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系統識別號 U0002-0507201020181700
中文論文名稱 局部偏離分析
英文論文名稱 Analysis of Locally Diversity
校院名稱 淡江大學
系所名稱(中) 數學學系碩士班
系所名稱(英) Department of Mathematics
學年度 98
學期 2
出版年 99
研究生中文姓名 廖家彥
研究生英文姓名 Chia-Yan Liao
學號 697190089
學位類別 碩士
語文別 中文
口試日期 2010-06-24
論文頁數 28頁
口試委員 指導教授-伍志祥
委員-張三奇
委員-楊恭漢
中文關鍵字 堅尼係數  漢明距離  局部偏離  科爾默戈羅夫-斯米爾洛夫檢驗  克魯斯卡爾-沃利斯檢驗  無母數統計 
英文關鍵字 Gini Index  Hamming Distance  Locally Diversity  Kolmogorov-Smirnov Test  Kruskal-Wallis Test  U-statistic 
學科別分類 學科別自然科學數學
中文摘要 本文目的在於檢定多個連續型隨機變數的母體分佈是否相同,首先對連續型隨機變數值域建構k個不同的分割,依據這些分割可以把連續型隨機變數轉換成高維度多項分佈的隨機向量,如此兩連續型隨機變數就能對應一個漢明距離,這是一個衡量離散資料的變異指標,P.K.Sen (2003)以漢明距離建構變異數分析來檢定隨機向量的邊際多樣性是否相同。但對於兩個連續型隨機變數用這個方法產生的漢明距離,會因為不同分割的選取而有不同的計算結果。針對此問題,我們讓k往無限大逼近,漢明距離會收斂到一正值,我們稱此正值為這兩隨機變數的局部偏離量。本論文是以局部偏離量建構變異數分析並稱為局部偏離分析。我們以模擬的方式比較探討局部偏離分析與Kolmogorov-Smirnov Test(1933)和 Kruskal-Wallis Test(1952)的檢定力。
英文摘要 In this thesis, a new procedure based on Hamming distance is proposed to test whether a collection of G independent continuous samples are drawn from the same population. For the i-th sample of size ni, i=1,2,…,G, we repeatedly partition the sample space into C cells for K times such
that every observation is transformed into an k-tuple to label its cell membership, in terms of the numbers 1,2,…,C, in k different partitions. Consequently, for the i-th sample we obtain ni such k-tuples. The proposed test statistics is then based on all the resultant k-tuples of all observations of the G samples. As k increases, the proposed test statistics becomes less sensitive to the choice of cell origin in each partition. And as k → ∞ , the test statistics converges to a positive constant, called local diversity, and is used to test the homogeneity of G samples. For the case of G=2, We compare the power of the proposed test with those of Kolmogorov-Smirnov and Kruskall-Wallis test.
論文目次 摘要
1. 介紹 1
2. 文獻回顧 2
2.1. Gini Index ...........2
2.2. Hamming Distance .....2
2.3. 統計方法 .............4
3. 檢定統計量 8
4 U統計量理論 13
5 模擬的結論 19
6 參考書目 27
參考文獻 1. Gini, C.W., 1912. Variabilita e mutabilita. Stud. Econom.-Giuridici della R. Univ. Cagliari 3 (2), 3 159. Hoeffding, W., 1948. A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293-325.
2. Hoeffding, W. (1948) A class of statistics with asymptotically normal distribution, Ann. Math. Statist., 19, 293 325.
3. Kolmogorov, A. (1933). Sulla determinazione empirica di une legge di distribuzione. Inst.Ital. Attwari 4:83 91.
4. Kruskal, W. H. (1952). A non-parameteric test for the several sample problem. Annals of Msthematical Statistics 23, 525-540.
5. Kruskal, W. H. and Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis, Journal of the American Statistical Association 47, 583-621.
6. Lee, A.J. , (1990) U-Statistic-Theory and Practice. Marcel Deckker, Inc., New York.
7. Paul H. Kvam , Brani Vidakovic (1962) Nonparametric Statistics with Applications to Science and Engineering., Hoboken, N.J. : Wiley-Interscience, c2007.
8. Sen,P.K. Gini Diversity Index, Hamming Distance and Curse of Dimensionality. METRON-International Journal of Statistics 2005, vol. LXIII, n. 3, pp. 329-349
9. Pinheiroa,P.H ,Pinheiroa,A.S. ,and Sen,P.K,. Comparison of genomic sequences using the Hamming distance . Journal of Statistical Planning and Inference 130 (2005) 325-339
10. Pinheiro, A. S., Sen, P. K., and Pinheiro, H. P. (2005) Decomposability of high-dimensional diversity measures: Quasi U-statistics, martingales and nonstandard asymptotics, submitted for publication.
11. Pinheiro, H. P., Seillier-Moiseiwitsch, F., Sen, P. K., and Eron, J. (2000) Genomic sequence and quasi-multivariate CATANOVA., In: Handbook of Statistics, Vol. 18: Bioenvironmental and Public Health Statistics, P.
K. Sen and C. R. Rao (eds.), Elsevier, Amsterdam, 713 746.
12. Pinheiro, H. P., Pinheiro, A. S., and Sen, P. K. (2005) Analysis of genomic sequence using Hamming distance, J. Statist. Plan. Infer., 130, 325 339.
13. Rizzo, Maria L. (2007), Statistical Computing with R, Chapman & Hall/CRC.
14. Smirnov, N. (1939a). On the estimation of the discrepancy between empirical curves of distribution for two independent samples. Bull. Mathematique de l Universite de Moscou 2:fsc.2.
15. Smirnov, N. (1939b). Sur lesecarts de la courbe de distribution empirique. Res. Math. CMat. SbornikD CNSD 6:3 26.
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