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系統識別號 U0002-2407201315065700
中文論文名稱 二維頻率多邊形圖之研究
英文論文名稱 Study on two dimension frequency polygon
校院名稱 淡江大學
系所名稱(中) 中等學校教師在職進修數學教學碩士學位班
系所名稱(英) Executive Master's Program In Mathematics for Teachers
學年度 101
學期 2
出版年 102
研究生中文姓名 方志元
研究生英文姓名 Chih-Yuan Fang
學號 700190027
學位類別 碩士
語文別 中文
口試日期 2013-06-29
論文頁數 30頁
口試委員 指導教授-伍志祥
委員-張三奇
委員-楊恭漢
委員-伍志祥
中文關鍵字 線性混合  直方圖  頻率多邊形  漸近變異數  漸近偏差量  均積分方差 
英文關鍵字 asymptotic bias  asymptotic variance  frequency polygon  histogram  linear blend  mean integral square error 
學科別分類
中文摘要 給定樣本數為n的雙變數隨機樣本,把樣本空間分割成等面積的正方形,並計算在每塊正方形上的樣本頻率,可以得到雙變數的直方圖。從直方圖得到頻率多邊形圖的作法有兩種:一種是Scotts (1985a,b)所考慮的,用形成三角形的相鄰3個直方圖區塊中心點的值差補而成。另一種是Terrell (1983)及Hjort (1986)所研究的混和線性差補。後者的優點是漸進偏差量只跟密度函數對x的二次偏微分與對y的二次偏微分有關,跟對xy的二次偏微分無關。
在這論文,我們改變樣本空間的分割方塊的中心點結構,且採用混和線性差補來建構多邊形圖。Terrell (1983)及Hjort (1986)的頻率多變形圖會成為這裡所考慮形式的特例。發現多邊形的銜接面為長方形時,其積分漸進變異數不會改變,但會減少積分平方漸進偏差量。若進一步適當的剪裁所要銜接成為多邊形的長方形面,在最佳化平滑參數值的情形下,會比Terrell (1983)及Hjort (1986)所討論的頻率多變形圖的平均積分均方差(MISE)降低了約10%。
英文摘要 Abstract:
Given n bivariate random sample, we cut the sample space into equal area of squares, then we can get bivariate histogram by calculating the frequency in every squares. There are two way to get frequency polygon: the first, consider by Scotts (1985a,b), is to interpolate the values at the centers of 3 adjacent histogram bins in a triangular ; the second way is the linear blend given by Terrell (1983) and Hjort (1986). The latter method has the advantage that the asymptotic bias is not appear the partial differential term fxy term which is the same as kernel density estimator by using product kernel.
In this paper, we vary the structure of the center of square, and interpolate the values at rectangular by using the linear blend, then Terrell (1983) and Hjort (1986) is a special case. The integral asymptotic variance of new estimators is the same as the estimator consider by Terrell (1983) and Hjort (1986), but decrease the integral square asymptotic bias. We further tailor our new estimator, we have the advantage, in the optimal smoothing parameter, the mean integral square error(MISE) is reduce about 10% then the Terrell (1983) and Hjort (1986) method.
論文目次 論文目錄
第1節 序論 1
第2節 用平行四邊形四頂點值作線性插補 2
第3節 用長方形分割樣本空間作線性插補 8
第4節 最佳化的比較 12
第5節 定理的證明 15
參考文獻 30
參考文獻 1. Dong, J. P. and Zheng, C. (2001). Generalized edge frequency polygon for density estimation. Statistics and Probability Letters, 55, 137-145.
2. Deng W. S., Wu J. S. and Chen L.C. (2013) A Note on frequency polygon based on weighted sum of binned data. Preprints and communications.
3. Hjort, N. L. (1986). On frequency Polygons and Averaged Shifted Histograms in Higher Dimensions. Technical Report 22, Stanford University.
4. Jones, M. C. (1989). Discretized and interpolated kernel density estimates. Journal of the American Statistical Association, 84, 733-741.
5. Jones, M. C., Samiuddin M., Al-Harbey, A.H. and Maatouk, T.A.H. (1998). The edge frequency polygon. Biometrika, 85, 235-239.
6. Lin, C. T., Wu J. S. and Yen C. H. (2006). A Note on kernel Polygons. Biometrika, 93, 228-234.
7. Scott, D. W. (1985a). Frequency polygons: theory and application. Journal of the American Statistical Association, 80, 348-354.
8. Scott, D. W. (1985b). Average shifted histograms: effective nonparametric density estimations in several dimensions. Annals of Statistics, 13, 1024-1040.
9. Scott, D. W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization. John, Wiley & Sons.
10.Terrel, G. R. (1983). The Multilinear Frequency Spline. Technical Report, Department of Math Sciences, Rice University.
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