給定一組單變數隨機樣本，若欲估計這n個隨機樣本的母體機率密度函數 ，則核機率密度函數估計量是一個非常受歡迎的估計方法。在實務上，我們會先在時數線上取一組等距離 分佈的分割點，然後使用核機率密度函數估計量來估計這些分割點上的機率密度函數，最後用直線線段連接起來，形成一條機率密度函數多邊形圖，以此多邊形圖作為機率密度函數估計曲線。此多邊形圖的漸進積分均方差(AIMSE)已由Jones(1989)計算出來。
賴家瑞、鄧文舜(1999)將核機率密度函數的構造方式與Jones (1998) Edge Frequency Polygon的觀念相結合，提出先將連續m個分割點以及該點上的核機密度函數估計量分別取平均值，作為新的等距分割點以及核機率密度函數估計量，再將這些新的核機率密度估計量每相鄰的兩點以直線線段連接，如此就構造出新的核機率密度函數多邊形圖。而在賴、鄧的文章中著重在 的討論，而沒有很深入的討論在多邊形圖及Edge Frequency Polygon之間存在著是否有更好的 值。
本文將討論當選用均勻函數來估計核機率密度多邊形圖時，核機率密度函數多邊形圖建立在最佳等距離h的分割點以及帶寬值 之上時，則均勻核函數與分割點間距及帶寬值的比值 兩者之間的關係，而建構多邊形圖的均勻核函數及其所對應的 值為何？

For a given one-variance random sample, kernel frequency polygon is a useful method to estimate the population density function . In practice, we use kernel estimator to estimate kernel density on a sequence of equally spaced partition points of real line, the kernel frequency polygon is constructed by these partition points and kernel estimates of PDF value on these partition points. The asymptotic integrated mean square error (AIMSE) of such frequency polygon has been computed out by Jones (1989).
Lai and Deng (1999) combine the Frequency Polygon and Jones (1998) Edge Frequency Polygon, take out a new polygon. It is constructed by taking the average of every m consecutive partition points and that of kernel estimates of PDF values on these partition points. The proposed polygon is constructed on these averages of kernel estimates of PDF value. However, in Lai and Deng (1999) is focus on m and no more conclusion on whether does there have an optimal between Frequency Polygon and Edge Frequency Polygon.
In this paper, we will discuss what is the relation between Uniform Kernel function and if Kernel Frequency polygon is constructed on equally partition and bin-width ; and what is the optimal when we use Uniform Kernel function to construct a Kernel Frequency polygon.

1. Introduction ．．．．．．．．．．．．．．．．　1-3　
2. 模型及估計量的介紹　．．．．．．．．．．．．　 4-8
3. 模擬結果　．．．．．．．．．．．．．．．．． 9-12
4. 定理證明 ．．．．．．．．．．．．．．．．．．　13-21
5. References ．．．．．．．．．．．．．．．．．　22　

1.David W. Scott(1985) ; Frequency polygons : Theory and Application ; Journel of the American Statistical Association ; Vol 80,pag 348-354
2.Hardle w.,(1990)Smoothing Techniques with Implementation in S
3.Jones(1998), M.C. ; The Edge Frequency Polygon ; Biometrika ,85,pag 235-239
4.Jones , M. C(1989). ;Discretized and Interpolated Kernel Density Estimates; Journel of the American Statistical Association ; Vol 84,pag 733-741
5.Silverman , B.W. (1986) Density estimation for statistics and data analysis. Chapman and Hall , London.
6.WEN-SHU ENN DENG and C.K. CHU*;On Study of Kernel Regression Function Polygons; Nonparameteric Statistics,Vol.12
pag 597-609
7.賴家瑞,鄧文舜(1999);核機率密度函數折線圖之研究;中國統計學報,第37卷第二期95-108頁