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Keywords:
$(h,\Phi )$-entropy measures; information metric; geodesic distance between probability distributions; maximum likelihood estimators; asymptotic distributions; Cramér-Rao inequality.; generalized entropies
Summary:
Burbea and Rao (1982a, 1982b) gave some general methods for constructing quadratic differential metrics on probability spaces. Using these methods, they obtained the Fisher information metric as a particular case. In this paper we apply the method based on entropy measures to obtain a Riemannian metric based on $(h,\Phi )$-entropy measures (Salicrú et al., 1993). The geodesic distances based on that information metric have been computed for a number of parametric families of distributions. The use of geodesic distances in testing statistical hypotheses is illustrated by an example within the Pareto family. We obtain the asymptotic distribution of the information matrices associated with the metric when the parameter is replaced by its maximum likelihood estimator. The relation between the information matrices and the Cramér-Rao inequality is also obtained.
References:
[1] S. I. Amari: A foundation of information geometry. vol. 66-A, , 1983, pp. 1–10.
[2] C. Atkinson, A. F. S. Mitchell: Rao’s distance measure. vol. 43, , 1981, pp. 345–365.
[3] S. Arimoto: Information-theoretical considerations on estimation problems. Information and Control 19 (1971), no. , , 181–194.
[4] J. Burbea: Informative geometry of probability spaces. vol. 4, , 1986, pp. 347–378.
[5] J. Burbea, C. R. Rao: Entropy differential metric, distance and divergence measures in probability spaces: A unified approach. J. Multivariate Analysis 12 (1982a), no. , , 575–596.
[6] J. Burbea, C. R. Rao: On the convexity of some divergence measures based on entropy functions. vol. IT-28, , 1982b, pp. 489–495.
[7] J. Burbea, C. R. Rao: On the convexity of higher order Jensen differences based on entropy functions. vol. IT-28, , 1982c, pp. 961–963.
[8] N. N. Cencov: Statistical Decision Rules and Optimal Inference. vol. , , 1982, pp. .
[9] I. Csiszár: Information-type measures of difference of probability distributions and indirect observations. vol. 2, , 1967, pp. 299–318.
[10] K. Ferentinos, T. Papaioannou: New parametric measures of information. vol. 51, , 1981, pp. 193–208.
[11] C. Ferreri: Hypoentropy and related heterogeneity divergence measures. vol. 40, , 1980, pp. 55–118.
[12] J. Havrda, F. Charvat: Concept of structural $\alpha $-entropy. vol. 3, , 1967, pp. 30–35.
[13] D. Morales, L. Pardo, L. Salicrú, M. L. Menéndez: New parametric measures of information based on generalized $R$-divergences. vol. , , 1993, pp. 473–488.
[14] R. J. Muirhead: Aspect of Multivariate Statistical Theory. vol. , , 1982, pp. .
[15] O. Onicescu: Energie Informationnelle. vol. 263, , 1966, pp. 841–842.
[16] C. R. Rao: Information and accuracy attainable in the estimation of statistical parameters. vol. 37, , 1945, pp. 81–91.
[17] C. R. Rao: Differential Metrics in probability spaces. vol. , , 1987, pp. .
[18] A. Rényi: On measures of entropy and information. vol. 1, , 1961, pp. 547–561.
[19] M. Salicrú, M. L. Menéndez, D. Morales, L. Pardo: Asymptotic distribution of $(h,\Phi )$-entropies. vol. 22(7), , 1993, pp. 2015–2031.
[20] C. E. Shannon: A mathematical theory of communication. vol. 27, , 1948, pp. 379–423.
[21] B. D. Sharma, I. J. Taneja: Entropy of type $(\alpha , \beta )$ and other generalized measures in information theory. vol. 22, , 1975, pp. 205–215.
[22] B. D. Sharma, P. Mittal: New non-additive measures of relative information. vol. 2, , 1975, pp. 122–133.
[23] I. J. Taneja: A study of generalized measures in information theory. vol. , , 1975, pp. .
[24] I. J. Taneja: On generalized information measures and their applications. vol. 76, , 1989, pp. 327–413.
[25] I. Vajda, K. Vašek: Majorization, concave entropies and comparison of experiments. vol. 14, , 1985, pp. 105–115.
[26] J. C. A. Van der Lubbe: $R$-norm information and a general class of measures for certainty and information. M. Sc. Thesis, Delf University of Technology, Dept. E.E., (1977), no. , , . (Dutch)
[27] J. C. A. Van der Lubbe: A generalized probabilistic theory of the measurement of certainty and information. Ph. D. Thesis, Delf University of Technology, Dept. E.E., (1981), no. , , .
[28] R. S. Varma: Generalizations of Renyi’s entropy of order $\alpha $. vol. 1, , 1966, pp. 34–48.
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