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題 名 | 核機率密度函數折線圖之研究=On Study of Kernel Probability Density Function Ploygons |
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作 者 | 賴家瑞; 鄧文舜; | 書刊名 | 中國統計學報 |
卷 期 | 37:2 1999.06[民88.06] |
頁 次 | 頁95-108 |
分類號 | 319.5 |
關鍵詞 | 核機率密度函數估計量; 折線圖; 漸近積分均方差; 最佳核函數; Kernal density estimator; Polygon; Asymptotic integrated mean square error; Optimal kernel; |
語 文 | 中文(Chinese) |
中文摘要 | 當我們使用核機率密度函數估計量 (kernel density estimator) 來估計機率密度函數曲線時,實務上,我們會先在實數線上找一組等距離分布的分割點,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值。最後將每相鄰兩個分割點上的核機率密度函數估計值用直線線段連接起來,形成一條機率密度函數折線圖,我們以此折線圖做為機率密度函數估計曲線。此折線圓的漸近積分均方差已由Jones (1989) 計算出來。然而有兩個問題其未能詳細說明,亦即構造此折線圓的最佳核函數為何?以及分割點距離與帶寬值大小,兩者會如何影響此折線圖的估計效果?本文將詳細回答這兩個問題。另外我們將進一步推廣折線圖的構造方式,我們將每連續q個分割點以及在其上面的核機率密度函數估計值分別取平均值,構造出新的等距離分布分割點以及核機率密度函數估計值。我們再將這些新的核機率密度函數估計值每相鄰兩點用直線線段相連,如此就構造出新的機率密度函數折線圖。我們發現新的機率密度函數折線圖具有下列特性。首先若分割點間的距離較小時,此新的機率密度函數折線圖 (當q>1時) 會較原來的機率密度函數折線圖 (當q=l時) 有更大的漸近積分均方差值,這意謂著取q>1時只會浪費計算。反之,若分割點距離較大時,前者會較後者有更小的漸近積分均方差。此時就能以很小的計算量,來改善後者的估計效果。 |
英文摘要 | The probability density function (PDF) estimate is produced practically by joining every two consecutive kernel estimates of the PDF values by a straight line segment. Here the PDF values are estimated on a sequence of equally spaced partition points of the real line. Hence such PDF estimate is of polygon type. The asymptotic integrated mean square error (AIMSE) of such PDF estimate has been studied by Jones (1989). However, there are two questions that concern us. One is what the optimal kernel function is for such PDF estimate. The other is how the performance of such PDF estimate is affected by the distance between the partition points. In this paper, the two questions are answered in detail. In this paper, we also propose a new PDF polygon. It is constructed by taking the average of every q consecutive partition points and that of kernel estimates of PDF values on these partition points. The proposed PDF polygon is constructed on these averages of kernel estimates of PDF values. It is shown that the new PDF polygon obtained in the case q > 1 has larger minimum AIMSE than that derived in the case q = 1, if the distance between partition points is much less than the value of bandwidth. On the other hand, the former has smaller minimum AIMSE than the latter, if the distance is larger than the value of bandwidth. In this case, the performance of the ordinary PDF polygon can be significantly improved by using a little computation. |
本系統中英文摘要資訊取自各篇刊載內容。