Parameter estimation for generalized Dirichlet distributions from the sample estimates of the first and the second moments of random variables

Generalized Dirichlet distributions have a more flexible covariance structure than Dirichlet distributions, and the computation for the moments of a generalized Dirichlet distribution is still tractable. For situations under which Dirichlet distributions are inappropriate for data analysis, generalized Dirichlet distributions will generally be an applicable alternative. When the expected values and the covariance matrix of random variables can be estimated from available data, this study introduces ways to estimate the parameters of a generalized Dirichlet distribution for analyzing compositional data. Under the assumption that the sample mean of every variable must be considered for parameter estimation, we present methods for choosing the statistics from a sample covariance matrix to construct a generalized Dirichlet distribution. Some rules for removing inappropriate statistics from a sample covariance matrix to speed up the estimation process are also established. An example for Taiwan's car market is introduced to demonstrate the applicability of the parameter estimation methods.