CHU_0 = 1 - \exp{(-Watts)}\). The \(\alpha\) parameter defines the sensivity towards regressive income transfers among the poor, such that the lower its value, larger is the regressive transfer impact on the index. When \(\alpha \rightarrow 1\), \(CHU_1 = FGT_0 \cdot I\), a measure insensitive to regressive income transfers among the poor.