傳統的統計方法檢定都假定資料來自於某個分配，但若假設檢定包含著不確定性時，有關模糊數的假設檢定有其重要性。由此可知，模糊統計推論已逐漸受到重視，這是符合現在複雜的社會現象所自然發展的結果。針對模糊資料，本文嘗試以簡易的計算配合模糊理論，定義出模糊數及模糊區間的排序方法，並將此方法應用在檢定上。即針對傳統無母數檢定方法，在無法解決參數假設為模糊數或是模糊區間值的情形下，為改進此一缺點，本文提出模糊Kruskal-Wallis檢定和Run test檢定。由實証的例子顯示，本文提出的檢定方法能有效解決模糊樣本問題。 再者，傳統的統計迴歸模式，假設觀察值的不確定性來自於隨機現象，但模糊迴歸則考慮不確定性來自於多重隸屬現象。因而以無母數統計方法，配合模糊迴歸理論，進而提出模糊無母數迴歸Theil法，並應用實際的例子，以顯示其存在的實質意義。 / Traditional statistical hypothesis testing is completely assumed that the data are from some statistical distribution. However if the data includes many uncertainties, fuzzy hypothesis testing will be useful in this condition. Thus it can be seen that fuzzy inferential statistics is gradually emphasized in modern world due to the development of complex social phenomenon. In this paper, the ordination technique, based on the fuzzy data, of fuzzy numbers and intervals will be defined by simple computations with fuzzy theories, and this technique will be applied to statistical testing. In another word, traditional nonparametric statistical hypothesis testing could not deal with the data from fuzzy numbers or intervals. To be successful for this, we provide Kruskal-Wallis Test and Run Test in this paper. The testing techniques mentioned by this paper could solve the limitation of fuzzy samples. Some empirical examples will be given to show for this. Furthermore, traditional statistical regression models assume that the uncertainty of the observed values is from random sampling. Nevertheless, fuzzy statistical regression models assume that the uncertainty of the observed data is from the phenomenon of Multiple Membership. Therefore we bring up Theil fuzzy nonparametric regression model considering nonparametric statistical techniques and fuzzy regression models. One practical example is given to show the application for this fuzzy nonparametric regression model in this paper.

當我們探討的是兩組樣本的變異是否有所差異時，常見的方法有以ANOVA 為 基礎的檢定與秩檢定，傳統的秩檢定需要假設兩母體具有相同的中位數或知道 其差異。本研究採用Moses (1963) 提出的rank-like 檢定方法，此方法在處理兩組樣本的變異問題時，優點是不需要估計任何中心參數，也不需要假設母體中心參數相同，在資料偏態的情況下也表現得很穩健，我們試圖在樣本數極小的情況下對此方法作修正，將此檢定方法與以ANOVA 為基礎的檢定和秩檢定進行模擬比較，以能夠良好的控制型一誤差與檢定力作為評斷標準。由模擬的結果可得知，rank-like 檢定方法與修正後的方法在不同的分配下皆表現的穩健而修正後的方法特別適用於小樣本的情形。 / We consider the problem of detecting variability change in the two-sample case.Several classical variability tests are investigated, including the ANOVA based tests and the rank tests. Traditional two-sample rank tests assume that the location parameters for both samples are identical or of known difference. In this thesis, a modified version of the distribution-free rank-like test proposed by Moses (1963) is proposed. Moses’s test has several advantages. It does not require location parameter estimation, is applicable without assuming that location parameter are identical, and is robust for skewed data. However, Moses’s test has no power when each of the two samples has size 5 or less. The modified version of Moses’s test proposed in this thesis has some power when the sample sizes are small. Comparative simulation results are presented. According to these results, both Moses’s test and the proposed test are robust under all conditions, and the proposed test works better when the sample sizes are small.

本文主要的目的是藉由The Geometer's Sketchpad (GSP)軟體的設計，幫助我們得到一組連續型模糊樣本。另外對於模糊數的無母數檢定我們提供了一個較為一般的方法，可以針對梯型、三角型，區間型的模糊樣本同時進行處理。 藉由利用GSP. 軟體所設計的模糊問卷，可以較清楚地紀錄受訪者的感覺，此外我們所提供之對於模糊數的無母數檢定方法比其他方法較為有效力。 在未來的研究裡，我們仍有一些問題需要解決，呈述如下：當所施測的樣本數很大時，如何有效率的在網路上紀錄受測者所建構的隸屬度函數？ / The purpose of this paper is to develop a methodology for getting a continuous fuzzy data by using the software The Geometer's Sketchpad (GSP). And we propose a general method for nonparametric tests with fuzzy data that can deal with trapezoid, triangular, and interval-valued data simultaneously. Using the fuzzy questionnaire designed by GSP. can help respondents to record their thoughts more precisely. Additionally our method for nonparametric tests with fuzzy data is more powerful than others. Additional research issues for further investigation are expressed by question such as follows: how to record the membership function on line, especially when the sample size is large?

本文將模糊理論的觀念，應用在估計、檢定及時間數列分析上。研究重點包括離散型及連續型模糊樣本的定義與度量，模糊參數的最佳估計，模糊排序方法應用於無母數檢定，模糊相似度的定義、性質，以及如何將其應用於辨識不同時間數列間的落差l期相似程度等。我們首先將常見的模糊資料分為離散型及連續型，並針對不同類型的資料，給定對應的模糊平均數、模糊變異數等模糊參數的概念與一些重要性質。接著我們提出幾種估計方法，針對不同的模糊參數進行最佳估計並提出可行的評判準則。進一步地，我們將模糊排序方法應用於無母數檢定推論。最後我們提出模糊相似度的定義與度量。經由系統性的模擬與分析，我們建立兩時間數列間模糊相似度演算法則。實證分析方面，我們利用提出的方法對台灣的股價加權指數、個股股價進行估計及檢定；同時，針對台灣歷年GDP、民間消費、毛投資間的相似性進行偵測，以驗證我們提出的模糊參數估計、模糊無母數檢定及模糊相似度演算法的效率性與實用性。 / In this paper, we apply fuzzy theory in estimation, nonparametric test, and time series analysis. Our focus is on: How to define and measure the discrete type fuzzy data and continuous one? How to find the optimal estimators for fuzzy parameters? How to apply fuzzy ranking methods in nonparametric test when the data is vague? How to define and find the degree of fuzzy similarity between two time series? First, fuzzy data is classified according to its type, discrete or continuous. Then we give some definitions and properties on fuzzy mean, fuzzy variance for different type of fuzzy data. Next, we proposed some estimating methods and evaluation rules. Moreover we apply fuzzy ranking methods in nonparametric test, such as Sign test, Wilcoxon signed rank test, Wilcoxon rank sum test, and so on. Finally, we suggest the definitions as well as the algorithm for computing the degree of fuzzy similarity between two time series. We also give some simulate and empirical examples to illustrate the techniques and to analyze fuzzy data. Results show that fuzzy statistics with soft computing are more realistic and reasonable for the social science research.