Summary
Comparisons among probability measures are rather frequent in many statistical problems and they are sometimes performed through the coefficients of divergence or the concentration functions with respect to a reference measure. Extending the notion of Lorenz-Gini curve, the concentration function studies the discrepancy between two probability measures Π and Π0.
In this paper, both the concentration function and the coefficients have been defined and studied for a signed measure Π, as an extension of the concentration curve for real valued statistical variables. Signed measures are relevant in statistical analysis, even if unusual, because real problems require them, especially in descriptive statistics, like the simple one presented here.
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References
Ali, S. M. andSilvey, S. D. (1966), A general class of coefficients of divergence of one distribution from another,J. R. Statist. Soc. B, 28, 131–142.
Ash, R. B. (1972),Real Analysis and Probability, Academic Press, San Diego.
Castellano, V. (1938), Sugli indici relativi di variabilità e sulla concentrazione dei caratteri con segno,Metron, 15, 31–49.
Cifarelli, D. M. andRegazzini, E. (1987), On a general definition of concentration function,Sankhyā B, 49, 307–319.
Csiszár, I. (1967), Information-type measures of difference of probability distributions and indirect observations,Studia Sci. Math. Hung., 2, 299–318.
Dey, D. K. andBirmiwal, L. R. (1990), Robust Bayesian analysis using entropy and divergence measures,Technical Report, Department of Statistics, University of Connecticut, Storrs.
Fortini, S. andRuggeri, F. (1990), Concentration function in a robust Bayesian framework,Quaderno IAMA 90.6. Milano: CNR-IAMI.
Fortini, S. andRuggeri, F. (1992), Infinitestimal properties of the concentration function,Unpublished Manuscript.
Fortini, S. andRuggeri, F. (1993a), Concentration functions and Bayesian robustness. To appear inJ. Statist. Plann. Inference.
Fortini, S. andRuggeri, F. (1993b), On defining neighbourhoods of measures through the concentration function. To appear inSankhyā A.
Gini, C. (1914), Sulla misura della concentrazione della variabilità dei caratteri,Atti del Reale Istituto Veneto di S.L.A., A.A. 1913–1914, 73, parte II, 1203–1248.
Istituto Centrale di Statistica (1989),Annuario Statistico Italiano; ISTAT, Roma.
Kolmogorov, A. N. andFomin, S. V. (1980),Elementi di teoria delle funzioni e di analisi funzionale, Mir, Moscow.
Lehmann, E. L. (1986),Testing Statistical Hypotheses, Wiley, New York.
Marshall, A. W. andOlkin, I. (1979),Inequalities: theory of majorization and its applications, Academic Press, New York.
Michetti, B. andDall'Aglio, G. (1957), La differenza semplice media,Statistica, 2, 159–255.
Pietra, G. (1915), Delle relazioni tra gli indici di variabilità,Atti del Reale Istituto Veneto di S.L.A. A.A. 1914–1915, 74, parte II, 775–792.
Regazzini, E. (1992), Concentration comparisons between probability measures. To appear inSankhyā B, 54.
Ruggeri, F. andWasserman, L. A. (1991), Density based classes of priors: infinitesimal properties and approximations,Technical Report # 528, Department of Statistics, Canergie Mellon University, Pittsburgh.
Ruggeri, F. andWasserman, L. A. (1993), Infinitestimal sensitivity of posterior distribution. To appear inCanad. J. Statist.
Strasser, H. (1985),Mathematical Theory of Statistics, Walter de Gruyter, Berlin.
Wold, H. (1935), A study on the mean difference, concentration curves and concentration ratio,Metron, 12, 39–58.
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Fortini, S., Ruggeri, F. Concentration function and coefficients of divergence for signed measures. J. It. Statist. Soc. 2, 17–34 (1993). https://doi.org/10.1007/BF02589073
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DOI: https://doi.org/10.1007/BF02589073