Abstract
Geometric and stochastic representations are derived for the big class of p-generalized elliptically contoured distributions, and (generalizing Cavalieri’s and Torricelli’s method of indivisibles in a non-Euclidean sense) a geometric disintegration method is established for deriving even more general star-shaped distributions. Applications to constructing non-concentric elliptically contoured and generalized von Mises distributions are presented.AMS subject classification Primary 60E05; 60D05; secondary 28A50; 28A75; 51F99
Highlights
The needs of statistical practice and challenging probabilistic questions in the interplay of measure theory and several other mathematical disciplines stimulate the development of statistical distribution theory
Geometric and stochastic representations are derived for the big class of p-generalized elliptically contoured distributions, and a geometric disintegration method is established for deriving even more general star-shaped distributions
1 Introduction The needs of statistical practice and challenging probabilistic questions in the interplay of measure theory and several other mathematical disciplines stimulate the development of statistical distribution theory
Summary
The needs of statistical practice and challenging probabilistic questions in the interplay of measure theory and several other mathematical disciplines stimulate the development of statistical distribution theory. The ipf is essentially based upon a suitably defined non-Euclidean surface content on a star sphere The latter notion needs the most effort in the present work. Theorem 5 proves that the local approach to the star generalized surface content results in the same quantity as a suitably defined non-Euclidean surface content in terms of an integral defined using a modified standard approach of differential geometry To this end, some more coordinate systems are introduced and exploited. For this specific class, all of the more general results of the preceding parts of Section 4, including the main results in Theorems 7 and 8, allow an additional interpretation which in each case is based upon a suitable non-Euclidean geometry.
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