Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 3, 345-365

DIFFERENTIAL PROPERTIES OF THE CONCENTRATION FUNCTION

By

SANDRA FORTINI, Universit\`a ``L. Bocconi'', Milano

And

FABRIZIO RUGGERI, CNR-IAMI, Milano

SUMMARY. T he Lorenz curve and its extensions, as the concentration function, are used in some statistical problems, mainly related to economics, to describe the discrepancies between a probability measure $P$ and a reference measure $P_0$. Some applications require the consideration of variations in the concentration of $P$ with respect to $P_0$ due to small changes of $P$. This leads to the study of the differential properties of the concentration function at a point, regarded as a functional defined on the space of the finite signed measures. This paper proves that the G\^ateaux differential of the concentration function exists in any direction and it provides its expression. Moreover, it shows that the concentration function is not Fr\'echet differentiable. Some applications are described.

 AMS (1991) subject classification. 62E10, 60E05

Key words and phrases. Concentration function, Ali-Silvy indices, Gateaux differential.

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