處理缺失之資料，已經有一些插補方法，但這些插補方法在不同情況下是否確實有效，仍有待探討pd Van Buuren et al.(2006) 對兩種不相容性模型 (一條件分配函數為線性，另一條件分配函數分別為平方及對數)進行討論，該論文依據模擬結果，僅表示在此兩不相容性模型下的插補方法似乎仍有效。本文則不僅嘗試解釋此兩模型為何有效，且進一步探討是否所有的不相容性模型插補後能與母體參數值相似而達到有效插補，並檢定其模擬後之結果，本文發現其答案為否定。 / There are some available imputation methods to deal with missing data. However, whether imputation methods based on conditional distributions are effective is still questionable. Van Buuren et al.(2006) discuss two incompatible conditional distributions models (one conditional distribution has a linear relation, the other conditional distribution has a squared or a logarithmic relation). According to their simulation results, Van Buuren et al.(2006) conclude that imputation methods based on these two incompatible models are effective. In this thesis, we try to explain why the two imputation models are effective. In addition, we discuss whether all imputation methods based on incompatible models give estimated parameter values close to the true values. The simulation results of these methods are also tested statistically to answer this question. In conclusion, we find the answer is negative.

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關於二維之變數，Arnold and Press (1989) 首先提出檢驗兩個條件分配是否滿足相容性的理論。本研究嘗試對n維之變數，探討n個條件分配滿足相容性的檢驗方式；並提出在三維聯合分配下，給定三個條件分配為常態(normal) 時，檢驗此三個條件分配滿足相容性的充分必要條件；最後，並推導出此三個條件分配滿足相容性時，其所對應的聯合機率密度函數之公式。若此三個條件分配其所對應的聯合機率密度函數進一步假設為常態時，檢驗其相容性的充分必要條件可更加以簡化。 / Arnold and Press (1989) first provide the theory about the compatibility of two conditional distributions in two dimensions. In this research, we extend the two dimensional cases to the high dimensional cases. In particular, we find the necessary and sufficient conditions of the compatibility of three conditional normal distributions in three dimensions. Furthermore, we also provide a formula to find the joint probability density function when three dimensional conditional normal distributions are compatible. Finally, simple sufficient and necessary conditions are also given when the joint distribution is further assumed to be normal.

根據Arnold and Press (1989) 提出檢驗兩個條件分配是否滿足相容條件的理論內容，本研究推論出，當給定二個條件機率密度函數的形式為常態（normal）時，如何判斷這兩個條件常態分配是否相容的充要條件，並進而推論出這兩個條件常態分配對應的聯合機率密度函數亦為常態分配的條件。我們更進一步透過電腦模擬方法，提供兩個不同聯合常態分配下所分別得到的兩組不同條件分配樣本，據以推得當對應的母數相差到何種程度時，可判定這兩組樣本其原始母體不同。 / Arnold and Press (1989) provide the theory about the compatibility of two conditional densities. In this research, we use their results to find the sufficient conditions of the compatibility of two conditional densities, which have the normal form. New sufficient conditions are also given if we further assume that corresponding joint density is normal. In addition, we use computer to generate two different samples from two different conditional normal distributions, which are from two different joint normal distributions. With the repeated samples, we provide the ranges of one population parameter when the other population parameters are fixed so that the samples are almost always incompatible.

電子產品及通訊產品具有相容性與網路外部性的產品特性，本文探討電信產業中，通訊產品與服務所特有的「網路外部性」以及「產品相容性」之間的關係，文中利用兩階段的雙占Hotelling模型，說明相容性與網內外費率之間的關係，而產品相容性問題導致電信業者的網內外費率有所不同，形成的網路外部性也各不相同，廠商如何利用此一特性進行訂價與競爭，並分析不同的技術、成本的廠商訂價，與相容程度大小的決策。本研究顯示廠商擁有較高的技術時，將會使相容性提高，使網內外費率的差距減少，所產生的網路外部性也越高，相較於對手廠商，產生更高的利潤並且產生較高的消費者剩餘。若消費者對於品牌有不同的偏好，廠商將會減少網路的相容性以提高利潤。最後並解釋廠商之間若有勾結之情形，將會依據消費者的願付價格來訂價，且社會總福利將會較雙占競爭高，雙占競爭時廠商因為提供了過高的相容性而損失了效率。

Chen(2010)提出如何用勝率函數來判斷給定的連續條件分配是否相容，以及 相容時如何求對應的聯合分配。本研究提出，在二維有限的情形下，如何用勝率 矩陣來判斷給定的條件機率矩陣是否相容，以及相容時如何求對應的聯合機率矩 陣。又給定的條件機率矩陣不相容時，我們介紹了四種修改勝率矩陣的方法，同 時在使用幾何平均法調整勝率矩陣的過程中，也發現選取最佳參考點以獲得最佳 近似聯合機率矩陣之方法，並且給予理論證明。最後以模擬的方式發現，在修改 勝率矩陣的四種方法中，以幾何平均法所得到的近似聯合機率矩陣，其條件機率 矩陣最常接近所給定的條件機率矩陣。 / Chen (2010) provides the representations of odds ratio function to examine the compatibility of conditional probability density functions and gives the corresponding joint probability density functions if they are compatible. In this research, we provide the representations of odds ratio matrix to examine the compatibility of two discrete conditional probability matrices and give the corresponding joint probability matrix if they are compatible. For incompatible situations, we offer four methods to revise odds ratio matrices to find near joint probability matrices so that their conditional probability matrices are not far from the two given ones. That is, we provide four methods so that the sums of error squares are small. For each method, the sum of error squares may depend on the same reference point of two odds ratio matrices. We first discover by example that only the geometric method out of these four methods has a pattern to get the best reference point so that the sum of error squares is smallest. We then prove this finding in general. In addition, through simulation results, the geometric method would provide the smallest sum of error squares most often among these four methods. Hence, we suggest using geometric method. Its strategy to find the best reference point is also given.

給定兩個條件機率分配，判斷他們是否相容？是否有唯一的聯合機率分配？以及相容時，如何找出所有可能的聯合機率分配？是研究相容性相當重要的課題。本文針對有限離散型二維條件機率分配，以Arnold and Press(1989) 最先提出的比值矩陣法，及由Song , Li, Chen, Jiang, and Kuo (2010) 所提出的檢驗法為架構，提出新演算法且利用此演算法來設計程式，使程式能判斷兩條件機率分配是否相容，以及相容後可求出對應的所有聯合機率分配。本文亦依據新演算法並應用MATLAB軟體設計程式，讓使用者可以很快地對上述三個問題得到答案。 / When two conditional distributions are given, the following three important questions are likely to be raised. Are they compatible? Is the corresponding joint distribution unique if they are compatible? How do you find all the corresponding joint distributions if they are compatible? In this thesis, basing on ratio matrix method given first by Arnold and Press (1989), and on the method for checking compatibility existence, for checking uniqueness, and for finding all possible joint distributions provided by Song, Li, Chen, Jiang, and Kuo (2010), we provide a new algorithm to answer these questions. Using this new algorithm, we also provide a MATLAB computer program so that any user could get the answer quickly for the above three questions.

單變量c-特徵函數已被證明可處理一些難以使用傳統特徵函數解決的問題， 在本文中，我們首先提出其反演公式，透過此反演公式，我們獲得（1）Dirichlet隨機向量之線性組合的機率密度函數；（2）以一些有趣測度為參數之Ferguson-Dirichlet過程其隨機動差的機率密度函數；（3）Ferguson-Dirichlet過程之隨機泛函的Lebesgue積分表示式。 本文給予對稱分配之多變量c-特徵函數的新性質，透過這些性質，我們證明在任何$n$維球面上之Ferguson-Dirichlet過程其隨機均值是一對稱分配，並且我們亦獲得其確切的機率密度函數，此外，我們將這些結果推廣至任何n維橢球面上。 我們亦探討條件分配相容性的問題，這個問題在機率理論與貝式計算上有其重要性，我們提出其充要條件。當給定相容的條件分配時，我們不但解決相關聯合分配唯一性的問題，而且也提供方法去獲得所有可能的相關聯合分配，我們亦給予檢驗相容性、唯一性及建構機率密度函數的演算法。 透過相容性的相關理論，我們提出完整且清楚地統合性貝氏反演公式理論，並建構可應用於一般測度空間的廣義貝氏反演公式。此外，我們使用廣義貝氏反演公式提供一個配適機率密度函數的演算法，此演算法沒有疊代演算法（如Gibbs取樣法）的收斂問題。 / 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.

近年來台灣政府積極推動「營造英語生活環境行動方案」，再加上「國小英語課程」的實施，使得國小校園中充斥各種英語印刷文字佈置。本研究旨在了解這些校園英語環境印刷文字佈置之設計與成效及如何有效運用這些英語教學資源。 本研究對象來自於台北縣一國小九位教職員(其中包括四位行政人員和五位英語教師)以及622位三到六年級的學童。資料收集主要來源為訪談及問卷。收集的資料(主要採用描述性統計及卡方檢定)分別進行分析比較。 本研究主要的發現如下: 1.校園英語環境印刷文字佈置設計規劃主要由教育主管機關以及學校行政人員決定和負責。 2.英語教師參與校園英語環境印刷文字佈置設計規劃的程度會影響她們是否使用這些英文印刷文字進行教學的意願。 3.80%以上的受訪學生都注意到校園中的英語環境印刷文字佈置，也顯示高度的學習意願。但是由於多數英語教師未使用這些英語環境印刷文字於教學，這些校園英語環境印刷文字佈置對於學生英語學習幫助不明顯。 4.大部分的受訪者贊同校園中英語環境印刷文字佈置的必要性，並提出建議。 本研究有助於初步瞭解校園中英語環境印刷文字佈置的設計使用情形和成效。文末作者進一步提供英語環境印刷文字佈置設計和應用之相關建議，作為未來校園英語環境印刷文字佈置設計之參考。 / In Taiwan, with the policy of creating an English-friendly environment and new elementary school English curriculum, different kinds of environmental print (EP) such as English signs and English learning related posters can be seen in nearly every elementary school. It is important to have a clear idea about the effectiveness of elementary school’s environmental print (ESEP) and how to make use of ESEP. The purpose of the present study is to investigate the design of the ESEP and its effect in language learning and teaching. Participants of the study were nine school staff and 622 students sampled from the third grade to the sixth grade in one elementary school in Taipei county. Two sets of interviews and questionnaires were employed to collect data. Both qualitative analysis of interviews and quantitative statistical analysis (descriptive statistics focusing on frequency and percentage examination, and chi-square test) of the questionnaire were employed. The major findings are summarized as follows. First, the design of ESEP is primarily a top-down process. The educational authority and the school’s administrative staff directed the design of ESEP. Second, the degree of the involvement of the design of ESEP influenced English teachers’ willingness to conduct ESEP instruction. Third, more than 80% of the students noticed the existence of ESPE and showed high interest in learning about ESEP. However, without English teachers’ instruction, the effectiveness of ESEP in students’ language learning was not obvious. Fourth, most participants appreciated the existence of ESEP, and they gave suggestions for future ESEP development. Finally, some implications and suggestions for future research were discussed.

給定兩個有限離散型條件分配，我們可以去探討有關相容性及唯一性的問題。Tian et al.(2009)提出一個統合的方法，將相容性的問題轉換成具限制條件的線性方程系統(以邊際機率為未知數)，並藉由 l_2-距離測量解之誤差，進而求出最佳解來。他們也提出了電腦數值計算法在檢驗相容性及唯一性時的準則。 由於 Tian et al.(2009)的方法是把邊際機率和為 1 的條件放置在線性方程系統中，從理論的觀點來看，我們認為該條件在此種做法下未必會滿足。因此，本文中將邊際機率和為 1 的條件從線性方程系統中抽離出來，放入限制條件中，再對修正後的問題求最佳解。 我們提出了兩個解決問題的方法：(一) LRG 法；(二) 干擾參數法。LRG 法是先不管機率值在 0 與 1 之間的限制，在邊際機率和為 1 的條件下，利用 Lagrange 乘數法導出解的公式，之後再利用 Rao-Ghangurde 法進行修正，使解滿足機率值在 0 與 1 之間的要求。干擾參數法是在 Lagrange 乘數法公式解中有關廣義逆矩陣的計算部份引進了微量干擾值，使近似的逆矩陣及解可快速求得。理論證明，引進干擾參數所增加的誤差不超過所選定的干擾值，易言之，由干擾參數法所求出的解幾近最佳解。故干擾參數法在處理相容性問題上，是非常實用、有效的方法。從進一步分析Lagrange 乘數法公式解的過程中，我們也發現了檢驗條件分配"理論"相容的充分條件。 最後，為了驗證 LRG 法與干擾參數法的可行性，我們利用 MATLAB 設計了程式來處理求解過程中的運算，並以 Tian et al.(2009)文中四個可涵蓋各種情況的範例來解釋說明處理的流程，同時將所獲得的結果和 Tian et al. 的結果做比較。 / Given two finite discrete conditional distributions, we could study the compatibility and uniqueness issues. Tian et al.(2009) proposed a unified method by converting the compatibility problem into a system of linear equations with constraints, in which marginal probability values are assumed unknown. It locates the optimum solution by means of the error of l_2 - discrepancy. They also provided criteria for determining the compatibility and uniqueness. Because the condition of sum of the marginal probability values being equal to one is in Tian et al.s’linear system, it might not be fulfilled by the optimum solution. By separating this condition from the linear system and adding into constraints, we would look for the optimum solution after modification. We propose two new methods: (1) LRG method and (2) Perturbation method. LRG method ignores the requirement of the probability values being between zero and one initially, it then uses the Lagrange multipliers method to derive the solution for a quadratic optimization problem subject to the sum of the marginal probability values being equal to 1. Afterward we use the Rao-Ghangurde method to modify the computed value to meet the requirement. The perturbation method introduces tiny perturbation parameter in finding the generalized inverse for the optimum solution obtained by the Lagrange multipliers method. It can be shown that the increased error is less than the perturbation value introduced. Thus it is a practical and effective method in dealing with compatibility issues. We also find some sufficient conditions for checking the compatibility of conditional distributions from further analysis on the solution given by Lagrange multipliers method. To show the feasibilities of LRG method and Perturbation method, we use MATLAB to device a program to conduct them. Several numerical examples raised by Tian et al.(2009) in their article are applied to illustrate our methods. Some comparisons with their method are also presented.