令f(x)表一機率密度函數，f'(x)及f'(x)分別表其第一、二階微 分，κ(x)表f(x)在x之曲率。f'(x)、f'(x)及k(x)提供f(x)之一些特性，為鑑別分布之重要參考。通常f(x)是未知的，所以f'(x)、f'(x)及κ(x)也未知，必需用樣本估計。本文討論f'(x)及f'(x)和κ(x)的核估計式。現有的文獻並未討論是否存在核函數使f'(x)及f'(x)之核估計式為漸近不偏，解決此問題為本文之首要目的，其結果將運用於估計κ(x)，並推出各估計式之中央極限定理。 In this paper, we find that the standard Guassian kernel function can be applied easily to construct the asymptotic unbiasedness of kernel estimators of the first two order derivatives of probability density function. we also find the central limit theorems for the kernel estimators mentioned above and the corresponding estimator of curvature.
