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系統識別號 U0026-0812200913334408
論文名稱(中文) 多維密度函數偏導數平方積分之 極優估計及其應用
論文名稱(英文) Root n estimates of integrated squared density partial derivatives and applications
校院名稱 成功大學
系所名稱(中) 統計學系碩博士班
系所名稱(英) Department of Statistics
學年度 95
學期 1
出版年 96
研究生(中文) 陳皇宇
研究生(英文) Huang-Yu Chen
學號 r2889101
學位類別 博士
語文別 英文
論文頁數 59頁
口試委員 指導教授-吳鐵肩
口試委員-任眉眉
口試委員-馬瀰嘉
口試委員-李隆安
口試委員-鄭明燕
中文關鍵字 特徵函數  收斂速度  多維度核密度估計  交叉確認法  無母數訊息下界 
英文關鍵字 Characteristic function  nonparametric information bound  cross-validation  multivariate kernel estimate  convergence rate 
學科別分類
中文摘要 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。
英文摘要 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.
論文目次 1 Introduction ---------------------------------------------------------------- 1
1.1 Background ------------------------------------------------------------ 1
1.2 The kernel method ----------------------------------------------------- 2
1.3 Error criteria -------------------------------------------------------- 3
1.4 Cross-validation and plug-in method ----------------------------------- 4
1.4.1 Least-squared cross-validation ---------------------------------- 4
1.4.2 Plug-in method -------------------------------------------------- 5
1.5 The problem ----------------------------------------------------------- 6
2 Literature ------------------------------------------------------------------ 7
2.1 In univariate cases --------------------------------------------------- 7
2.2 In multivariate cases -------------------------------------------------10
3 Estimator -------------------------------------------------------------------12
3.1 The proposed estimator ------------------------------------------------12
3.2 Main theoretical results ----------------------------------------------15
3.3 The modification of the proposed estimator ----------------------------17
3.4 Simulation results for integrated squared density partial derivatives -19
4 Application -----------------------------------------------------------------22
4.1 The rate of convergence -----------------------------------------------22
4.2 Simulation results for plug in bandwidth selector ---------------------23
5 Conclusion ------------------------------------------------------------------25
6 Appendix: proof -------------------------------------------------------------26
Bibliography ------------------------------------------------------------------40
Tables ------------------------------------------------------------------------43
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