Some improvement on non-parametric estimation of income distribution and poverty index

In this paper, we propose an estimator of Foster, Greer and Thorbecke class of measures $\displaystyle P(z,\alpha) = \int_0^{z}\Big(\frac{z-x}{z}\Big)^{\alpha}f(x)\, dx$, where $z>0$ is the poverty line, $f$ is the probabily density function of the income distribution and $\alpha$ is the so-called poverty aversion. The estimator is constructed with a bias reduced kernel estimator. Uniform almost sure consistency and uniform mean square consistenty are established. A simulation study indicates that our new estimator performs well.