令X_1，X_2…X_n表一組獨立同分布之隨機變數且其共同機率密度函數為f(x)，則常用之f(x)之估計式為核估計式$hat{f}(x)$. 核估計式具有許多好的性質，密度函數之泛函如密度函數之眾數(mode)，微分及積分均有深入之研究( 參考 Pagan and Ullah (1999) , Silverman (1986) , Prakasa Rao (1983)及Tapia and Thompson (1977) ) . 本文研究對稱型機率密度函數 之一些尚未討論之泛函H(f)的核估計$H(hat{f})$，即關鍵點，反曲點，斜率，曲率及概似函數之核估計.

Kernal density estimator $hat{f}$ is by far the most popular estimator of probability density function f .It is interesting to find performances of $H(hat{f})$ for functionals H(f) of f.Well known results cover a great many H(f) include $f^{(k)}(x)$
, the k-th derivatives of f,integral of f like $int_{-infty}^{x}f(s)ds$ , the distribution function , evaluated at x , and modes of x . In this paper , we investigate $H(hat{f})$ for functionals
H(f) that represent critical points and reflection points of f , slopes and curvatures of f evaluated at fixed points , and likelihood functions , topics that are not discussed yet.

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