泊松分佈(Poisson distribution)是一經常被配適於稀有事件建模的機率分配，其應用領域相當的廣泛，如生物，商業，品質控制等。其中許多的應用均為兩群體均數的比較，如欲檢測一新的處理是否較原本的處理俱優越性(superiority)，或者欲驗證一新的方法相較於舊的方法是否俱有不劣性(non-inferiority)。因此，此研究的目標為發展假設檢定的方法，用於比較兩獨立的泊松樣本是否有優越性及不劣性。一般探討假設檢定方法時，均因干擾參數的出現而導致理論探討及計算上的困難。為因應此困境，本研究由簡入繁，亦即先探討相等式的虛無假設(the null hypothesis of equality)，繼而，再推展至非優越性的虛無假設(the null hypothesis of non-superiority)，最後將這些探究的假設檢定方法應用至檢定不劣性並驗證這些方法的適用性。 兩種Wald 檢定統計量是本研究主要的研究興趣。對應於這兩種檢定統計量的近似的假設檢定法，是利用其極限分配為常態分配的特性而衍生的。此研究裡，可推導得到近似的檢定法的檢定力函數及欲達成某一檢定力水平時所需的樣本數公式。並依據此檢定力函數檢驗此檢定法的有效性(validity)及不偏性(unbiasedness)。並且推廣一連續修正的方法至任何的樣本數組合。另外一方面，此研究亦介紹並推廣兩種p-值的正確(exact)檢定法。其中一種為信賴區間p值檢定法(Berger和Boss, 1994), 另一種為估計的p值檢定法(Krishnamoorthy和Thomson, 2004)。一般正確檢定法較需要繁瑣的計算，故此研究將提出某些步驟以降低計算的負擔。就信賴區間p值檢定法而言，其首要工作為縮減求算p值的範圍，並驗證所使用的檢定統計量是否滿足Barnard凸面(convexity)的條件。若此統計量符合凸面convexity的條件，且在Poisson 的問題上，則此正確的信賴區間p值將出現在屬於虛無假設的參數空間的邊界上。然而，對於估計的p值檢定法而言，因在虛無假設的參數空間上求得Poisson均數的最大概似估計值，並不簡單及無法直接求得，故在此研就，將以一Poisson均數的點估計值代替。對於正確的假設檢定方法，此研究亦提出一個欲達成某一檢定力水平時所需的樣本數的步驟。 此研究將透過一個廣大的數值分析來驗證之前所提出的假設檢定方法。其中，可發現這些近似的假設檢定法之間的差異會受到兩群體之樣本數之比率的影響，而連續性的修正於某些情況下確實能夠使型I誤差較能夠受到控制。另外，當樣本數不夠多時，正確的假設檢定法是較近似的假設檢定法適當，尤其在型I誤差的控制上更是明顯。最後，此研究所提出的假設檢定方法將實際應用於一組乳癌治療的資料。 / The Poisson distribution is a well-known suitable model for modeling a rare events in variety fields such as biology, commerce, quality control, and so on. Many applications involve comparisons of two treatment groups and focus on showing the superiority of the new treatment to the conventional one, or the non-inferiority of the experimental implement to the standard implement upon the cost consideration. We aim to develop statistical tests for testing the superiority and non-inferiority by two independent random samples from Poisson distributions. In developing these tests, both computational and theoretical difficulties arise from presence of nuisance parameters. In this study, we first consider the problems with the null hypothesis of equality for simplicity. The problems are extended to have a regular null hypothesis of non-superiority next. Subsequently, the proposed methods are further investigated in establishing the non-inferiority. Two types of Wald test statistics are of our main research interest. The correspondent asymptotic testing procedures are developed by using the normal limiting distribution. In our study, the asymptotic distribution of the test statistics are derived. The asymptotic power functions and the sample size formula are further obtained. Given the power functions, we justify the validity and unbiasedness of the tests. The adequate continuity correction term for these tests is also found to reduce inflation of the type I error rate. On the other hand, the exact testing procedures based on two exact $p$-values, the confidence-interval $p$-value (Berger and Boos (1994)), and the estimated $p$-value (Krishnamoorthy and Thomson (2004)), are also applied in our study. It is known that an exact testing procedure tends to involve complex computations. In this thesis, several strategies are proposed to lessen the computational burden. For the confidence-interval $p$-value, a truncated confidence set is used to narrow the area for finding the $p$-value. Further, the test statistic is verify whether they fulfill the property of convexity. It is shown that under the convexity the exact $p$-value occurs somewhere of the boundary of the null parameter space. On the other hand, for the estimated $p$-value, a simpler point estimate is applied instead of the use of the restricted maximum likelihood estimators, which are less straightforward in this problem. The estimated $p$-value is shown to provide a conservative conclusion. The calculations of the sample sizes required by using the two exact tests are discussed. Intensive numerical studies show that the performances of the asymptotic tests depend on the fraction of the two sample sizes and the continuity correction can be useful in some cases to reduce the inflation of the type I error rate. However, with small samples, the two exact tests are more adequate in the sense of having a well-controlled type I error rate. A data set of breast cancer patients is analyzed by the proposed methods for illustration.

本研究之樣本取自1991年7月1日至2010年3月之月資料，探討各總體經濟變數包括：利率、匯率(美元對新台幣)、M1B、出口、GDP、領先指標綜合指數與大陸及美國兩股市，對台股指數之影響。實證結果顯示，道瓊工業指數為影響台股加權指數最具代表性與領先的指標，大陸股市則非如一般所預期對台股指數變動有重要解釋能力。且道瓊工業指數、利率、M1b、GDP對台股具有領先的單向因果關係。 在衝擊反應函數及變異數分解中，除了道瓊工業指數為判斷台股指數變動最重要因素外，利率與貨幣供給則扮演著解釋台股變動另一重要的角色，利率調升對台股指數之影響為先正後負，當利率調升前，投資者會事先反應，但調升後便會開始調節，反而對台股造成負向影響；而GDP及出口在變異數分解中占台股變異數比例是相對次高的比重，說明台股的變動反應了經濟的基本面因素，台股的變動亦會受其影響，惟此二項變數屬於落後指標，只能用在事後分析。而(美元兌新台幣)匯率及領先指標綜合指數則對台股變動無顯著解釋能力。

在逐漸高齡化的社會中，關注老人的生活議題並加以分析益顯重要。在研究老人問題時，由於研究對象均曾經歷不同的時空背景與人生閱歷，各個體間存在的差異極大；不同族群的老人對其所慣用語彙的理解與表達亦不盡相同。故若利用傳統的統計分析研究結果，強迫人們採用二元邏輯的方式思考與解釋，可能會導致偏差或錯誤的結論。且傳統的統計檢定方法，往往假定取樣的樣本滿足某個分配，因而導致過多的解釋，影響決策品質。 因此，為避免因誤解老人而造成虛耗社會成本，使有限的社會資源得以充分運用，本文於研究老人身心特質與個人期待時，嘗試以模糊理論的軟計算，提出反模糊化轉換。並應用中位數檢定及變異數檢定，建立當統計參數為模糊數或模糊區間時之小樣本無母數模糊統計檢定方法模型。由實證例子分析結果顯示，我們提出的檢定方法，能有效分析模糊樣本的問題，並進而期望能對老人議題的分析和決策有所貢獻，及將此方法運用於其它模糊性議題之研究。 / In a gradually aging society, it is important to pay attention to and then analyze elderly people’s life issues. When a study about elderly people is undertaken, the subjects are very inconsistent given their diverse life experience, and various subgroups of subjects have quite different understanding of and way of expressing a vocabulary. Therefore, analyzing study result with conventional statistical analysis method, which forces thinking in a binary logic way, may lead to biased or erroneous conclusion. Furthermore, a conventional statistical test, usually assuming a certain distribution for its samples, may lead to exaggerated explanation, which is detrimental to the quality of a decision. So, in order to avoid the waste of our social cost from misunderstanding of the elderly, and to make the most use of our limited social resource, when we investigated the elderly people’s personal characteristics and expectations, we tried to apply the soft calculation of the fuzzy theory, proposed the counter-fuzzy transformation, and, by using the median test and variance test, established a nonparametric fuzzy statistical model for small-sized samples and for parameters of the fuzzy number or fuzzy interval types. The analyses of real-world examples demonstrated that this method of statistical test can analyze the problems of fuzzy samples, and can hopefully contribute to improved analysis and decision making of the elderly people’s issues, and apply this method on the investigation of other fuzzy social issues. Key words: counter-fuzzy transformation, fuzzy statistical analysis, median test, variance test, aging society.

臺灣的癌症發生率及死亡率有連年增加的趨勢，研究指出原因可能與環境中的污染物質有關，檢測可能的污染源附近是否存在癌症群聚(Cluster)，將有助於未來的癌症防治。在空間統計(Spatial Statistics)有不少方法可用於檢測群聚現象，其中用來檢測某個特定位置周圍是否發生群聚的方法被稱為焦點檢定(Focused Test)，本文介紹及評估常用的焦點檢定方法，並使用較佳方法探討臺灣地區疑似污染源的地區。 首先本文使用電腦模擬，在不同情境假設下比較焦點檢定方法的檢定力(Power)，例如研究區域大小、群聚形狀等不同的情境，以判斷檢定方法之間的優劣。最後本文分析臺灣鄉鎮市(Township)層級癌症死亡資料，應用焦點檢定方法分析石門核一廠、恆春核三廠及麥寮六輕周圍的癌症死亡率，檢定結果發現核一廠及麥寮六輕附近有較高的癌症死亡率。 / The cancer incidence and mortality rate in Taiwan have been increasing over the past 30 years. Previous studies indicate that the pollution sources, especially for those creating air pollution and excess radiation, are one of the potential causes for the increment. Correctly, detecting the location of possible sources of contaminants can help for cancer prevention. In spatial statistics, focused test can be used to determine if the intensity rate are higher around a possible pollution source. We will introduce and evaluate frequently used focused tests and apply them in Taiwan. First we use computer simulation to compare the power of focused tests in different scenarios, such as study region and cluster shape. Next, we apply the focused tests to Taiwan cancer mortality data, in order to decide if the cancer mortality rates are higher around Chinshan nuclear power plant, Maanshan nuclear power plant, and Mailiao sixth naphtha cracker. The results show that the cancer mortality rates around Chinshan nuclear power plant and Mailiao sixth naphtha cracker are significantly higher.

本篇探討在2010年4月16日滬深300股指期貨正式上市到2012年9月18日止的連續近月每日收盤日資料，進行區域內金融期貨市場連動關係的研究，試圖發現兩岸三地之股價指數期貨市場在亞太地區的金融主導地位，以作為國內外投資者在區域內的投資決策參考。 實證結果顯示，從共整合及向量誤差修正模型檢定發現，兩岸三地股指期貨具有長期均衡及短期的互動關係，因此可以視此三地為單一區域市場。在Granger因果檢定上，台股指數期貨雖無法預測恆生指數期貨，但仍明顯領先滬深300股指期貨且程度大於恆生指數期貨，或可推測兩岸因ECFA的簽訂使實體經濟的關聯性更為緊密，至於恆生指數期貨大多以金融、地產股為其主要成分，與大陸主要以實體經濟為主的金融市場，其Granger預測滬深300股指期貨的能力因此相對較弱。另由衝擊反應檢定得知恆生指數期貨為一獨立的市場，不受台灣及大陸指數期貨市場衝擊的影響；滬深300指數期貨因大陸金融市場逐漸開放，也會受到香港及台灣金融期貨市場之衝擊而產生影響；至於台股指數期貨則在兩岸三地，最易受到其他市場影響。最後由預測變異數分解檢定發現，台股指數期貨及滬深300股指期貨的波動皆易受到恆生股價指數期貨變異的影響，而恆生指數期貨在兩岸三地間之解釋能力最強，於兩岸三地間具金融主導地位。至於台股指數期貨對大陸金融期貨的影響也有突出的表現，因此若政府有心推展亞太金融中心之營運，勢必得加強區域間整合的力度，提出有利吸引外資之最政策，以增加台灣股市於國際間之競爭力。 / This study conducts analysis of regional linkage between financial futures market by examining consecutive daily closing information from April 16, 2010 (the official list date of CSI 300 index futures) to September 18, 2012. This study tries to find the financial dominance of these index futures market in the Asia Pacific region and hopefully it may be used as an investment decision reference for domestic and foreign investors. The empirical results show that from the total integration and vector error correction model tests and three places all indicate long-run equilibrium stock index futures and short-term interaction. Therefore, these three places can be viewed as a single regional market. In the Granger causality test on the TAIEX futures and Hang Seng Index futures, in spite of TAIEX futures can’t predict Hang Seng Index futures, it is significantly ahead of the CSI 300 index futures. TAIEX futures on the CSI 300 index futures even more impact than the Hang Seng Index Futures. It can explain that the ECFA has been signed and results show closely-related economy. Since the Hang Seng Index futures are mainly from financial and real estate stocks while the mainland-based financial market is mainly from the real economy, Granger predicts ability of CSI 300 index futures is relatively weak. Another test on the impulse response shows that (1) Hang Seng Index Futures is an independent market and is not affected by shocks from Taiwan and the mainland index futures markets, (2) CSI 300 index futures is affected by shocks from Hong Kong and Taiwan because of the gradually open financial markets, and (3) TAIEX futures can be seen as a potential Taiwanese dish economy because it is most vulnerable to other market influences among the three places. To sum up, the forecast variance decomposition tests show that TAIEX futures and the CSI 300 stock index futures are vulnerable to fluctuations in the Hang Seng index futures. In order words, the Hang Seng Index futures have the strongest explanatory power among the three places and shows financial dominance. The TAIEX futures also show its significant impact on the mainland China financial futures index. If the Government decides to promote the operation of the Asia-Pacific financial center and to increase competitiveness of Taiwan stock market, it will inevitably have to strengthen inter-regional integration efforts and make the most favorable policies to attract foreign investment.

十二年國教及中小學教師評鑑即將上路，教學方向的調整與教師能力的提升在不久的將來將列為重要的教師績效指標之一。教師應如何轉型，及如何提升學生的上課狀況，都可透過教學問卷的回饋來作參考；教學問卷可直接且快速的反應學生想法並成為師生溝通的交流管道，使教師反省自我教學方式及技巧，進而改善；因此，在使用問卷時，若利用傳統的統計分析方式來研究結果，強迫學生採用二元邏輯的方式思考與解釋問卷結果，將可能會導致偏差或錯誤的結論。本論文應用模糊理論的概念，以模糊問卷為工具，利用模糊德菲法探討學生喜歡的數學老師類型，再提出新反模糊化值，並藉由模糊威克生等級和檢定及變異數檢定方法，分析學生滿意度是否會因性別、年紀、成績、背景而有所不同，最後討論學校老師及校外老師間的滿意度是否有差別。由實證例子分析結果顯示，我們提出的檢定方法，能有效分析模糊樣本的問題，進而期望能對教學問卷的分析和決策有所貢獻，並將此方法運用於其它模糊性議題之研究。 / The twelve-year compulsory education and the evaluation of the primary and secondary school teachers are brought into practice. The way teachers organizing their teaching strategies and improving their capability will be the key indicators of teachers’ performance review in the coming future. How will teachers fine-tune their instructional skill to attract students’ focus and then to boost students’ learning motivation and academic performance, is binding to the result of a practical Teaching Assessment System. The satisfaction questionnaires designed to students as teachers’ Teaching Assessment is a good evaluation tool to have student’s feedback on teachers’ performance. The questionnaires can quickly and directly reflect the thoughts of the students and serve as a communication channel between the teachers and the students, which can help teachers examine and ameliorate how and what they teach. The result could be used as reference for teachers to enhance the quality and effectiveness of teaching. In those varies of statistical methods used as analysis, if conventional statistical analysis is adapted to analyze the questionnaires and force the students to think and to explain through binary logic, it may result in deviations or erroneous consequences. Furthermore, it may drive to the exaggerated interpretation and detrimental decision. The study, based on Fuzzy Delphi Methods, aims to apply the concept of fuzzy theory and uses fuzzy questionnaires as a tool to analyze what kinds of mathematics teachers the students like. We propose the counter-fuzzy transformation, by using the Fuzzy Wilcoxon Rank-Sum Test and variance test to assay whether the students’ satisfaction differ owing to gender, age, grade, or their family background. Lastly, we will discuss whether the satisfaction is different between the school teachers and the teachers in other schools. The result demonstrates that assaying method, as using fuzzy statistics analysis, is a functional and competent way to analyze fuzzy sampling data through its aims and objectives. We believe it could sustain to support related analysis and decision making on Teaching Assessment, and also could be used to other fuzzy test study.

關聯結構(Copula)這個字最早由Sklar(1959)以法文所提出，在邊際分配未知的假設下，透過關聯結構的特性，可以容易的建立聯合機率分配，所以關聯結構的觀念廣泛應用在財務領域中。對於資料在配適關聯結構的同時，要如何知道哪種關聯結構函數是最符合資料型態的分配呢?為解決這個問題，本文中參考Dobric and Schmid (2005)所提出的方法--卡方適合度檢定，來看資料配適關聯結構函數是否配適的恰當。所以本文的研究重點就是在利用卡方適合度檢定來探討各類股間日報酬率資料配適關聯結構的情形。在5種不同關聯結構(Normal關聯結構、t關聯結構、Clayton 關聯結構、Frank關聯結構、Gumbel關聯結構)，利用蒙地卡羅模擬方法，來做關聯結構在卡方適合度檢定之模擬，以及檢定力曲線。在檢定統計量、參數估計、顯著水準的估計都還不錯，只有當切割數越大時參數估計會和設定值差異較大。從檢定力曲線可看出這些檢定的檢定力都很好，代表有足夠能力能去辨別出分配的差異性。實證的部份，從台灣上市公司選取4個內需概念股報酬率的日內資料。結果可看出在Normal、Clayton、Frank、Gumbel這4個關聯結構，是不適合用來描述實際報酬的日資料。而當t關聯結構自由度較小時來描述資料型態是表現的不錯。

在研究工作的範疇□，探討一件事物的因果關係時，考慮的變數太多，常是導致研究 效率無法提高的主因。而主成份分析與因子分析便是解決此種問題的主要方法；它們 最重要的功能，便是能將為數眾多的相關觀察變數，精簡成少數幾個互不相關的變數 ，而此少數的變數對原來的問題仍具有相當高的解釋能力。 然而，布進行精簡變數的過程中，取樣的多寡成了如何有效進行分析的經濟效益問題 。因此，本文最主要是研究，在某些控制條件之下，對所而樣本數之決定提出參考的 依據。在此，因其變異矩陣（Σ）為因子分析模型中主要關鍵所在，而其固有值（E- igenvalue ）為主要成份的判足依據，所以在控制型Ｉ誤差（α），對所有固有值均 相等進行檢足時，若達到預定檢定力，則可控制樣本數的大小；也就是當重要因素出 現時，能夠很可靠的挑選出來。 本文預計分五章討論，如下： 第一章緒論：將詳述研究動機、目的、限制及結構。 第二章因子分析模型之原理：將介紹因子分析的主要理論與特質，並導出對主要成份 選取之有關檢定。 第三章樣本數之決定：利用上述檢定之檢定力函數，進而控制檢定力強弱以決定樣本 數與變數個數之關係。 第四章模擬分析。 第五章結論。

本文主要探討卡方分布檢定統計式中期望次數少於五時，此檢定統計式漸近卡方分布 不佳，故將此檢定式修改，使其漸近卡方分布，並將修改後之漸近氏做一比較。 共分五章 第一章 緒論 第一節 研究動機 第二節 研究目的 第三節 本文結構 第二章 真正卡方分布之期望值及變方 第一節 真正卡方分布之期望值 第二節 真正卡方分布之變方 第三章 有關卡方檢定當期望次數少時之漸近式 第一節 Loghormal 漸近式 第二節 Ｃ（m） 漸近式 第三節 連續校正 第四節 以上三種方法之比較 第四章 結論 第五章 實例

統計上的資料，在許多是依性質加以分類，再計算出每一個類別的次數，在社會學 及醫學這種資料尤其多。列聯表就是將同一母體依不同性質分類，再計算各個類別 次數的表，一般而言，列聯表上的次數均大於零，如果發生次數為零的情況，一為 抽樣次數不夠，二為某些巢其次數先天為零，沒有發生的可能，後者，吾人稱之為 不完全列聯表，本文即探討如何估計不完全列聯表上非先天為零的次數，並據之以 作統計上的檢定。 全文分為六章： 第一章 簡介不完全列聯表。 第二章 敘述有關的基本定義及定理。 第三章 建立Log-Linear模式，討論模式和列聯表的關係，巢的次數估計及獨立性 檢定，在不完全列聯表的情況下，如何修改或補充模式。 第四章 討論何種情況下的不完全列聯表可用直接法來估計巢的次數。 第五章 三角形不完全列聯表中非先天為零的巢，其次數估計的方法及比較。 第六章 主軸為零的不完全列聯表，比較估計非先天為零的巢之次數的方法。