Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series A, Pt. 3 , pp. 444-457

ON DEFINING NEIGHBOURHOODS OF MEASURES THROUGH THE CONCENTRATION FUNCTION

By

SANDRA FORTINI and FABRIZIO RUGGERI, CNR-IAMI

SUMMARY. Statistical procedures are often interested in comparing probability measures by means of distances defined over the space $\clp$ of all probability measures, endowed with some classical topology, like the variational or the Prohorov ones. Other topologies can be obtained by means of the concentration function, which extends the notion of Lorenz curve. Hence, neighbourhood classes $\Gamma$ of probability measures, including well-known ones, are defined and a representation theorem is proved. Finally, ranges of functionals over $\Gamma$ are found, restricting the search among the extremal measures in~$\Gamma$.

AMS (1980) subject classification. Primary 62E10; secondary 60E05.

Key words and phrases. Neighbourhoods of probability measures; concentration function; mixtures of probability measures; extremal probability measures.

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