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%。

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

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