定義在「非負」與「總和為一」的單位體之多變量分配，常採用狄氏分配，因為狄氏分配在運算上較為簡單。然而，狄氏分配受限於任兩變數必須為負相關的限制，當參數向量存有顯著正相關的因子時，狄氏分配便不適用。因此基於此考量，所發展出另一更具彈性的機率分配，為廣義狄氏分配。廣義狄氏分配是一個比狄氏分配更具彈性的分配，因此廣義狄氏分配除了包含狄氏分配的優點之外，也彌補了其任兩變數就必為負相關的缺點。廣義狄氏分配的參數數量是其所擁有的隨機變數數量的兩倍，必須由給定的統計量(例如：期望值、變異數或共變異數)中去設法捨棄雜亂的資訊，以求能得到最精確描述廣義狄氏分配的參數組合。本研究在必須選入所有隨機變數期望值的前提下，提出兩種求解參數的方法，一為線性求解參數法，二為參數配適最佳化。線性求解參數法為利用共變異數矩陣中的數值去求解參數，選擇不同的數值會得到不同的參數值，也有可能會產生退化的情況使得問題無法求解，因此存在很多不同的挑選方法。參數配適最佳化為以樣本共變異數矩陣與求解參數後之矩陣兩差異最小化為目的，去求解所有參數。本研究提出數個在共變異數矩陣中挑選時，能快速找到適當參數組合的準則，以及利用使共變異數矩陣差異最小化去找出與樣本資料最符合的廣義狄氏分配。本研究的結果提供了在不同限制下可使用的方法，以得到較恰當的參數組合，讓分析人員可以找到最適合描述實際情況的廣義狄氏分配，提早做出防範或擬定因應策略。

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 its moments is simple. When any two variables are significantly positively correlated, the Dirichlet distribution becomes an inappropriate choice. Generalized Dirichlet distribution is more practical than Dirichlet distribution, because the generalized Dirichlet distribution has a more flexible covariance structure. When any two variables in a random vector tend to be positively correlated, a generalized Dirichlet distribution can be an appropriate choice for the random vector. The main purpose of this thesis is to estimate the parameters of a generalized Dirichlet distribution from the available statistics, such as expected values, variances, and covariances, derived from data. When the expected values of all variables have to chosen in estimating the parameters of a generalized Dirichlet distribution, the number of degrees of freedom for adjusting the spread of this distribution is equal to the number of variables. We propose a linear method and a nonlinear method to solve the parameters when the covariance matrix is available. The linear method is faster, while the nonlinear method can find the fittest parameter set to the original covariance matrix. Some principles for removing inappropriate statistics from the covariance matrix to speed the estimation process are also established.

中文
洪明君，Dirichlet分配、非短視行為與演化性賽局，國立政治大學經濟學研究所碩士論文，1999年。

郭家菱，帶有分型誤差之染病同胞對資料之貝氏調整連鎖分析方法，國立臺灣大學流行病學研究所碩士論文，2003年。

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