Geometrical properties of Pareto distribution

摘要

The differential-geometrical framework for analyzing statistical problems related to Pareto distribution, is given. A classical and intuitive way of description the relationship between the differential geometry and the statistics, is introduced [Publicationes Mathematicae Debrecen, Hungary, vol. 61 (2002) 1–14; RAAG Mem. 4 (1968) 373; Ann. Statist. 10 (2) (1982) 357; Springer Lecture Notes in Statistics, 1985; Tensor, N.S. 57 (1996) 282; Commun. Statist. Theor. Meth. 29 (4) (2000) 859; Tensor, N.S. 33 (1979) 347; Int. J. Eng. Sci. 19 (1981) 1609; Tensor, N.S. 57 (1996) 300; Differential Geometry and Statistics, 1993], but in a slightly modified manner. This is in order to provide an easier introduction for readers not familiar with differential geometry. The parameter space of the Pareto distribution using its Fisher’s matrix is defined. The Riemannian and scalar curvatures to parameter space are calculated. The differential equations of the geodesics are obtained and solved. The J-divergence, the geodesic distance and the relations between of them in that space are found. A development of the relation between the J-divergence and the geodesic distance is illustrated. The scalar curvature of the J-space is represented.