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.

本文探討如何將近似無關迴歸模型(SUR)與分量迴歸結合,聯合多條迴歸方程式,估計不同分量下自變數對應變數的影響效果。本文提出資料重排分量迴歸,以SUR模型的資料排列方式堆疊在一起,再以分量迴歸進行估計,估計方法容易理解,實際計算也較易操作,不僅可以考慮不同方程式間的同期相關性,也可觀察不同分量下的邊際效果,使估計結果更為準確,同時,本文以模擬方式比較分量迴歸、Zhao(2001)的加權分量迴歸、Jun and Pinkse(2009)的近似無關分量迴歸及資料重排分量迴歸等估計方法,結果顯示資料重排分量迴歸的估計式同時兼具準確性與精確性,為一良好的估計方法。接著,本文延伸Frankel and Poonawala(2010)的研究使用資料重排分量迴歸進行分析,實證結果顯示高階市場貨幣與新興市場貨幣以遠期溢酬作為未來即期匯率報酬的預測上都是偏誤的,且高階市場貨幣的偏誤大於新興市場貨幣,當匯率變化較大時,亦即高分量時,不論是高階市場或是新興市場匯率偏誤都會變小。

傳統迴歸分析是假設觀測值的不確定性來自於隨機現象，本文則應用模糊隨機變數概念於迴歸模式的架構，考慮將隨機現象和模糊認知並列研究。針對樣本模糊數(x_i, Y_i)，我們進行模糊迴歸參數估計，並稱此為模糊迴歸模式分析。模糊迴歸參數估計大都採用線性規劃，求出適當區間，將觀測模糊數Y_i的分佈範圍全部覆蓋。但是此結果並不能充分反映觀測樣本Y_i的特性。本研究提出一套模糊迴歸參數的估計方法，其結果對觀測樣本的解釋將更為合理，且具有模糊不偏的特性。在分析過程中，我們亦提出一些模糊統計量如模糊期望值、模糊變異數、模糊中位數的定義，以增加對這些參數的模糊理解。最後在本文中也針對台灣景氣指標與經濟成長率作實務分析，說明模糊迴歸模式的適用性。 / Conventional study on the regression analysis is based on the conception that the uncertainty of observed data comes from the random property. However, in this paper we consider both of the random property and the fuzzy perception to construct the regression model by using of fuzzy random variables. For the fuzzy sample (x_i,Y_i), we will process the parameters estimation of the fuzzy regression, and we call this process as fuzzy regression analysis. The parameters estimation for a fuzzy regression model is generally derived by the linear programming scheme. But it's result usually doesn't sufficiently reflect the characteristics of the observed samples. Hence in this paper we propose an alternative technique for parameters estimation in constructing the fuzzy regression model. The result will describe the observed data better than the conventional method did, moreover it will have the fuzzy unbiased properties. For the purpose of fuzzy perception on the fuzzy random variables, we also give definitions for certain important fuzzy statistics such as fuzzy expected value, fuzzy variance and fuzzy median. Finally, we give an example about the Taiwan Business Cycle and the Taiwan Economic Growth Rate for illustration.