# 臺灣博碩士論文加值系統

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 過去已有許多文獻是關於機率密度函數的峰數檢定，而這其中包含Dip test, Calibrated excess mass test, 以及Adjusted K-S test等等。有些方法都和核密度函數估計有關且這些方法在計算上是比較複雜的。這個研究的目的是去找到一個容易藉由現有的軟體去計算且方便的方法。這裡提出的方法是先去對資料找出一個比較接近其機率密度函數的單峰分配，再把這個分配當作理論分配去繪製Q-Q 圖。之後再對Q-Q圖上的點配適一條迴歸線，並取所有點和配適的迴歸線距離最大者視為檢定統計量。之後利用拔靴法(Bootstrap)去找到在各個顯著水準上的臨界點。模擬結果顯示在某些情況下，提出的方法檢定力比Calibrated excess mass test和Adjusted K-S test來的好。最後，使用一組血癌病人的資料利用在這三種方法去做說明。
 There were many literatures about modality test of probability density function in the past. They included the dip test, the calibrated excess mass test, the adjusted K-S test and so on. Some methods are related to kernel density estimate, but they are more complicated on calculation. The purpose of this study is to find a method which is easy to compute and convenient by using current software. The proposed method is to select a closer unimodal distribution for the data and regard it as theoretical distribution to construct the Q-Q Plot. Then we fit a regression line for the points on the Q-Q Plot and take maximum vertical distance between the regression line and the points on the Q-Q Plot as test statistic. The bootstrap method is used to find the critical points in different significance levels. The simulation results show that the power of proposed method is greater than calibrated excess mass test and the adjusted K-S test in some situation. Finally, an acute lymphoblastic leukemia data is used to illustrate the proposed method, calibrated excess mass test and the adjusted K-S test.
 Chapter 1 Introduction 1Chapter 2 Literature Review 32.1 The Calibrated Excess Mass Test 32.2 The Adjusted K-S test 52.3 The Threshold regression models 7Chapter 3 Proposed Method 9Chapter 4 Simulation Study and Results 134.1 The Simulation process 13(1) The calibrated excess mass test 13(2) The adjusted Kolmogorov-Smirnov test 14(3) Simulation process 15(4) Proposed method 234.2 Results 234.3 Real example 29Chapter 5 Discussions 32References 34Appendix A 35Appendix B 36Appendix C 38Appendix D 46
 [1] Chan, K.S, (1993), “Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model, The Annals of Statistics 21, 520-533.[2] Cheng, M.-Y. and Hall, P. (1998), “Calibrating the excess mass and dip tests of modality, Journal of the Royal Statistical Society Series B, 60, 579 - 589.[3] Hartigan, J. A. and Hartigan, P. M. (1985), “The Dip test of unimodality, The Annals of Statistics, 13, 70-84.[4] Hansen, B.E. (1999), “Threshold effects in non-dynamic panels：Estimation, testing and inference, Journal of Econometrics, 93, 345-368.[5] Muller, D.W and Sawitzki, G. (1991), “Excess mass estimates and tests for multimodality, Journal of the American Statistical Association,86, 738-746.[6] Silverman, B. W. (1981), “Using kernel density estimates to investigate Multimodality, Journal of the Royal Statistical Society Series B , 43, 97-99.
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