| Summary: | 博士 === 國立政治大學 === 應用數學研究所 === 96 === The univariate c-characteristic function has been shown to be important in cases that are hard to manage using the traditional characteristic function. In this thesis, we first give its inversion formulas. We then use them to obtain (1) the probability density functions (PDFs) of a linear combination of the components of a Dirichlet random vector; (2) the PDFs of random functionals of a Ferguson-Dirichlet process with some interesting parameter measures; (3) a Lebesgue integral expression of any random functional
of the Ferguson-Dirichlet process.
New properties of the multivariate c-characteristic function with a spherical distribution are given in this thesis. With them, we show that the random mean of a Ferguson-Dirichlet process over a spherical surface in n dimensions has a spherical distribution on the n-dimensional ball. Moreover, we derive its exact PDF. Furthermore, we generalize this result to any ellipsoidal surface in n-space.
We also study the issue of compatibility for specified conditional distributions. This issue is important in probability theory and Bayesian computations. Several necessary and sufficient conditions for the compatibility are provided. We also address the problem of uniqueness of the associated joint distribution when the given conditionals are compatible. In addition, we provide a method to obtain all possible joint distributions that have the given compatible conditionals. Algorithms for checking the compatibility and the uniqueness, and for constructing all associated densities are also given.
Through the related compatibility theorems, we provide a fully and cleanly unified theory of inverse Bayes formula (IBF) and construct a generalized IBF (GIBF) that is applicable in the more general measurable space. In addition, using the GIBF, we provide a marginal density fitting algorithm, which avoids the problems of convergence in iterative algorithm such as the Gibbs sampler.
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