Abstract
Multivariate parametric statistical uncertainty relations are proved to specify multivariate basic parametric statistical models. The relations are expressed by inequalities. They generally show that we cannot exactly determine simultaneously both a function of observation objects and a parametric statistical model in a compound parametric statistical system composed of observations and a model. As special cases of the relations, statistical fundamental equations are presented which are obtained as the conditions of attainment of the equality sign in the relations. Making use of the result, a generalized multivariate exponential family is derived as a family of minimum uncertainty distributions. In the final section, several multivariate distributions are derived as basic multivariate parametric statistical models.
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References
Aitken, A. C. and Silverstone, H. (1942). On the estimation of statistical parameters, Proc. Roy. Soc. Edinburgh, Sect. A, 61, 186–194.
Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory, Wiley, New York.
Bohr, N. (1928). The quantum postulate and the recent development of atomic theory. Nature, 121, 580–590.
Brown, L. D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory, IMS, Lecture Notes-Monograph Series, Vol. 9, Hayward, California.
Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics, Philos. Trans. Roy. Soc. London Ser. A, 222, 309–368.
Fisher, R. A. (1936). Uncertain inference, Proceedings of American Academy of Arts and Sciences, 71, 245–258.
Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretishen Kinematik und Mechanik, Zeitschrift fur Physik, 43, 172–198.
Kapur, J. N. (1989). Maximum-Entropy Models in Science and Engineering, Wiley Easter, New Delhi.
Matsunawa, T. (1994). Origin of distributions —Nonparametric statistical uncertainty relation and a statistical fundamental equation—, Proc. Inst. Statist. Math., 42, 197–214 (in Japanese).
Matsunawa, T. (1995). Development of distributions —The Legendre transformation and canonical information criteria—, Proc. Inst. Statist. Math., 43, 293–311 (in Japanese).
Matsunawa, T. (1997). Basic parametric statistical model building without assuming existence of true distributions, Proc. Inst. Statist. Math., 45, 107–123 (in Japanese).
Matsunawa, T. and Zhao, Y. (1994). Specification of models for observations in multivariate case, Research Memo., No. 500, The Institute of Statistical Mathematics, Tokyo.
Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory, Wiley, New York.
Pearson, K. (1895). Contribution to the mathematical theory of evolution II, Skew variation in homogeneous material, Philos. Trans. Roy. Soc. London Ser. A, 186, 343–414.
Pearson, K. (1916). Contribution to the mathematical theory of evolution XIX, Second supplement to a memoir on Skew variation, Philos. Trans. Roy. Soc. London Ser. A, 216, 429–457.
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Matsunawa, T. Parametric Statistical Uncertainty Relations and Parametric Statistical Fundamental Equations. Annals of the Institute of Statistical Mathematics 50, 603–626 (1998). https://doi.org/10.1023/A:1003715611735
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DOI: https://doi.org/10.1023/A:1003715611735