Inference for Lorenz curves requires its rigorous definition and good specifications. This post briefly discusses alternative definitions and ways of specifying Lorenz curves:

Let and denote income level, CDF, PDF and Inverse-CDF functions respectively.

- An intuitive definition of the Lorenz curve is through

where *c* is the population proportion and *L* is the Lorenz curve. This definition is not the most general definition because it requires existence of a density function and existence of a density function itself requires absolute continuity of the CDF function. The good thing about this definition however is that it can be used to derive the Lorenz curve for well-known density functions such as Lognormal, Singh-Maddala, GB2, etc.

- A more general way of defining a Lorenz curve is through

This definition is preferred from a technical point of view because it requires only existence of a cumulative density function. It can also be used for specification of a Lorenz curve if one has a good specification for an Inverse-CDF function.

- It has been shown that any continuous convex function
*L* from [0,1] to [0,1] with *L*(0)=0 and *L*(1)=1 can a Lorenz curve. Using this result, one can directly specify a Lorenz curve without the need for starting from a distribution function. A host of functional forms have been proposed using this approach. See *Sarabia et al.* (2008) and references cited therein for more on specification of Lorenz curves and this book for technical issues related to the definition of Lorenz curves.

Reference:

Sarabia, J. M. (2008). Parametric Lorenz curves: Models and applications. In D. Chotikapanich (Ed.), Modeling Income Distributions and Lorenz Curves, Chapter 9, pp. 167–190. Springer.

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Posted by oosorio456 on October 10, 2015 at 5:43 pm

Inequality is becoming increasingly popular in Western Hemisphere

Posted by oosorio456 on October 10, 2015 at 5:43 pm

It’s time to discuss social justice along with tax cuts and start thinking about it

Posted by Reza on October 11, 2015 at 7:08 am

I think it is good thing that inequality is becoming prominent but, for now, this blog is mainly concerned with its measurement not the causes or consequences.