About some difficulties with the functional forms of Lorenz curves

We study to what extent some functional form assumption on the Lorenz curve are amenable to calculating headcount poverty, or poverty threshold, the key concept to determine a poverty index. The difficulties in calculating it have been underestimated. We must choose some functional forms for the Lorenz concentration curve. We examine three families of one-parameter functional forms to estimate Lorenz curves: power (elementary and Pareto), exponential (elementary and Gupta) and fractional (Rohde). Computing these numerical functions may be difficult and impose some restrictions on their domain of definition, may impose to use some numerical approximation methods. The elementary power and exponential forms are not a problem. However, Pareto raises the problem of a restricted domain of definition for its parameters. The exponential form of Gupta leads to a Lambert function that poses multiple problems, including a restricted field of definition. The fractional form of Rohde has also a restricted domain of definition. It is probably time to choose functional forms not only according to their ability to fit the data, but also according to their ability to calculate poverty indices.