本文考慮 $d$ 維度密度函數偏導數積分平方之估計問題。我們提出以樣本特徵函數為基礎的估計量。樣本特徵函數的高頻 (high-frequency) 區受樣本變異的影響，並沒有包含很多 $f$ 的訊息。藉由廣義交叉確認法 (generalization of cross-validation) 選取截斷頻率，捨去此截斷頻率 (cut-off frequency) 之後的樣本特徵函數以降低估計量的變異數。對於任意維度 $d$ 及 $f$
具有某些平滑特性的情況下，本文證明了估計 $d$
維度密度函數偏導數積分平方之訊息界限 (information bound) 。除此之外，對於 $f$ 及核函數滿足某些平滑的條件之下，提出的估計量具有漸近常態 (asymptotically normal)、 $sqrt{n}$ 的收斂速度和變異數達到訊息下界等大樣本性質。在本論文中，模擬研究呈現此估計量在有限的樣本數下的良好表現。最後，我們也考慮此估計量的應用問題。

Based on a random sample of size $n$ from an unknown $d$-dimensional density $f$, the nonparametric estimation of integrated squared density partial derivatives is considered.
These functionals are important in a number of contexts. The proposed estimator is constructed in the frequency domain by using the sample characteristic function. It is known that the sample characteristic function at high frequency is dominated by sample variation and does not contain much information about $f$. Hence, the variation of the estimator can be reduced by modifying the
sample characteristic function beyond some cut-off frequency. It is proposed to select adaptively the cut-off frequency by a generalization of cross-validation. For every $d$ and sufficiently
smooth $f$, the information bounds of estimating integrated squared density partial derivatives are established. Furthermore, for sufficiently smooth $f$ and kernel function, it is shown that the
proposed estimator is asymptotically normal, attains the optimal $sqrt{n}$ convergence rate and achieves the information bound. In simulation studies the superior performance of the proposed
procedures is clearly demonstrated. Finally, an application based on the plug-in method of bandwidth selection in kernel estimation of
$f$ is also considered.

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