局部偏離分析

Analysis of Locally Diversity

廖家彥(Chia-Yan Liao)

指導教授 ： 伍志祥

淡江大學/理學院/數學學系碩士班/碩士(2010年)

https://doi.org/10.6846/TKU.2010.00139

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。

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

並列摘要

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.

並列關鍵字

Gini Index ； Hamming Distance ； Locally Diversity ； Kolmogorov-Smirnov Test ； Kruskal-Wallis Test ； U-statistic

參考文獻

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2. Hoeffding, W. (1948) A class of statistics with asymptotically normal distribution, Ann. Math. Statist., 19, 293　325.

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.

7. Paul H. Kvam , Brani Vidakovic (1962) Nonparametric Statistics with Applications to Science and Engineering., Hoboken, N.J. : Wiley-Interscience, c2007.