| Summary: | 碩士 === 國立成功大學 === 工業與資訊管理學系碩博士班 === 94 === Collecting data for a sufficient statistic is generally much easier and less expensive than
recording the details of the available data. When the posterior distributions of a quantity of interest given the aggregate and disaggregate data are identical, perfect aggregation is said to hold, and in this case the aggregate data is a sufficient statistic for the quantity of interest. The Dirichlet distribution is one of the most popular multivariate distributions defined on unit simplex (i.e., all variables are nonnegative and their sum equals one), because the computation for the moments of the Dirichlet distribution is simple. However, the Dirichlet distribution can be used only when all variables are negatively correlated. When some of the variables are significantly positively correlated, the Dirichlet distribution will be an inappropriate prior for Bayesian analysis. The multivariate distributions generated from stochastic bifurcation processes governed by independent beta laws, which will be called independent beta bifurcation distributions, allow the correlation sign to vary within a row. Hence, independent beta bifurcation distributions are more appropriate for analyzing compositional data. In this thesis, when the quantity of interest is the sum of some parameters in a vector having an independent beta bifurcation distribution, the necessary and sufficient conditions for perfect aggregation are established. The methods by considering the means together with either variances or covariances of variables to construct an independent beta bifurcation distribution are also presented.
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